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For this reason, inhibition of GK expression in β1-tanycytes suppressed the glucosensing network, altering satiation and/or satiety signal processing by neurons from the AN. These results, together with those shown by Elizondo et al.29 in MCT1 knockdown animals, represent a clear evidence of the metabolic coupling existing between tanycytes and neurons. In this metabolic interaction, tanycytes would metabolize glucose to lactate, and the lactate released by tanycytes could be sensed by neurons as a measure of blood glucose concentration (Fig. 7). It is possible that the astrocyte-neuron lactate shuttle could work as a compensating mechanism for GK inhibition. However, our results indicate that it fails to completely compensate for the effect of GK inhibition.Figure 7Hypothalamic glucosensing and its inhibition by GK knockdown in tanycytes. Model of glucosensing based on the metabolic interaction between tanycytes and AN neurons. Tanycytes are a specialized ependymal cell, localized in the lower parts of the ventricular walls and the floor of the 3V. α and β1-tanycytes line the infundibular recess, and their basal projections reach orexigenic and anorexigenic AN neurons. β2-tanycytes cover the floor of the 3V and extend their projections inside the ME and regulate the permeability properties of the fenestrated capillaries of the ME. Hyperglycemia results in increased glucose concentrations in the 3V, which affects GK-positive β1-tanycytes. These glial cells sense the increased glucose by the presence of GLUT2 and GK. The GK activity could be regulated by GKRP and its nuclear compartmentalization. Lactate released by tanycytes could activate neurons of AN, generating a satiety response. GK inhibition represses the glucosensing circuit, causing an altered feeding behavior. 3V, third ventricle; ME, median eminence; CSF, cerebrospinal fluid; GK, glucokinase; AN, arcuate nucleus; MCT, monocarboxylate transporter.
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| 100.0 |
Hypothalamic glucosensing and its inhibition by GK knockdown in tanycytes. Model of glucosensing based on the metabolic interaction between tanycytes and AN neurons. Tanycytes are a specialized ependymal cell, localized in the lower parts of the ventricular walls and the floor of the 3V. α and β1-tanycytes line the infundibular recess, and their basal projections reach orexigenic and anorexigenic AN neurons. β2-tanycytes cover the floor of the 3V and extend their projections inside the ME and regulate the permeability properties of the fenestrated capillaries of the ME. Hyperglycemia results in increased glucose concentrations in the 3V, which affects GK-positive β1-tanycytes. These glial cells sense the increased glucose by the presence of GLUT2 and GK. The GK activity could be regulated by GKRP and its nuclear compartmentalization. Lactate released by tanycytes could activate neurons of AN, generating a satiety response. GK inhibition represses the glucosensing circuit, causing an altered feeding behavior. 3V, third ventricle; ME, median eminence; CSF, cerebrospinal fluid; GK, glucokinase; AN, arcuate nucleus; MCT, monocarboxylate transporter.
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In conclusion, this is the first evidence of a crucial role for tanycytic GK in feeding behavior and represents an important step in unraveling the hypothalamic glucosensing mechanism36 and the way this glucose sensor modulates food intake. These results open a new window in the search for therapeutic strategies for controlling feeding behavior and the treatment of obesity, including the function of the glial element.
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| 99.94 |
All animals were handled in strict accordance with the Animal Welfare Assurance (permit number 2010101 A), and all animal work was approved by the appropriate Ethics and Animal Care and Use Committee of the Universidad de Concepción, Chile. Fifty-six male adult Sprague-Dawley rats of 250–300 g were used in all experiments. Animals were housed in a separate animal room with constant temperature (21 ± 2 °C) and a controlled 12-h light/12-h dark cycle; lights were turned on every day at 7:00 a.m. Animals had free access to a standard rodent diet (Lab Diet, 5P00 Prolab RMH 3000, Purina Mills, St. Louis, MO) and tap water.
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| 99.9 |
Serotype 5 ΔE1, E3 based replication-deficient adenoviruses were generated as previously described29. Briefly, oligonucleotides targeting rat GK were designed and selected using the Genebank accession number, L38990. Sense shRNA-GK 5′-TCA GAG TGA TGC TGG TCA A-3′ and antisense shRNA-GK 5′-TTG ACC AGC ATC ACT CTG A-3′ sequences shared no homology with other rat coding sequences by BLAST analysis. A ring sequence of nine base pairs (TTC AAG AGA) existed between the sense and antisense strands. Control shRNA oligonucleotides were designed and selected to target β-galactosidase from E. coli: sense5′-CGC GCC AAG GCC AGA CGC GAA TTA TTT CAA GAG AAT AAT TCG CGT CTG GCC TTT TTT TTT TAA T-3′ and antisense 5′-TAA AAA AAA AAG GCC AGA CGC GAA TTA TTC TCT TGA AAT AAT TCG CGT CTG GCC TTG G-3′. The expression cassette was then cloned into the adenoviral shuttle vector. All shRNAs (Invitrogen, Rockville, MD, USA) were synthesized and designed to contain both AscI and PacI restriction enzyme sites (New England Biolabs, Ipswich, MA, USA), which were used to ligate them into the plasmid, pDC311.2-OFF-EGFP downstream of the human H1 promoter, as previously described29. The plasmid was then cotransfected with the Ad genomic plasmid, pBHGlox∆E1,3Cre (Admax system, Microbix Biosystems, Ontario, Canada) into HEK293A cells. Virus particles were released by heat shock, and cell debris was removed by centrifugation for 5 min at 5000× g. The particles were recovered from the supernatant by filtration through a 0.45-µm filter. The resulting adenoviral expression vectors (Ad-shGK-EGFP and Ad-shβGal-EGFP) were tittered by EGFP expression using the Adeno-XTM Rapid Titer Kit Protocol (Clontech, Mountain View, CA, USA).
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| 100.0 |
Rats were handled everyday for one week prior to the experiments to get used to the researchers and the experimental procedures. Rats were anesthetized with an intraperitoneal injection of ketamine (90 mg/kg) and xylazine (10 mg/kg). The fur at the top of the head was removed to expose the area to be incised. A hole was drilled in the skull, and a guide cannula (28 gauge stainless steel; Plastics One, Roanoke, VA) was stereotactically implanted in the 3V of the rat (anterior-posterior from bregma −3.14 mm, medial-lateral from midsaggital sinus 0.0, and dorsal-ventral from the top of the skull 9.2 mm). The guide cannula was secured to the skull using 3/32 mm mounting screws and dental acrylic. A removable dummy cannula (28 gauge stainless steel; Plastics One,) fit into the cannula guide and sealed the opening of the guide cannula throughout the experiments except when it was removed for the injections. After the surgery, rats were single-housed and allowed to recover for 5 days before adenovirus injection.
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| 99.94 |
Rats were anesthetized with isoflurane and then injected into the 3V with either 25 µL of the control Ad-shβGal-EGFP stock solution (108 IFU/mL) or 25 µL of the Ad-shGK-EGFP stock solution (2 × 108 IFU/mL) for the experimental group at 2.5 µL/min. We were not able to further concentrate the adenovirus in order to inject it in a lower volume. Brains were collected for immunohistochemistry at 18, 24 and 48 h. The rat brains were fixed in 4% paraformaldehyde (PFA) by immersion for 48 h. After fixation, thick frontal sections of the hypothalamus (40 µm) were cut with a cryostat and subsequently processed free-floating. Tissues were immunostained with mouse anti-vimentin (1:200, DAKO, Carpinteria, CA, USA), rabbit anti-glial fibrillary acidic protein (GFAP; 1:200, DAKO) and rabbit anti-NeuN (1:1000, Abcam, Cambridge, MA, USA) antibodies diluted in Tris-HCl buffer (pH 7.8) containing 8.4 mM sodium phosphate, 3.5 mM potassium phosphate, 120 mM sodium chloride, and 1% bovine serum albumin. Sections were incubated with the primary antibodies overnight at room temperature in a humid chamber. After extensive washing, sections were incubated for 2 h at room temperature with Cy2-,Cy3- or Cy5-labeled secondary antibodies (1:200; Jackson ImmunoResearch, West Grove, PA, USA). These samples were counterstained with the DNA stain TOPRO-3 (1:1000; Invitrogen). The slides were analyzed using confocal laser microscopy (LSM 700, Zeiss). Colocalization of different markers was assessed by measuring the Pearson’s correlation coefficient, Rr. An Rr value of ‘1’ indicates complete colocalization, and a Rr value of ‘0’ indicates no specific colocalization.
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The tissues were processed for qRT-PCR analysis as previously described28, 29. Briefly, 48 h after adenovirus injection, rats were fasted for 24 h. After that, rats were anesthetized with isoflurane and i.c.v. injected with 10 μL of saline buffer (128 mM NaCl, 3 mM KCl, 1.3 mM CaCl2, 1.0 mM MgCl2, 1.3 mM NaH2PO4, 21 mM Na2HPO4, pH 7.4 and 320 mOsm) or 10 μL of 50 mM D-glucose diluted in the same buffer (320 mOsm, pH 7.4) at a rate of 2.5 μL/min. Hypothalamic samples were collected 2 h post-glucose or saline injection for the mRNA analysis expression. qRT-PCR analysis was used to measure the expression of hypothalamic cyclophilin (the housekeeping gene), GK, NPY, CART, POMC and AgRP. After the experimental treatments, the brain of each rat (six per condition) was removed, and the hypothalamic area was isolated and further dissected to obtain a region close to the 3V ependymal layer. All microdissections were performed closest to the basal 3V and under a stereomicroscope Leica M80 (Leica, Wetzlar, Germany). Total RNA from hypothalamus was isolated using TRIzol (Invitrogen) and treated with DNase I (Fermentas International INC, Burlington, Ontario, Canada) to remove genomic DNA contamination. A total of 2 µg of RNA from each sample was reverse transcribed into cDNA according to the manufacturer’s protocol of M-MULV reverse transcriptase (Fermentas International INC). Parallel reactions were performed in the absence of reverse transcriptase to control for the presence of genomic DNA. qRT-PCR reactions were prepared with a Brilliant II SYBR Green QPCR Master Mix kit (Agilent Technologies, Inc., Santa Clara, CA, USA) in a final volume of 12.5 µL containing 1x SYBR green Master Mix, 1 µL cDNA sample and 500 nM of the following sets of primers: cyclophilin, sense 5′-ATA ATG GCA CTG GTG GCA AGT C-3′ and antisense 5′ATT CCT GGA CCC AAA ACG CTC C3′ (expected product of 239 bp); GK, sense 5′AAA GAT GTT GCC CAC CTA CGT GCG3′ and antisense 5′ATC ATG CCG ACC TCA CAT TGG C3′ (expected product of 510 bp); NPY, sense 5′-TGT TTG GGC ATT CTG GCT GAG G-3′ and antisense 5′- CTG GGG GCA TTT TCT GTG CTT TC-3′ (expected product of 203 bp); AgRP, sense 5′GCA GAC CGA GCA GAA GAT GTT C3′and antisense 5′GTA GCA CGT CTT GAA GAA GC GG3′ (expected product of 186 bp); POMC, sense 5′CTC CTG CTT CAG ACC TCC ATA GAC3′ and antisense 5′AAG GGC TGT TCA TCT CCG TTG3′ (expected product of 164 bp) and CART, sense 5′TCT GGG AAG AAG AGG GAC TTT CGC3′and antisense 5′TCC ATT TGT GTT GCT TTG GGG TG3′ (expected product of 137 bp). All reactions were performed with an initial denaturation of 5 min at 95 °C, followed by 40 cycles of 30 s at 95 °C, annealing for 30 s at 55 °C, and extension for 1 min at 72 °C in an Mx3000 P qPCR System (Agilent Technologies). Ct values of GK mRNA obtained from three different experiments were normalized according to the 2−ΔΔCt method, using cyclophilin as a reference gene45.
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| 100.0 |
For protein analysis, hypothalamic samples were collected 48 h post-adenoviral injection. Rat hypothalamus was homogenized in buffer A (0.3 mM sucrose, 3 mM DTT, 1 mM EDTA, 100 μg/mL PMSF, 2 μg/mL pepstatin A, 2 μg/mL leupeptin and 2 μg/mL aprotinin) and sonicated three times on ice at 300 W (Sonics & Material INC, VCF1, Connecticut, USA) for 10 s. After centrifugation at 4,000× g for 10 min, the proteins were resolved by SDS-PAGE (50 µg/lane) in a 12% (w/v) polyacrylamide gel, transferred to PVDF membranes (0.45 µm pore, Amersham Pharmacia Biotech., Piscataway, NJ, USA), and probed with rabbit anti-GK (1:100, Santa Cruz Biotechnology, CA, USA), rabbit anti-GKRP (Santa Cruz Biotechnology), rabbit anti-GLUT2 (Alpha Diagnostics, 1:1000), rabbit anti-EGFP and anti-β-actin (1:1000) antibodies. After extensive washing, the PVDF membranes were incubated for 1 h at 4 °C with peroxidase-labeled anti-rabbit IgG (1:5000; Jackson ImmunoResearch). The reaction was developed using an enhanced chemiluminescence (ECL) Western blotting analysis system (Amersham Biosciences). Negative controls consisted of incubating the membrane in the absence of anti-GK. The images shown are representative of at least three analyses performed on samples from at least three separate experiments. β-actin expression levels were used as a loading control, and GFP expression levels were used as a transduction control.
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At 24 h post-adenoviral injection, rats were subjected to a 24 h fasting period followed by a 12 h refeeding period for the feeding behavior analysis. Food intake was monitored during the refeeding period by providing rats with pre-weighed rat chow and weighing it again after 12 h (Fig. 6, schematic representation). Food intake was expressed as g consumed per 200 g of body weight. Every interaction with the feeder was recorded by a computerized data acquisition system (VitalView, Respironics, Inc., Murraysville, PA, USA), registering frequency and time of permanency in the feeder.
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| 100.0 |
A meal was considered when the bouts were larger than 10 s into the feeder, and these meals were separated from other feeding bouts by more than 10 min of inter-meal interval46, 47. Except for the first meal, when a bout was longer than 30 min, two meals were considered. The meal patterns calculated included the following: duration of the first meal (in min), frequency of meals (in number of meals in 12 h), cumulative eating time as the total time spent in the feeder (in min), duration of inter-meal intervals (in min), and number of meal intervals. The inter-meal interval was calculated as the time period between the end of one meal and the initiation of the next one. The mean meal size was determined as the total food intake (g) divided by frequency. The mean meal duration was calculated by dividing the total meal duration (in min) by meal frequency, and the eating rate was estimated dividing total food intake (mg) by meal duration (min).
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| 100.0 |
For each data group, results were expressed as mean ± standard error of the mean (SEM), and n refers to the number of animals that were used. For statistical analysis, each treatment was compared with its respective control. Differences between two groups were assessed using the Student t-test. Differences between more than two groups were assessed using ANOVA. Differences were considered significant when P < 0.05. The statistical analyses were performed using GraphPad Prism 5.0 Software (GraphPad Software Inc., San Diego, CA, USA).
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| 99.94 |
Telomere is a repeat of specific short sequences of nucleotides found at the end of chromosomes. The length of telomere is so considerable, hence telomere shortening has been correlated with various pathological disorders and processes such as biological aging, oxidative stress and inflammation [1–6]. The stability of telomere is vital for the conservation of genetic information and maintaining cellular stability [7–9]. Additionally, distortion of telomere can result in cellular abnormality and organ failure [10, 11].
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| 99.7 |
There is great interest in studying telomere length in relation with kidney health [12, 13], but clinical and epidemiological studies remain scanty. The limited existing evidence suggest that patients with end-stage renal disease (ESRD) may have shorter telomere and accelerated telomere shortening compared with the general population [14, 15]. Data for subjects with chronic kidney disease (CKD), derived mainly from two studies of severe heart failure patients, also suggest a strong correlation between reduced kidney function and shorter telomere length (TL), even after adjustment for age [16, 17].
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| 99.8 |
The mechanism through which creatinine is regulated by the kidney, and the relationship between kidney function and TL are not fully understood, while existing data have been controversial . However, clarifying the association of kidney function with TL is imperative, to confirm if this pathway holds promise for CKD risk evaluation and/or reduction.
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| 99.75 |
Of the 10568 eligible participants, 48.0% (n = 5020) were men. The mean age was 44.1 years overall, 43.5 years in men and 44.8 years in women (p = 0.063). With regard to education 52.5% (n = 4022) of the participants had completed more than high school, 25.6% (n = 2178) had completed high school, while 21.7% (n = 3248) had completed less than high school. White (non-Hispanic) represented 70.4% (n = 4864) of the participants, African-Americans - 10.9% (n = 2103) and Mexican-Americans - 6.9% (n = 2690). Overall, 20.8% were current smokers (24.8% of the men and 16.6% of the women). The mean and standard error of mean (SEM) for the TL in the overall sample was 1.08±0.015 (1.07±0.014 in men and 1.08±0.016 in women).
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| 100.0 |
The distribution of participants by stage of CKD based on eGFR and ACR levels was the following: CKD stage 1 - 65.3%, CKD stage 2 - 26.5%, CKD stage 3 - 7.1%, CKD stage 4 - 0.6%, and CKD stage 5 - 0.4%. Overall, 8.1 % of the participants had eGFR less than 60 ml/min/1.73m2. Table 1 shows the characteristics of the participants according to their status for CKD. The lipid profile including triglyceride, total cholesterol, high-density lipoprotein cholesterol, and low-density lipoprotein cholesterol, was better in participants without CKD than in those with CKD (all p < 0.001).
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| 100.0 |
The association age- and sex-adjusted cardiometabolic factors, kidney function tests across quarters of TL analyzed are summarized in Table 2. Mean body mass index, fat-free mass, fat mass, triglyceride, total cholesterol and C-reactive protein significantly decreased across increasing TL quarters (all p < 0.001), while high-density lipoprotein cholesterol significantly increased across increasing TL quarters (p < 0.001, Table 2). eGFR significantly decreased and ACR significantly increased across increasing quarters of TL (both p < 0.001).
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| 100.0 |
Further univariable and multivariable (age-, sex-, race-, smoking-, fasting blood glucose-, systolic and diastolic blood pressure-, body mass index-, and C-reactive protein) regression analysis were performed to examine the association of TL with kidney function (Table 3). Univariable models revealed that TL was negatively associated with urea albumin and ACR (both p < 0.001), and positively associated with serum creatinine and eGFR (both p < 0.001). In multivariable adjusted models, the association remained significant between TL and eGFR, and borderline significant between TL and Urea Albumin (β-coefficient = -0.012, p = 0.056). Logistic regression was used to determine the association between quartile of the TL and chance of CKD, however we have failed to find any significant association between quartile of the TL and odds of CKD neither in crude nor in adjusted (age-, sex-, race-, smoking-, fasting blood glucose-, systolic and diastolic blood pressure-, body mass index-, and C-reactive protein) models.
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| 100.0 |
In this large representative sample of adults Americans, eGFR decreased while urinary albumin excretion increased across decreasing TL quarters. These patterns were robust to adjustment for potential confounding factors. Although, these findings did not translate into significant association between TL and prevalent CKD, our study findings suggest that telomere shortening could be an independent predictor of deteriorating kidney function.
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| 100.0 |
In accordance with our findings, recent systematic review proposed that shortening TL might be related with CKD prevalence/occurrence or declining kidney function, however this relation is probably balanced by the cellular telomere reparative process in those surviving longer with CKD, furthermore stated that Short TL was independently related with increased risk of prevalent micro albuminuria in diabetic men with CKD . Furthermore, recent Japanese investigation among persons with increased cardiovascular risk, telomere length indicated a relationship of longer telomere length to better renal function .
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| 99.9 |
In line with our study, previous reports have suggested that TL is shorter in patients with end-stage renal disease patients on dialysis compared with the general population. For example, in a study of 15 patients on haemodialysis and 15 age-matched controls, the authors found an accelerated telomere shortening in patients on dialysis . Another study in 18 diabetic patients on dialysis and 20 controls found an inverse correlation between TL and length of time in dialysis . A study of 42 haemodialysis patients found reduced telomerase activity compared with non-haemodialysis patients .
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| 99.9 |
Interestingly, recent investigation conducetd by Karin Luttropp concluded that Telomere attrition after 12 months was significantly greater in patients with renal replacement therapy compared to dialysis patients in addition non-CKD patients had meaningfully longer telomeres than CKD patients .
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| 99.94 |
regarding to the evidence from pre-ESRD and CKD patients, earlier studies in patients with heart failure and normal range kidney function have reported a strong correlation between shortening TL and declining kidney function, even after adjustment for age [16, 17]. In this regard, Pim et al, have evaluated association of TL with renal function in 610 patients with heart failure (aged 40 to 80 years), and found that age-and sex-adjusted TL decreased steadily across decreasing quarters of eGFR . Another study by Wong et al, explored the association between TL and renal function in patients with chronic heart failure (n = 866, median age was 74) . They reported that TL was associated with renal function, even after adjustment for age, gender, age at chronic heart failure onset, and severity of chronic heart failure . In contrast, the Heart and Soul study which is longitudinal study of patients with stable coronary heart disease, found that kidney function was not independently associated with shortened TL or telomere shortening over 5 years . However their findings should be considered with caution, considering the focus on relatively old predominantly subjects (mean age was 66.7), with stable coronary heart disease .
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review
| 99.7 |
This study has several strengths. It is one of the largest studies of the association of TL with kidney impairment. Kidney impairment was assessed by both eGFR and proteinuria. The selection of participants was based on random sampling of the general population and therefore the results can be extrapolated to the general population. As the data collection was performed on all days of the week throughout the year in NHANES, the potential for selection bias is very low [25, 26]. The findings from our study should be considered in the context of some limitations. The cross-sectional nature does not allow inference about causality. We did not have available any repeated measure of TL with quantitative polymerase chain reaction in the same subjects after several follow-up years to elucidate temporality of these findings.
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| 99.94 |
This study has important clinical and public health implications. Understanding the interplay between TL and kidney function is a necessary and important step toward any application of the resulting knowledge for public health policy and action. Moreover we knew that CKD may increase the risk of CVD as leading cause of the death which may could be explain by role of the TL [27–29]. Our study provides a comprehensive snapshot of the relationship of kidney function with TL at the national level in the US.
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| 99.94 |
The NHANES are ongoing repeated cross-sectional surveys conducted by the US National Center for Health Statistics (NCHS). The NCHS Research Ethics Review Board approved the NHANES protocol and consent was obtained from all participants [30, 31]. Data collection on demographic, diets, and behaviours occurred through questionnaires administered during home visits, while anthropometrics and biomarkers data were collected by trained staff using mobile examination units [30, 32]. The interview consisted of questions on socio-demographic characteristics (age, gender, education, race/Hispanic origin, and health insurance) and questions on previously diagnosed medical conditions. More detailed information on the NHANES protocol is available elsewhere [30, 33, 34]. This study was based on analysis of data from the 1999-2002 NHANES cycles. Analyses were restricted to participants aged 18 years and older. Fasting blood glucose (FBG), total cholesterol (TC), low density lipoprotein cholesterol (LDL-C), high density lipoprotein cholesterol (HDL-C), triglycerides (TG) levels and telomere length were assayed using methods described in the NHANES Laboratory/Medical Technologists Procedures Manual [30, 35, 36]. Complete laboratory procedures for collection, storage, calibration and quality control of blood samples for determination of hsCRP and other inflammatory markers are available elsewhere. Creatinine was measured by the Jaffe reaction and standardized by methods described previously . A random urine specimen was collected from participants, and urinary creatinine was measured by the Jaffe rate reaction, urinary albumin was measured by solid-phase fluorescent immunoassay . Albuminuria was measured by urinary albumin-creatinine ratio (ACR) . Glomerular filtration rate [eGFR, (ml/min/1.73m2)] was estimated using the CKD Epidemiology Collaboration (CKD-EPI) equation. CKD was defined as eGFR less than 60 (ml/min/1.73m2) .
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| 99.94 |
Aliquots of purified DNA, isolated from whole blood using the Puregene (D-50 K) kit protocol (Gentra Systems, Inc., Minneapolis, MN, USA), were obtained from participants. TL assay was performed using the quantitative polymerase chain reaction method to measure TL relative to standard reference DNA (also known as the telomere-to-single copy gene (T/S) ratio) [35, 40]. Each sample was assayed 3 times on 3 different days. The samples were assayed on duplicate wells, resulting in 6 data points. Control DNA values were used to normalize between-run variability [40, 41]. Runs with more than 4 control DNA values falling outside 2.5 standard deviations from the mean for all assay runs were excluded from further analysis (6% of runs). For each sample, any potential outliers were identified and excluded from the calculations (2% of samples). The inter-assay coefficient of variation was 6.5%. The Centers for Disease Control (CDC) conducted a quality control review before linking the TL data to the NHANES data files.
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| 100.0 |
We conducted the analyses according to the CDC guidelines for analysis of complex NHANES data, accounting for the masked variance and using the proposed weighting methodology . To investigate the association between TL and kidney function, univariable and multivariable (age-, sex-, race-, smoking-, fasting blood glucose-, systolic and diastolic blood pressure-, body mass index-, C-reactive protein, diabetes and hypertension) regressions were applied. Adjusted (age- and sex-) mean of cardiometabolic factors and kidney function were compared across quarters of TL using the analysis of co-variance (ANCOVA) with Bonferroni correction. Variables were compared by using analysis of variance (ANOVA) and Chi-square tests. All tests were two sided, and p < 0.05 used to characterise statistically significant results. Data were analysed using SPSS® complex sample module version 22.0 (IBM Corp, Armonk, NY, USA).
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| 100.0 |
Bones provide mechanical protection for the body (such as protecting internal organs and blood forming marrow), facilitate locomotion, and serve as a reservoir for calcium, magnesium and phosphate minerals . Osteogenesis often requires a replacement graft to restore the function of damaged tissue. Scaffolds for bone tissue engineering offer a promising alternative treatment for medical use, as well as a controllable system for studies of biological function, development of biology and pathogenesis [2, 3]. The materials for scaffolds exhibit many of the mechanical properties of the engineered graft. Inorganic and organic scaffolds are easily fabricated into different structures, but the compressive modulus of organic scaffolds is often unsatisfactory. Alternatively, ceramic scaffolds have excellent stiffness, but are fragile and have low porosity, resulting in loosening of fractured implants in clinical applications. Combining organic and inorganic materials to form composite scaffolds can enhance the mechanical and biochemical properties of scaffolds for bone tissue regeneration [4–6].
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| 99.9 |
Numerous research efforts have addressed the development of an ideal scaffold for bone tissue engineering [7, 8]; however, they still have several limitations. Due to its biocompatibility, biodegradability, controllable strength, and good oxygen and water permeability, silk fibroin (SF) originated from Bombyx mori has been fabricated for various tissue engineering scaffolds with various chemical, structural and biochemical modifications. SF has been investigated with regard to applications of tissue engineered blood vessels, skin, bone, and cartilage [9–13]. Porous 3-D scaffolds are suitable for bone tissue engineering, as they enhance cell viability, proliferation, and migration. Furthermore, highly porous scaffolds (up to 92% porosity) facilitate nutrient and waste transport into and out of the scaffolds . Physically crosslinked SF hydrogels have been produced through the induction of the β-sheet structure in SF solutions. However, due to the β-sheet formation, the SF exhibits relatively slow degradation in vitro and in vivo. To improve the degradability and strength of hydrogels, the SF has been crosslinked in recent years via a number of methods. Chemically crosslinked SF hydrogels using chemical crosslinkers, such as genipin and glutaraldehyde [10, 15, 16], ionizing irradiation , nitrate salts , and enzymatic crosslinker including tyrosinase have also been studied. However, these crosslinking methods were found to be time-consuming and cytotoxic. Therefore, it is very important to establish a rapid crosslinking method to develop chemically crosslinked SF hydrogels.
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review
| 99.9 |
Ionizing radiation, like gamma ray (γ-ray), electron beam, and ion beam has been used as an initiator for the preparation of hydrogel from unsaturated compounds. The irradiation results in the formation of radicals on the unsaturated polymer chain and water molecules, which attack the polymer chains and thus induce intermolecular crosslinking [20, 21]. The ionizing radiation would be an excellent pathway for the preparation of uniformly dispersed organic/inorganic composite hydrogels, because polymer solutions easily undergo chemical crosslinking and solidify immediately. In addition, potentially toxic initiators and crosslinkers do not need to be used for the synthesis of organic/inorganic composite scaffolds for tissue engineering .
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| 99.9 |
This study employed SF and HAP NPs due to the composite hydogel’s biocompatibility and osteoconductivity, and easy reproducibility of fabrication. The SF hydrogels were prepared via a chemical crosslinking reaction using γ-ray irradiation. Also, the effects of HAP content on the morphological, structural, and mechanical properties of porous SF hydrogels were examined. In addition, the effect of SF/HAP composite hydrogel on the osteogenic responses of hMSCs was assessed with respect to bone tissue regeneration.
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| 100.0 |
SF solution was prepared according to the previously established protocol [17, 23]. Briefly, the scoured Bombyx mori (B. mori) SF fiber was dissolved in a ternary solvent composed of calcium chloride, ethanol and water (1:2:8 M ratio) at 85 °C for 4 h. The dissolved SF solution was dialyzed in distilled water for 72 h using dialysis cellulose tubular membranes (250-7 μ, Sigma, St. Louis, MO, USA) to remove the salts. After dialysis, the solution was centrifuged at 3000 rpm for 10 min to remove the insoluble impurities. The final concentration of the resultant aqueous SF solution was approximately 2.3 wt%, which was determined by weighing the remaining sponge weight after lyophilization. A higher concentration SF solution was prepared by reverse dialysis against 25 wt% polyethylene glycol (PEG, Mw 20,000) solution at room temperature [24, 25]. The SF concentration after reverse dialysis was approximately 7.9 wt%. The regenerated SF solution was stored at 4 °C for further use.
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| 100.0 |
SF/HAP composite hydrogels were prepared as shown in Fig. 1. Freshly regenerated 7.9 wt% SF solution was blended with poly(vinyl pyrrolidone) (PVP) to improve the dispersity of HAP NPs. SF/HAP aqueous solution was prepared by adding HAP NPs (particle size <200 nm, Sigma Aldrich, St. Louis, MO) with various concentration directly into the SF aqueous solution. SF/HAP aqueous solution was poured into a petri dish and irradiated by γ-ray from a Co-60 source. The irradiation dose varied to 60 kGy and the dose rate was 15 kGy/h. The irradiated samples were cut into small pieces and then lyophilized for 3 days to analyze various properties.Fig. 1Schematic illustration of the preparation method of the SF/HAP composite hydrogels
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| 100.0 |
SF/HAP composite hydrogels with different HAP contents (0–3 wt%) were named as SF-0, SF-1, SF-2, and SF-3 respectively. Table 1 shows the compositions of SF/HAP composite hydrogels.Table 1Sample code and composition of SF/HAP composite hydrogelsSample nameSF concentration (wt%)HAP concentration (wt%)PVP concentration (wt%)SF only7.90.01.0SF-1% HAP7.91.01.0SF-2% HAP7.92.01.0SF-3% HAP7.93.01.0
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study
| 100.0 |
The pore structure, morphology, and distribution of HAP NPs of SF/HAP composite hydrogels were observed by field emission scanning electron microscopy (FE-SEM) (JSM-7000F, JEOL, Japan) and energy dispersive X-ray spectroscopy (EDX) equipment. The pore parameters including surface area, pore volume, pore size and porosity were characterized by mercury porosimetry (Micromeritics, ASAP 2020). The crystalline structure of SF/HAP composite hydrogels was measured by X-ray diffraction (XRD) (D8 Discover, Bruker, USA) in the range of 2θ from 5 to 50° (λ = 0.154 nm, 40 kV, 40 Ma). The compressive strength of composite hydrogels was measured using a cube-shaped sample (10 mm × 10 mm × 10 mm) by Instron 5848 mechanical tester machine with a crosshead speed of 5 mm/min and 50% strain using a 500 N load cell.
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study
| 100.0 |
To evaluate the biocompatibility of composite hydrogel, hMSCs were purchased from the American Type Culture Collection (ATCC, Manassas, VA, USA). The cells were cultured in α-MEM (Gibco-BRL, Gaithersbug, MD, USA) containing 10% fetal bovine serum (FBS) and 1% antibiotics at 37 °C under 5% CO2 and 100% humidity. Osteoblast differentiation was induced using osteoblast differentiation reagents (10 mM β-glycerophosphate, 50 μg/mL ascorbic acid, and 100 nM dexamethasone (Sigma-Aldrich, St. Louis, MO, USA). The number of viable cells was determined using the CellTiter96® aqueous one solution kit (Promega, Madison, WI, USA). Briefly, cells were seeded to the hydrogel. At a predetermined time point (6 days), 200 μL of MTS reagent was mixed with 500 μL of culture media and added to each well. After incubation for 2 h, absorbance of the supernatant was measured at 490 nm using an ELISA reader (SpectraMAX M3; Molecular Devices, Sunnyvale, CA, USA). After 6 days of cultivation, cell-loaded hydrogels were rinsed with PBS to remove the phenol red, and were with PBS. In addition, the Live/Dead® Viability/Cytotoxicity staining kit (Molecular Probe, Eugene, OR, USA) reagent solution was added. After incubation for 30 min in a CO2 incubator, the samples were observed using an inverted fluorescence microscope (DM IL LED Fluo; Leica Microsystems, Wetzlar, Germany). SEM was used to observe cell adhesion to the hydrogels. After 6 days of culture, the cell-loaded hydrogels were fixed with 2.5% glutaraldehyde, and additional-fixation was performed with 0.1% osmium tetroxide (Sigma-Aldrich, St. Louis, MO, USA). After dehydration with a graded ethanol series (50%, 75%, 95% and 100%), the samples were sputter-coated with gold, and observed by SEM (EM-30; Coxem, Daejeon, Korea) .
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study
| 100.0 |
The degree of osteoblast differentiation in the cells was evaluated by determining the alkaline phosphatase (ALP) activity. After 7 days of culture using osteogenic induction medium, the adherent cells were removed from the hydrogel by homogenization in PBS with 1% Triton X-100. Then, the suspension was mixed with 0.1 M glycine NaOH buffer (pH 10.4) and 15 mM p-nitrophenyl phosphate (p-NPP; Sigma, St. Louis, MO, USA). After 30 min incubation at 37 °C, the reaction was terminated by adding 0.1 N NaOH, and the p-NPP hydrolysis was determined by ELISA reader (Spectra MAX M3) at 410 nm. Protein concentrations were measured by bicinchoninic acid (BCA) protein assay reagent kit (Pierce, Rockford, IL, USA), and normalized. To determine the hydroxyapatite nucleation on the surface of hydrogel, simulated body fluid (SBF) was used. Briefly, the fabricated hydrogels were immersed in 1× SBF (Biosesang, Sungnam, Korea), and maintained at 37 °C. After immersion period of 7 days, the hydrogels were removed from the fluid, gently rinsed with distilled water, and dehydrated with a graded ethanol series. After the sample was sputter-coated with gold, the behavior of hydroxyapatite crystal growth was observed by SEM (EM-30).
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study
| 100.0 |
hMSCs were cultured with continuous treatment with osteoblast differentiation reagents contained media. After 21 days, the cell-loaded hydrogels were fixed with 70% ice-cold ethanol for 1 h at 4 °C. After the ethanol was removed, calcium accumulation was measured by staining with 40 mM Alizarin Red-sulfate (AR-S; Sigma-Aldrich, St. Louis, MO, USA) solution, and normalized with non-cultured scaffold, respectively. The stained portions were photographed by digital camera. The deposited stain was then dissolved using 10% cetylpyridinium chloride solution and the absorbance was read at 562 nm by ELISA reader.
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study
| 100.0 |
The fabrication of 3-dimensional porous SF/HAP composite hydrogels was prepared by γ-ray irradiation process. The pore structure of each hydrogel was observed by FE-SEM (Fig. 2). Each hydrogel had uniform pore size and interconnected pore structure, in particular, HAP concentration did not affect the pore size within the hydrogels. HAP NPs were uniformly dispersed on the pore wall of composite hydrogels, and incorporated NPs were increased with increasing HAP concentration. Therefore, the distribution of pores was uniform and this morphology resembles that of previously studied pore structures obtained by radiation technique . The pore size of various hydrogels ranged between 130 and 250 μm (average pore size 161 ± 42 μm). To corroborate the presence of HAP NPs in SF/HAP composite hydrogels, EDX mapping equipment was used. Figure 3 shows the results of EDX mapping for the hydrogels. The green marked points in the images represent the site of detected Ca elements in HAP NPs. As shown in Fig. 3, Ca elements were not observed in SF-0 (Fig. 3a), but Ca element (green intensity) was well dispersed, and was increased with increasing incorporated HAP NPs contents (Fig. 3b-d). These findings indicate that HAP NPs were appropriately incorporated and well dispersed into the composite hydrogels. In order to further confirm the presence of HAP NPs, SF/HAP composite hydrogels (SF-0, SF-1, SF-2, and SF-3) were characterized by XRD. The XRD spectrum of SF/HAP composite hydrogels showed amorphous silk I conformation. The specific HAP NPs peaks also appeared in all composite hydrogels. The results show that all SF composite hydrogels were successfully generated by intermolecular chemical crosslinking reaction, instead of secondary structural change of SF. Figure 4 shows the XRD spectrum of SF based composite hydrogels.Fig. 2Representative FE-SEM images of a SF only, b SF-1% HAP, c SF-2% HAP, and d SF-3% HAP Fig. 3Distribution of calcium element in SF/HAP composite scaffolds; a SF only, b SF-1% HAP, c SF-2% HAP, and d SF-3% HAP Fig. 4X-ray diffraction of SF/HAP composite scaffolds
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study
| 100.0 |
Figure 5 describes the porosity and mechanical properties of SF/HAP composite hydrogels. The appropriate pore size and interconnected pores of hydrogels provide sufficient opportunity for cell proliferation. The porosities of SF-0, SF-1, SF-2, and SF-3 were similar (Fig. 5a), and there was no significant difference in the porosity among the hydrogels. Therefore, SF composite hydrogels could provide a good environment for cell migration and differentiation. These results were also related to the pore structure on FE-SEM. Also, Fig. 5b shows the maximum compressive strength of composite hydrogels with/without HAP. Interestingly, SF-0 had the highest compressive strength compared with HAP incorporated SF hydrogels, and also the maximum compressive strength of composite hydrogels decreased as the HAP NPs content increased up to 3 wt% because of the lack of organic/inorganic interaction. Furthermore, during the irradiation, gelation did not occur when more than 3% HAP was added (data not shown). These results were also related to decrease in the compressive strengths of SF/HAP composite scaffolds.Fig. 5Physical properties of SF/HAP composite scaffolds; a porosity and b compressive gel strength respectively
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study
| 100.0 |
The proliferation and cytotoxicity of the SF/HAP composite hydrogels were determined using the standard MTS assay with hMSCs to evaluate the potential of these materials as a scaffold for bone regeneration. Figure 6 shows that the MTS assay revealed increased cell proliferation rate as the HAP concentration increased, which indicated that HAP supported the proliferation of hMSCs. However, there was no significant difference in proliferation between SF-2 and SF-3. After 6 day of culture, hMSCs were found to have attached and distributed evenly on all hydrogel samples and a small number of hMSCs filled the pores, and formed a continuous monolayer in all hydrogel samples (Fig. 7). The cell monolayer density was increased with increasing HAP NPs concentration. The hMSCs were stained with a Live-Dead™ kit after 4 days of culture, and then observed with confocal microscopy. Green color represents the live cells, while red color represents the dead cells . After 4 days culture, most cells presented green fluorescence, which indicated no significant cell death in the hydrogels under culture, as shown in Fig. 8. The SF/HAP composite hydrogels induced by γ-ray irradiation have noteworthy potential as bone tissue scaffolds, because they showed no significant cytotoxicity against hMSCs.Fig. 6Proliferation of human mesenchymal stem cells on the SF/HAP composite scaffolds evaluated by MTS assay at day 6 Fig. 7Representative FE-SEM images of hMSCs cultured on a SF only, b SF-1% HAP, c SF-2% HAP, and d SF-3% HAP scaffolds at day 6 Fig. 8Viability and cytotoxicity staining of cells cultured on a, e SF only, b, f SF-1% HAP, c, g SF-2% HAP, and d, h SF-3% HAP scaffolds at day 4
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study
| 100.0 |
To investigate the osteogenic differentiation of hMSCs seeded on composite hydrogels, ALP activity was assessed. The ALP activity of hMSCs cultured on different types of hydrogel was assessed at 7 days. The ALP activity has been implicated as an early marker of osteogenic differentiation [28–30]. As shown in Fig. 9a, the ALP activity increased as the HAP NPs concentration increased up to 2%. However, there was no significant difference between 2 and 3% HAP concentration. It is considered that the HAP NPs affected osteogenesis and osteogenic differentiation of the hMSCs. Figure 9b-e show SEM imagery of the surface immersed in SBF. After 7 day, the HAP nuclei were formed on the surface of the hydrogels, and then the HAP nuclei grew and the amount of HAP increased with increasing HAP concentration. Figure 10 shows the calcium accumulation of hMSCs-loaded SF/HAP composite hydrogels. The stained Alizarin red-sulfate (AR-S) intensity was increased with increasing HAP concentration. From the results, the SF/HAP composite hydrogels showed excellent cell proliferation, osteogenic differentiation, and calcium accumulation, which are highly desirable properties for bone tissue engineering scaffolds.Fig. 9 a ALP activity of SF/HA hybrid scaffolds and hydroxyl apatite nucleation of b SF only, c SF-1% HAP, d SF-2% HAP, and e SF-3% HAP scaffolds in SBF solution at day 7 Fig. 10Calcium accumulation of SF/HAP composite scaffolds at day 21
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study
| 100.0 |
In this study, the SF/HAP composite hydrogels for bone tissue engineering were prepared by gamma-ray irradiation. The morphology and distribution of HAP NPs in the SF hydrogels were investigated by FE-SEM, EDX and XRD. From the results, the SF/HAP composite hydrogels had highly porous structure, and HAP NPs were evenly dispersed in the SF hydrogel. Compared with pure SF hydrogel, the maximum compressive strength of composite hydrogels was decreased with increasing HAP content due to insufficient organic/inorganic interaction. The SF/HAP composite hydrogels also showed increased cell proliferation and adhesion. Furthermore, these hydrogels enhanced in vitro hMSCs osteogenic differentiation. Therefore, these results indicate that the 3D porous SF/HAP composite hydrogel offers promise as a biomaterial for bone tissue engineering.
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study
| 100.0 |
Chemical Reaction Networks (CRNs) are traditionally used to capture the behaviour of inorganic and organic chemical reactions in a well-mixed solution. Recently, a paradigm shift in the scientific community has seen the use of CRNs extend to that of a high-level programming language for molecular computing devices (Cook et al. 2009), where the fundamental computational process differs from conventional digital electronics in that it involves transformation of input chemicals into output via reaction rules, as opposed to processing discrete signals (voltage bands) interpreted as Boolean values. Several digital and analogue circuits (Magnasco 1997; Soloveichik et al. 2008) have been designed in CRNs and their computational power studied (Soloveichik et al. 2010; Chen et al. 2013). It has also been demonstrated in principle that any CRN can be physically realised in DNA (Soloveichik et al. 2010; Cardelli 2010; Chen et al. 2013). CRNs are therefore particularly attractive as a programming language for use in nanotechnology and biomedical applications, where it is difficult to integrate traditional electronics.
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review
| 99.75 |
Chemical systems can store and process information in several ways. We focus on finite systems of molecules interacting in a well-mixed solution under mass-action kinetics and emulate Boolean circuits by encoding information through molecular concentrations reaching a particular threshold. The computation proceeds by transforming input species concentrations into outputs according to the reactions of a finite CRN. It is known that the computational power of CRNs is affected by the choice of the semantics, deterministic or stochastic. In particular, assuming a small probability of error, (finite) stochastic CRNs have been shown to be Turing universal (Soloveichik et al. 2008). The deterministic semantics interprets the reactions as a system of differential equations, which describe the evolution of the system as a vector of real-valued species concentrations over time (Chen et al. 2013). The stochastic semantics, on the other hand, views the state of the system as a vector of (non-negative) integer molecular counts and state transitions as a reaction which has a non-zero probability of occurring (Cook et al. 2009). The stochastic evolution of the system over time is obtained as a solution of the Chemical Master Equation (CME) (Kampen 1992). It is well known that the deterministic semantics is not accurate for small populations. While the stochastic semantics is exact, it is infeasible for large molecular counts. One scalable alternative is the Linear Noise Approximation, which is a real-valued approximation of the CME (Cardelli et al. 2015). The correctness of the behaviour of a circuit described by a finite CRN can be analysed by inspecting its stochastic and deterministic evolution over time. In addition, techniques such as model checking can be employed to analyse the temporal ordering of events.
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study
| 100.0 |
While CRN designs for synchronous sequential logic circuits have been proposed, to mention (Magnasco 1997; Soloveichik et al. 2010, 2008), a physical realisation of these devices is challenging because of their reliance on a clock to synchronise events in order to ensure the correct temporal order of the phases of the computation. Clocks are difficult to make, since they arise from unique conditions of chemical concentrations and kinetic constants, and must control a large number of events. In electronics, an alternative circuit design technology is asynchronous sequential logic (Spars and Furber 2002; Myers 2004), which instead of a clock relies on handshaking protocols to synchronise events. Asynchronous circuits are widely used for low-power microprocessor designs, e.g., by ARM, though require a larger circuit area. The key component is the Muller C-element, which is used to synchronise multiple independent processes in a manner insensitive to the delays on wires and individual components. To ensure Turing completeness of asynchronous circuits, we also require an isochronous fork in addition to the Muller C-element. An isochronous fork is a component which produces a fan-out of signals that reach the target at virtually the same time. This assumption is difficult to achieve in conventional electronics, because of the need to make the wires the same length, but is straightforward in chemical kinetics because of the well-mixed assumption.
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other
| 99.25 |
This paper provides novel CRN designs for the construction of an asynchronous computing device based on a bi-molecular reaction motif inspired by the Approximate Majority network (Angluin et al. 2008; Cardelli and Csikász-Nagy 2012). The motif employs catalytic reactions to achieve bistable switching of molecular concentrations, which emulates high and low voltage signals in digital electronics. All components are produced with simple reactions and uniform reaction rates, where we assume a well-mixed solution under mass action kinetics, and are independent of a universal clock. Moreover, any design provided in this paper could in principle be realised as a DNA strand displacement device (Cardelli 2010). We work with the dual-rail design methodology and employ a variant of the diagrammatic language of Cardelli (2014) to represent the designs at the high level. Starting from the Muller C-element, we design the main components of a complete asynchronous computing device in terms of CRNs in a principled way, including logic gates, control flow and basic arithmetic, as well as more complex structures such as queues. We validate the designs by exploring their time evolution for all possible combinations of inputs using Microsoft’s Visual GEC tool,1 with the latter also approximated using an experimental implementation of the Linear Noise Approximation (LNA) of Cardelli et al. (2015) provided by Visual GEC that offers better scalability. We use the LNA to highlight a flaw with a key design component. Further, we demonstrate the correct behaviour of the circuits against temporal logic specifications with the probabilistic model checker PRISM2 (Kwiatkowska et al. 2011). Our designs constitute the first feasible implementation of asynchronous computational components as CRNs, and are relevant for a multitude of applications in synthetic biology and biosensing.
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study
| 99.94 |
The computational power of Chemical Reaction Networks, viewed as a programming language for engineering biochemical systems, has been studied by a number of authors, to mention Cook et al. (2009) and Chen et al. (2013). There are a number of ways in which chemical systems can encode and process information. This includes simulating Boolean circuits, where information is encoded in binary form using high and low concentrations similarly to this paper, e.g. Magnasco (1997), Soloveichik et al. (2010) and Soloveichik et al. (2008), as well as geometric arrangements, for example self-assembly (Rothemund et al. 2004) and molecular walkers (Dannenberg et al. 2015) not considered here. Researchers have also investigated the power of CRNs to model distributed algorithms (Angluin et al. 2008).
|
review
| 99.56 |
Regarding synchronous logic circuits, much of the work to date considered abstract CRN schemes. One exception is Silva and McClenaghan (2004), where a system of actual chemical reactions is given, together with a precise molecular implementation for gates complete with a thermodynamic analysis of how the system would evolve, though only for simple gate designs. In Cook et al. (2009) we see the construction and composition of simple logic gates based upon catalytic reactions, but they do not mention control flow or systematic component design in a dual rail setting. In Senum and Riedel (2011) the authors propose CRNs for an inverter, an incrementer, a decrementer and a copier; their designs are based on two rate constants, “fast” and “slow”, and thus are not rate-independent, in contrast to the designs presented here.
|
review
| 99.5 |
CRNs can also be viewed as computing functions over reals or Booleans. A single CRN computes a function over a finite domain, which is analogous to Boolean circuits in the sense that any given circuit computes only on inputs of a particular size (Soloveichik et al. 2008). An implementation of dual-rail logic gates that are rate-independent is given in Chen et al. (2014). In contrast, our designs are composable and capable of performing non-trivial computation.
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other
| 99.9 |
Since the behaviour of CRNs is asynchronous, a fact evident through their equivalence with Petri net models (Cook et al. 2009), the main difficulty with programming them is the need to control the order of reactions. In Cook et al. (2009) it is suggested that this “uncontrollability” can be handled by changing rate constants, an idea followed up in Napp and Adams (2013), where CRN designs for basic arithmetic are given based on two rate constants, “fast” and “slow”. Our designs, on the other hand, exploit the asynchrony of the underlying CRN model and work with uniform rates.
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study
| 83.94 |
Designs for the Muller C-element, though not the remaining components of an asynchronous device, have been constructed from genetic logic gates (Nguyen et al. 2010) and a genetic toggle switch (Nguyen et al. 2007), but we are not aware of any other nanoscale designs for asynchronous circuits. Soloveichik et al. (2010) shows that any CRN, including those presented in this paper, can theoretically be implemented as a DNA Strand Displacement device. These devices have been demonstrated in the lab (Qian and Winfree 2011, 2011; Chen et al. 2013), and thus provide an indication of experimental feasibility of our designs.
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study
| 99.94 |
A Chemical Reaction Network (CRN) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=(\varLambda ,R)$$\end{document}C=(Λ,R) is a pair of finite sets, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda$$\end{document}Λ is a set of chemical species and and R is a set of reactions. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varLambda |$$\end{document}|Λ| denotes the size of the set of species. Reactions in R describe how species interact. Formally, a reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in R$$\end{document}τ∈R is a triple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =(r_{\tau },p_{\tau },k_{\tau })$$\end{document}τ=(rτ,pτ,kτ), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\tau } \in {\mathbb {N}}^{|\varLambda |}$$\end{document}rτ∈N|Λ| is the vector of molecular counts of the reactants, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\tau } \in {\mathbb {N}}^{|\varLambda |}$$\end{document}pτ∈N|Λ| is the vector of molecular counts of the products and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{\tau } \in {\mathbb {R}}_{>0}$$\end{document}kτ∈R>0 are the coefficient associated to the rate of the reaction. We assume ordering of species within vectors is alphabetical. Given a reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1=( ^T,^T,k_1 )$$\end{document}τ1=(T,T,k1), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdot ^T$$\end{document}·T is the transpose of a vector, we often refer to it as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1 : A + B \, \overset{k_1}{\rightarrow } \, 2C$$\end{document}τ1:A+B→k12C, where A, B and C are generic species.
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other
| 97.6 |
In this paper we are only concerned with uni-molecular reactions, i.e. those which have only one reactant, and bi-molecular, i.e. those with two reactants. The “reversible reaction” notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A + B \leftrightharpoons 2C$$\end{document}A+B⇋2C is a shorthand for the two reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A + B \overset{k_1}{\rightarrow } 2C$$\end{document}A+B→k12C and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2C \overset{k_2}{\rightarrow } A + B$$\end{document}2C→k2A+B, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1$$\end{document}k1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_2$$\end{document}k2 are not necessarily equal.
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study
| 99.8 |
We assume that the system is well stirred, that is, the probability of the next reaction occurring between two molecules is independent of the location of those molecules, at fixed volume V and temperature; under these assumptions a configuration or state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in {\mathbb {N}}^{|\varLambda |}$$\end{document}x∈N|Λ| of the CRN is given by the number of molecules of each species. Given a configuration x we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=\frac{x}{N}$$\end{document}z=xN, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=V \cdot N_A$$\end{document}N=V·NA is the volumetric factor, V is the volume and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_A$$\end{document}NA Avogadro’s number. We write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(\lambda _i)$$\end{document}x(λi) for the number of molecules of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _i$$\end{document}λi in the configuration x and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z(\lambda _i) = \frac{x(\lambda _i)}{N}$$\end{document}z(λi)=x(λi)N to denote the concentration of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _i$$\end{document}λi in the same configuration.
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study
| 99.94 |
We will sometimes distinguish between CRNs with different initial configurations, and to this end define a chemical reaction system (CRS) as a tuple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=(\varLambda ,R,x_0)$$\end{document}S=(Λ,R,x0) where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varLambda ,R)$$\end{document}(Λ,R) is a CRN and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in {\mathbb {N}}^{|\varLambda |}$$\end{document}x0∈N|Λ| represents its initial configuration, and we sometimes use the terms CRN and CRS interchangeably.
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other
| 78.7 |
Diagrammatic CRN notation To better visualize a CRN \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=(\varLambda ,R)$$\end{document}C=(Λ,R) as a circuit, we employ a directed multi-edge graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varLambda ,E)$$\end{document}(Λ,E) based upon a fragment of the diagrammatic notation for influence graphs Cardelli (2014). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda$$\end{document}Λ, the nodes, represent the species of the CRN C and edges E are derived from reactions R as follows. A reaction is represented as a directed multi-edge with sets of species as source and target. Each edge is either a pointed arrow (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uparrow$$\end{document}↑) or a rounded arrow (), with the source represented by the flat edge and the target represented by the arrow head. A reaction that produces a species as a product is connected to it by a directed edge.
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study
| 99.7 |
All reactions within our diagrams are catalytic and are bi-molecular reactions of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X + Y \rightarrow X + Z$$\end{document}X+Y→X+Z, meaning that Y is transformed to Z and X is a catalyst, that is, X influences the transformation of Y to Z. The edges represent that a source species is catalytic to a target reaction.
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other
| 75.75 |
(CRN Diagrams Example) We illustrate the flexibility of the diagrammatic notation with three CRN examples. Figure 1a shows the CRN with the single reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C + A \rightarrow C + B$$\end{document}C+A→C+B. The ball () indicates that C is a catalyst to the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \rightarrow B$$\end{document}A→B represented by the arrow (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uparrow$$\end{document}↑). A species can act as both a reactant or catalyst in the same reaction, and similarly for a product. In Fig. 1b we depict the CRN with two reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ A + A \rightarrow A + B, B + B \rightarrow B + A \}$$\end{document}{A+A→A+B,B+B→B+A}, in which both A and B catalyse themselves. Figure 1c depicts the CRN with reaction set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ A + B \rightarrow B + B, C + B \rightarrow B + B \}$$\end{document}{A+B→B+B,C+B→B+B}, with species B catalysing multiple reactions, which is represented by a multiheaded edge with multiple s. This CRN transforms species A and C into species B.Fig. 1Diagrammatic notation for CRNs. a CRN with the single reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C + A \rightarrow C + B$$\end{document}C+A→C+B, in which species C catalyses the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \rightarrow B$$\end{document}A→B. b CRN with reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ A + A \rightarrow A + B, B + B \rightarrow B + A \}$$\end{document}{A+A→A+B,B+B→B+A}, in which species A and B are both reactants, products and catalysts. c CRN with reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ A + B \rightarrow B + B, C + B \rightarrow B + B \}$$\end{document}{A+B→B+B,C+B→B+B}, which demonstrates that B can be catalytic to multiple reactions
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| 99.94 |
Diagrammatic notation for CRNs. a CRN with the single reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C + A \rightarrow C + B$$\end{document}C+A→C+B, in which species C catalyses the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \rightarrow B$$\end{document}A→B. b CRN with reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ A + A \rightarrow A + B, B + B \rightarrow B + A \}$$\end{document}{A+A→A+B,B+B→B+A}, in which species A and B are both reactants, products and catalysts. c CRN with reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ A + B \rightarrow B + B, C + B \rightarrow B + B \}$$\end{document}{A+B→B+B,C+B→B+B}, which demonstrates that B can be catalytic to multiple reactions
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| 97.94 |
Dual-rail representation We represent a Boolean circuit with inputs I and outputs O, denoted B(I, O), as follows. Firstly, a Boolean variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=\{0,1\}$$\end{document}b={0,1} could be encoded in a single species X, where 0 would be encoded as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[|X|] = 0$$\end{document}E[|X|]=0 and 1 as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[|X|] \ge M$$\end{document}E[|X|]≥M, where E[|X|] denotes an expectation of the number of molecules of X and M is a molecular population threshold.
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| 99.8 |
The CRN computes by transforming an input concentration into an output concentration, which reaches the appropriate level upon convergence. However, since absence of molecules cannot be measured, we employ dual-rail methodology and represent every Boolean variable with two species, denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, X_{lo}$$\end{document}Xhi,Xlo. Just like we cannot represent both 0 and 1 on an electrical wire, we restrict our CRNs such that either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[|X_{hi}|] \ge M$$\end{document}E[|Xhi|]≥M or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[|X_{lo}|] \ge M$$\end{document}E[|Xlo|]≥M, but not both, can be present when a CRN has stabilised and no further reactions occur. We consider a high concentration output as correct if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[|X_{hi}|] > 0.8 max(X_{hi}) - 1SD(X_{hi})$$\end{document}E[|Xhi|]>0.8max(Xhi)-1SD(Xhi), where 0.8 is a threshold normalised between the values [0, 1], max() is a function that returns the maximum molecular concentration of a species within the CRN and 1SD computes 1 standard deviation from the mean concentration of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[X_{hi}]$$\end{document}E[Xhi]. 1SD returns 0 under the deterministic semantics. Similarly, we consider a low concentration as correct if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[|X_{lo}|] < 0.2 * max(X_{lo}) + 1SD(X_{lo})$$\end{document}E[|Xlo|]<0.2∗max(Xlo)+1SD(Xlo).
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| 99.94 |
For simplicity, we apply the dual rail methodology only to the variables in the input and output sets I and O. The circuits may contain additional variables, which will be considered internal and assumed not to catalyse with any species outside of the CRN circuit. We will encode these with single species and use the naming convention of referring to these internal species as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda , \lambda _1, \cdots , \lambda _i$$\end{document}λ,λ1,⋯,λi. When composing two circuits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1(I_1,O_1)$$\end{document}B1(I1,O1) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2(I_2,O_2)$$\end{document}B2(I2,O2) in series, we define their composition as a circuit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(I_1,O_2)$$\end{document}B(I1,O2), in which all variables in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_1 \cup I_2$$\end{document}O1∪I2 have been made internal.
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| 93.56 |
(Dual-Rail CRN ‘Motif’) We introduce a simple two reaction CRN which forms a ‘motif’ common to all our CRN circuit diagrams. The CRN is given by the set of reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ X_{hi} + Y_{lo} \rightarrow X_{hi} + Y_{hi}, X_{lo} + Y_{hi} \rightarrow X_{lo} + Y_{lo} \}$$\end{document}{Xhi+Ylo→Xhi+Yhi,Xlo+Yhi→Xlo+Ylo} shown in Fig. 2a, where the input set I contains \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and the output set contains \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}, Y_{hi}$$\end{document}Ylo,Yhi. In this CRN the input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi influences the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo} \leftrightharpoons Y_{hi}$$\end{document}Ylo⇋Yhi to produce as output the same Boolean value. We also include a diagrammatic CRN showing the composition of two such motifs in series. Here the inputs are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},X_{lo}$$\end{document}Xhi,Xlo and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}, Y_{lo}$$\end{document}Yhi,Ylo.Fig. 2 a The CRN ‘motif’, in which the input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi influences the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo} \leftrightharpoons Y_{hi}$$\end{document}Ylo⇋Yhi to produce as output the same Boolean value. b CRN obtained by composing two ‘motif’s depicted in a in series. The inputs are now \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},X_{lo}$$\end{document}Xhi,Xlo, outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}, Y_{lo}$$\end{document}Yhi,Ylo, and the remaining species have been made internal through renaming with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ.
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a The CRN ‘motif’, in which the input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi influences the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo} \leftrightharpoons Y_{hi}$$\end{document}Ylo⇋Yhi to produce as output the same Boolean value. b CRN obtained by composing two ‘motif’s depicted in a in series. The inputs are now \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},X_{lo}$$\end{document}Xhi,Xlo, outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}, Y_{lo}$$\end{document}Yhi,Ylo, and the remaining species have been made internal through renaming with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ.
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| 99.75 |
(Dual-rail CRN with Internal (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ) Species) The CRN shown in Fig. 3 represents a circuit B(I, O) with internal species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ which does not belong to the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I \cup O$$\end{document}I∪O. Therefore dual rail methodology is not used for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ as it is not catalytic to any species outside of this CRN circuit. I comprises \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, X_{hi}$$\end{document}Xlo,Xhi and O comprises \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}, Y_{lo}$$\end{document}Yhi,Ylo. The CRN is simplified to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ X_{hi} + \lambda \rightarrow X_{hi} + Y_{hi}, X_{lo} + \lambda \rightarrow X_{lo} + Y_{lo}, \lambda + Y_{hi} \rightarrow \lambda + \lambda , \lambda + Y_{lo} \rightarrow \lambda + \lambda \}$$\end{document}{Xhi+λ→Xhi+Yhi,Xlo+λ→Xlo+Ylo,λ+Yhi→λ+λ,λ+Ylo→λ+λ}. This CRN converts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ, assuming a non-zero initial concentration of molecules, unless there is either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi present, in which case a deadlock occurs. We use the term conversion to mean the occurrence of a reaction where there are non-zero molecular counts of reactants present.Fig. 3Example CRN with internal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ species, which converts outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ, assuming a non-zero initial concentration of molecules, unless there is input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi present
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| 99.8 |
Example CRN with internal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ species, which converts outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ, assuming a non-zero initial concentration of molecules, unless there is input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}$$\end{document}Xlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi present
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| 99.5 |
Deterministic semantics Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=(\varLambda ,R)$$\end{document}C=(Λ,R) be a CRN. The net change associated to a reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in R$$\end{document}τ∈R is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon _{\tau }=p_{\tau } - r_{\tau }$$\end{document}υτ=pτ-rτ. The deterministic semantics models the concentration of the species in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda$$\end{document}Λ over time as a set of autonomous polynomial first order differential equations (ODEs):1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\rm d} \varPhi (t)}{{\rm d}t}= & {} \nonumber \\ F(\varPhi (t))= & {} \sum _{\tau =(r_{\tau },p_{\tau },k_{\tau }) \in R} \upsilon _{r} \cdot \left( k_{\tau }\prod _{i=1}^{|\varLambda |}{\varPhi _i(t)}^{r_{\tau ,i}}\right) . \end{aligned}$$\end{document}dΦ(t)dt=F(Φ(t))=∑τ=(rτ,pτ,kτ)∈Rυr·kτ∏i=1|Λ|Φi(t)rτ,i.where function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi : {\mathbb {R}}_{\ge 0} \rightarrow {\mathbb {R}}^{\varLambda }$$\end{document}Φ:R≥0→RΛ describes the behaviour of the system assuming a continuous state-space semantics, and therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi (t) \in {\mathbb {R}}^{|\varLambda |}$$\end{document}Φ(t)∈R|Λ| is the vector of the species concentrations at time t and F is simply the derivative of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi$$\end{document}Φ with respect to time. Assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0=0$$\end{document}t0=0, the initial condition is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi (0)=[x_0]$$\end{document}Φ(0)=[x0], where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document}x0 is the initial configuration (vector of concentrations of molecules) of the CRN. It is well known that the deterministic semantics may be imprecise for low molecular counts, but is accurate in the limit for high populations Kampen (1992). However, the deterministic semantics produces the same proportion of molecules, regardless of total concentration.
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| 99.25 |
Stochastic semantics The stochastic semantics is represented through a continuous-time Markov chain (CTMC), whose transient evolution can be given via the Chemical Master Equation (CME) (Kampen 1992). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=(\varLambda ,R)$$\end{document}C=(Λ,R) be a CRN. The propensity rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\tau }$$\end{document}ατ of a reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document}τ is a function of the current configuration of the system x such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\tau }(x)dt$$\end{document}ατ(x)dt is the probability that a reaction event occurs in the next infinitesimal interval dt. We assume mass action kinetics, therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\tau }(x)=k_{\tau } \frac{\prod _{i=1}^{|\varLambda |} r_{i,\tau } ! }{N^{|r_{\tau }|-1}}\prod _{i=1}^{|\varLambda |} \left( {\begin{array}{c}x(\lambda _i)\\ r_{i,\tau }\end{array}}\right),$$\end{document}ατ(x)=kτ∏i=1|Λ|ri,τ!N|rτ|-1∏i=1|Λ|x(λi)ri,τ, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{i,\tau }$$\end{document}ri,τ is the i-th component of the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\tau }$$\end{document}rτ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|r_{\tau }|=\sum _{i=1}^{|\varLambda |}r_{i,\tau }$$\end{document}|rτ|=∑i=1|Λ|ri,τAnderson and Kurtz (2011). We define a time-homogeneous CTMC \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X^C(t),t \in {\mathbb {R}}_{\ge 0})$$\end{document}(XC(t),t∈R≥0) with state space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \subseteq {{\mathbb {N}}}^{|\varLambda |}$$\end{document}Q⊆N|Λ| as follows. Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in Q$$\end{document}x0∈Q, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document}x0 is the initial configuration of the system, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X^C(0)=x_0)=1$$\end{document}P(XC(0)=x0)=1. The transition rate from state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document}xi to state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_j$$\end{document}xj is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(x_i,x_j) =\sum _{\{\tau \in R | x_j=x_i+ v_{\tau }\}} N\alpha _{\tau }(x_i)$$\end{document}r(xi,xj)=∑{τ∈R|xj=xi+vτ}Nατ(xi). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^C(t)$$\end{document}XC(t) describes the stochastic evolution of the molecular populations of each species in C at time t. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in Q$$\end{document}x∈Q, we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(t)}(x)=P(X(t)=x|X(0)=x_0)$$\end{document}P(t)(x)=P(X(t)=x|X(0)=x0), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document}x0 is the initial configuration. The CME describes the time evolution of X as:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\rm d}{{\rm d}t} \left( P^{(t)}(x)\right)= & {} \sum _{\tau \in R} \left\{ N\alpha _{\tau }(x-\upsilon _{\tau })P^{(t)}(x-\upsilon _{\tau })-N\alpha _{\tau }(x)P^{(t)}(x)\right\} . \end{aligned}$$\end{document}ddtP(t)(x)=∑τ∈RNατ(x-υτ)P(t)(x-υτ)-Nατ(x)P(t)(x).The solution of the CME is computed through numerical simulation or discretisation techniques such as uniformization (Kwiatkowska et al. 2007), and is generally feasible only for small populations. The CTMC is often represented as a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \times Q$$\end{document}Q×Q rates matrix, which can be viewed as a state transition graph and subjected to model checking against temporal logic properties (Kwiatkowska et al. 2011).
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Linear noise approximation The LNA approximates the CTMC as a continuous-state Gaussian process, given in the form of a set of ODEs that describe the time evolution of expectation and variance of the species. The error of approximation is dependent upon the volumetric factor N, the structure and the rates of the CRN. Given a CRN \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}=(\varLambda ,{\mathcal {R}})$$\end{document}C=(Λ,R) with initial configuration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in {\mathbb {N}}^{|\varLambda |}$$\end{document}x0∈N|Λ| and in a system of volume size N, we define the stochastic process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y=N\cdot \varPhi + \sqrt{N}\cdot Z$$\end{document}Y=N·Φ+N·Z, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi$$\end{document}Φ is the deterministic process given in Eq. 1, and Z is a zero-mean Gaussian process (since we assume the initial condition is a fixed value), and with covariance C[Z(t)] described by the solution of the following ODEs with initial condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[Z(0)]=0$$\end{document}C[Z(0)]=0:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\rm d} C[Z(t)] }{{\rm d} t}= & {} F'(\varPhi (t), C[Z(t)]) \nonumber \\= & {} J_F(\varPhi (t))C[Z(t)] + C[Z(t)]J^T_F(\varPhi (t))+W(\varPhi (t)) \end{aligned}$$\end{document}dC[Z(t)]dt=F′(Φ(t),C[Z(t)])=JF(Φ(t))C[Z(t)]+C[Z(t)]JFT(Φ(t))+W(Φ(t))where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${J}_F(\varPhi (t))$$\end{document}JF(Φ(t)) is the Jacobian of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(\varPhi (t))$$\end{document}F(Φ(t)), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^T_F(\varPhi (t))$$\end{document}JFT(Φ(t)) its transpose, and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W(\varPhi (t))= \sum _{\tau \in {\mathcal {R}}} \upsilon _{\tau } {\upsilon _{\tau }}^T k_\tau \prod _{S \in \varLambda }\varPhi _{S}^{r_{S,\tau }}(t). \end{aligned}$$\end{document}W(Φ(t))=∑τ∈RυτυτTkτ∏S∈ΛΦSrS,τ(t).Expected value and covariance matrix of Y(t) are completely characterized by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi (t)$$\end{document}Φ(t) and C[Z(t)] since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Y(t)]=N\varPhi (t)$$\end{document}E[Y(t)]=NΦ(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[Y(t)]=\sqrt{N} C[Z(t)] \sqrt{N}=N C[Z(t)]$$\end{document}C[Y(t)]=NC[Z(t)]N=NC[Z(t)].
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The LNA requires solving a number of ODEs quadratic in the number of species (Cardelli et al. 2015) and is therefore a scalable alternative to the solution of the CME. In contrast to the deterministic semantics, which considers average concentrations, the LNA does not compromise stochasticity.
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Tool support A number of software tools exist for examining the behaviour of CRNs. We employ Microsoft’s Visual GEC, which provides a programming language, LBS, for designing and simulating a given CRN under the deterministic or stochastic semantics, including also the LNA approximation of the stochastic semantics. The tool is capable of producing plots of expected or average species concentrations over time. This functionality is used extensively within this paper to validate our circuit designs. In addition, Visual GEC exports models to the probabilistic model checker PRISM (Kwiatkowska et al. 2011), which then enables verification of the induced continuous-time Markov chain against temporal logic properties. We use PRISM to verify the correctness of the temporal ordering of events occurring as the CRN circuit executes.
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(CRN Validation Under Different Semantics) We show the operation of our dual-rail CRN ‘motif’ given in Example 2 under the deterministic and stochastic semantics. The input configuration is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_{hi}| = 10$$\end{document}|Xhi|=10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Y_{lo}| = 10$$\end{document}|Ylo|=10 molecules. Figure 4 demonstrates that, after 0.4 s, the CRN stabilises reaching the concentrations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Y_{hi}| = 10$$\end{document}|Yhi|=10 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Y_{lo}| = 0$$\end{document}|Ylo|=0 as desired. With regards to simulations provided, concentrations (given in nanomolars) are directly correlated to concentrations as we assume the volumetric factor N is fixed.Fig. 4Simulation of the motif given in Example 2 under different semantics. The input is initially \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and the output is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo, both starting at a concentration of 10 molecules. In a, b, c we respectively show the deterministic solution, stochastic simulation and LNA plot with variance, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo is seen in blue and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi is seen in yellow. We can observe that in all cases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi is present after 0.4 s. In d we show the state transitions of the induced CTMC that correspond to the output switching from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi. The computation reaches the correct output state and stabilises, with no more transitions enabled. (Color figure online)
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Simulation of the motif given in Example 2 under different semantics. The input is initially \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and the output is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo, both starting at a concentration of 10 molecules. In a, b, c we respectively show the deterministic solution, stochastic simulation and LNA plot with variance, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo is seen in blue and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi is seen in yellow. We can observe that in all cases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi is present after 0.4 s. In d we show the state transitions of the induced CTMC that correspond to the output switching from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{lo}$$\end{document}Ylo to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi. The computation reaches the correct output state and stabilises, with no more transitions enabled. (Color figure online)
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Asynchronous computation is a model of computation that relies on transitions via local input signals rather than transitions via a global clock. Asynchronous computation (Spars and Furber 2002), just like its synchronous counterpart, is Turing complete (Manohar and Martin 1996), meaning that any bounded-tape Turing machine can be implemented with an asynchronous circuit, providing that the implementation of that circuit has isochronous forks. An isochronous fork is the propagation of a signal from a single source to multiple receivers, with the important constraint that the signal must reach the receivers at precisely the same time. In classical digital circuitry this corresponds to the propagation of a signal down wires of exactly the same length from one component to another. In CRNs this could be seen as two species reaching a threshold N at precisely the same time.
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Asynchronous circuits, which are designed based upon the theoretical principles of asynchronous computation, are widely used for low-power microprocessor designs, e.g., by ARM, and are increasing in popularity with the increase in distributed computing (Myers 2004). Asynchronous designs offer a number of advantages, the main one being correctness independent of timing, although they require a greater overhead in terms of silicon area.
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Muller C-element The cornerstone of asynchronous computation is the Muller C-element. A Muller C-element has two Boolean inputs, X and Y, and one output Z. By definition these inputs can either be low or high (represented by 0 or 1). When both inputs are low the output is low. Similarly, when both inputs are high the output is high. The variation from a classical logic gate, however, is that if the inputs are high, or low, and one of them changes, it ‘remembers’ the last state. In other words, it retains the last 0 or 1 state. This is summarised in Fig. 5a. An important property of the C-element is that it allows an observer to conclude on seeing output change from 0 to 1 that both inputs are now 1, and similarly for input change from 1 to 0.
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The table specification indicates that asynchronous circuits exhibit concurrency and causality, and hence their specifications need to reflect these characteristics. A common way is as a timing diagram, seen in Fig. 5d for the C-element, which represents a set of signals and their interactions over time. Each row of a timing diagram represents one signal and how it switches from low to high over time. If a signal displays a change before another signal on another line then this signal must precede the other. An arrow represents that one signal change triggers the change of another. In the C-element diagram, note that X and Y have to precede Z, both in the transition to 1 and down to zero. However, there is no causal dependency between X and Y.
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Asynchronous diagrams, and in particular the C-element, are accurately described using Petri nets (Spars and Furber 2002, p. 86) or process algebras (Wang and Kwiatkowska 2007). We present a (1-bounded) Petri net for the C-element in Fig. 5c, in which transitions are interpreted as signal transitions and places and arcs capture the causal relations between the signal transitions. Following the usual convention, the Petri net is drawn in simpler form where most places have been omitted. We can observe that both tokens are needed in order to excite the transitions that cause the event \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi, which in turn will require the events \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi to be triggered. The same is true for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo.
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When considering circuit synthesis, we typically employ a state graph specification, which can be obtained from the Petri net representation (Myers 2004) and is given for the C-element in Fig. 5b. The values in each state correspond to the values of inputs X, Y and output Z, respectively. A * symbol indicates that the corresponding variable is excited by the outgoing transition (and will be changed in the following state). Observe how we can only transition to a state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\text {*} 1\text {*} 1$$\end{document}1*1*1 from a state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$110\text {*}$$\end{document}110* requiring X as 1 and Y as 1. Because this is derived from a 1-bounded Petri net, we can assume that the transitions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\text {*}10 \rightarrow 110\text {*}$$\end{document}0*10→110* and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10\text {*}0 \rightarrow 110\text {*}$$\end{document}10*0→110* do not conflict.Fig. 5Four specifications of a C-element with inputs X, Y and an output Z. a Conventional logic table, where ‘unchanged’ means that the state of the output is the last stable configuration of 1 or 0. b State graph, in which each state denotes a configuration and a transition is caused by the presence of a signal, where * indicates that the signal is excited. c 1-bounded Petri net specifying the Muller C-element. d Timing diagram for the C-element
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Four specifications of a C-element with inputs X, Y and an output Z. a Conventional logic table, where ‘unchanged’ means that the state of the output is the last stable configuration of 1 or 0. b State graph, in which each state denotes a configuration and a transition is caused by the presence of a signal, where * indicates that the signal is excited. c 1-bounded Petri net specifying the Muller C-element. d Timing diagram for the C-element
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Muller C-pipeline In order to replace the need for a global clock, asynchronous computation relies on ‘local cooperation’ in the form of handshaking protocols. These protocols exchange completion signals in order to establish when a computation has terminated. These handshaking protocols rely heavily on the C-element described above.
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The Muller pipeline, shown in Fig. 6, is constructed by the composition of Muller C-elements (depicted by the gate symbol labelled with C) and classical NOT-gates, which receive and send data to/from the environment (Left, Right in the figure). Its function is to propagate a high and low signal along the pipeline, emulating the ‘wave’ of high and low signals of a classical synchronous clock. Initially, all C-elements are set to a value of 0. The ith C-element C[i] will propagate a 1 from its predecessor, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[i-1]$$\end{document}C[i-1], only if its successor, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[i+1]$$\end{document}C[i+1], is 0. Similarly, it will propagate a 0 from its predecessor only if its successor is 1. Eventually the first request initialised on the left hand side of our pipeline is propagated to the final request on the right.
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The protocol enacted upon this pipeline uses request and acknowledge rails that can be set to high or low. The Muller pipeline implements a basic four phase protocol, which is as follows. Firstly, the sender sends data and sets request to high, viewed in Fig. 6 as the signal Req being high. The receiver then records this data and sets acknowledge to high (Ack). Then the sender responds by setting request to low (Req), and finally the receiver acknowledges this by setting acknowledgement to low (Ack). If at any point a handshake along the pipeline is slower than another, the pipeline will behave like a FIFO queue with data preserved. Herein lies the important purpose of the pipeline: it allows for the delay-insensitive transfer of information from one place to another. In combination with a latch we can create the propagation of information across latches using the pipeline as a control structure.Fig. 6Muller pipeline. Signals are propagated from left to right using request and acknowledge signals. The pipeline effectively queues data, only allowing a transition to occur when a further signal has been acknowledged
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Timing properties of asynchronous circuits Asynchronous circuits can be classified as being self-timed, speed independent or delay-insensitive, depending upon delay assumptions that are made. Assume a circuit is composed of gates and wires. A self-timed circuit operates correctly if both gates and wires experience measurable and fixed delays. A speed independent circuit operates correctly if gates exhibit some unknown time delay within gates but exhibits no time delay on wires. A delay-insensitive circuit operates correctly if there is both unknown delay within gates but also unknown delay within wires. The set of delay insensitive circuits is small, essentially those built from the Muller C-element and NOT gates, and so a broader class of quasi-delay insensitive circuits are identified. Quasi-delay insensitive circuits, which can be composed of purely C-elements, NOT-gates and forks, are Turing complete Manohar and Martin (1996). They are not possible to achieve without an isochronous fork.
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In this section we present our dual-rail designs for an asynchronous computing device in CRNs. The key component is the Muller C-element, whose design is inspired by the well known Approximate Majority (AM) CRN (Angluin et al. 2008). We begin by providing a detailed justification for our C-element design, and then describe the remaining simple components, including latches, logic gates and control flow. Finally, we present complex circuits such as the pipeline, queue and adder.
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To justify the designs, we demonstrate that each component we design exhibits correct behaviour. Considering as an example the C-element, this amounts to working with an informal specification of the C-element in terms of high/low signals as in Fig. 5, and then showing that our (continuous) CRN empirically satisfies that specification according to appropriate thresholding for high/low signals, as normally done in electronics for transistor logic. To this end, we explore the time evolution of the components under the deterministic and stochastic semantics, including also LNA for scalability. We additionally employ probabilistic model checking with PRISM, where temporal logic is used to express the temporal ordering of events. We remark that, although we validate the components for all possible input configurations, this does not amount to full verification of the correctness of the designs. We discuss the challenges of achieving full verification in Sect. 6.
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| 88.94 |
The C-element design is based on the AM CRN, which computes the majority of two finite populations by converting the minority population into the majority population, so that a single population emerges as output. It uses a third ‘undecided’ state of the population, from where catalysis can drive the individuals into either of the final states. Interestingly, since approximate majority cannot be exactly computed by a bi-molecular CRN with less than 4 reactions (Mertzios et al. 2014), below we present the bi-molecular AM CRN with exactly four reactions:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X + Y \rightarrow X + \lambda \\ Y + X \rightarrow Y + \lambda \\ X + \lambda \rightarrow X + X \\ Y + \lambda \rightarrow Y + Y \end{aligned}$$\end{document}X+Y→X+λY+X→Y+λX+λ→X+XY+λ→Y+Ywhere X, Y are both the input and output species and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ is the aforementioned catalytic driver. The intuition behind this reaction network is that we have two competing initial populations of X and Y, both of which try to eliminate the other by transforming their counterpart into the intermediary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ then interacts with X, it transforms itself into X, else into Y. Presented below in Fig. 7a is the same AM CRN in our diagrammatic notation.Fig. 7Two diagrammatic CRNs which are capable of computing Approximate Majority Angluin et al. (2008). In a we present the original in which the inputs X, Y, dependent on which species has the majority, influence outputs X, Y. b Shows a similar AM circuit, but now the input species are catalysed to arbitrary output species W, Z
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Two diagrammatic CRNs which are capable of computing Approximate Majority Angluin et al. (2008). In a we present the original in which the inputs X, Y, dependent on which species has the majority, influence outputs X, Y. b Shows a similar AM circuit, but now the input species are catalysed to arbitrary output species W, Z
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Deconstructed, if we consider the left-hand side of the diagram, the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \rightarrow \lambda$$\end{document}X→λ is catalysed by the species Y and so yields the bi-molecular reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y + X \rightarrow Y + \lambda$$\end{document}Y+X→Y+λ. Similarly, X catalyses the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow X$$\end{document}λ→X in the other direction and yields the reaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X + \lambda \rightarrow X + X$$\end{document}X+λ→X+X. On the right-hand side of the diagram, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ is again catalysed by X and Y to produce Y, yielding the other two reactions from the AM CRN. The CRN in Fig. 7b, with inputs X and Y and outputs W, Z, is similar to the AM CRN, except that we produce new arbitrary outputs W, Z instead of producing greater quantities of X and Y.
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Since we wish to apply the dual-rail methodology, we represent each signal as a pair of species and encode the value 1 (0) as a molecular population of at least M for some population threshold M (population 0). We thus present in Fig. 8a a dual-rail CRN which computes approximate majority with four inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, X_{lo}, Y_{lo}, Y_{hi}$$\end{document}Xhi,Xlo,Ylo,Yhi and outputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}, Z_{lo}$$\end{document}Zhi,Zlo. Like before, our input catalyses an intermediary species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ, but this is split over two reactions. In addition, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi catalyse with reactions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo} \leftrightharpoons \lambda$$\end{document}Zlo⇋λ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi} \leftrightharpoons \lambda$$\end{document}Zhi⇋λ. This has the effect that, if there are larger numbers of either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi, the network enlarges its majority by converting the other into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ. Two rounded arrows over a reaction (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uparrow$$\end{document}↑) indicates that either species can act as a catalyst to that reaction. We demonstrate this AM effect in Fig. 8b, where, given the initial configuration of inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo of 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi, we observe under deterministic semantics that, because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_{lo}|, |Y_{lo}| > |X_{hi}|,|Y_{hi}|$$\end{document}|Xlo|,|Ylo|>|Xhi|,|Yhi|, an output of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Z_{lo}| = 10$$\end{document}|Zlo|=10 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Z_{hi}| = 0$$\end{document}|Zhi|=0 is produced after 0.5 s.Fig. 8A dual-rail Approximate Majority CRN. a Circuit diagram. b Deterministic simulation of the CRN in a. Given inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi, we observe that, because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_{lo}|, |Y_{lo}| > |X_{hi}|,|Y_{hi}|$$\end{document}|Xlo|,|Ylo|>|Xhi|,|Yhi|, an output of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Z_{lo}| = 10$$\end{document}|Zlo|=10 molecules (seen in blue) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Z_{hi}| = 0$$\end{document}|Zhi|=0 (seen in red) is expected to be produced after 0.5 s. (Color figure online)
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A dual-rail Approximate Majority CRN. a Circuit diagram. b Deterministic simulation of the CRN in a. Given inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi, we observe that, because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_{lo}|, |Y_{lo}| > |X_{hi}|,|Y_{hi}|$$\end{document}|Xlo|,|Ylo|>|Xhi|,|Yhi|, an output of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Z_{lo}| = 10$$\end{document}|Zlo|=10 molecules (seen in blue) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Z_{hi}| = 0$$\end{document}|Zhi|=0 (seen in red) is expected to be produced after 0.5 s. (Color figure online)
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We now need to justify the correctness of the design against the C-element specification in Fig. 8a. Given a starting configuration of input X, Y both 1, and any starting output, the C-element should eventually output 1. Similarly, if X, Y are both 0, on any initial output our final output should be 0. For any other configuration of X, Y, the output signal should remain the same. Thus, given an input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},Y_{hi}$$\end{document}Xhi,Yhi, and crucially any starting output configuration Z, we wish to reach a state where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi has at least M molecules where M is a population threshold, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo has 0 molecules. Similarly, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo and any Z we should to see a presence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo after some time t. For all other configurations of the inputs we wish the output species to remain the same.
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When validated against this informal specification with a starting configuration of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi at 10 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules, our CRN unfortunately fails, seen in Fig. 9. More specifically, we observe that species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi is at a concentration of 6 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ has a concentration of 4 molecules after 0.3 s. This simulation was produced using LNA, see Sect. 3.1, which outputs standard deviation of the mean concentrations of species, seen in the shaded regions. This output configuration is incorrect since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi is below the required threshold, see Sect. 3.1, of 8 molecules to represent an output of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z = 1$$\end{document}Z=1.Fig. 9LNA plot for the candidate CRN for the dual-rail Muller C-element design in Fig. 8. With a starting configuration of inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi at a concentration of 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at a concentration of 10 molecules, we can observe that after 0.3 s \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi (seen in red) is at a concentration of 6 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ is at a concentration of 4 molecules (seen in green). The shaded regions represent standard deviation. As we can see, with a non-zero probability we cannot distinguish the signals. (Color figure online)
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LNA plot for the candidate CRN for the dual-rail Muller C-element design in Fig. 8. With a starting configuration of inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi at a concentration of 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at a concentration of 10 molecules, we can observe that after 0.3 s \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi (seen in red) is at a concentration of 6 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ is at a concentration of 4 molecules (seen in green). The shaded regions represent standard deviation. As we can see, with a non-zero probability we cannot distinguish the signals. (Color figure online)
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We present an amended CRN that resolves this issue in Fig. 11a. This CRN is composed of two approximate majority circuits connected to each other, with the outcome of the first AM amplifying the outcome of the second. As we can see from the plot in Fig. 9, we need to amplify the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi species and suppress the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ species. The second AM corrects exactly this issue. Figure 11b is a simulation with inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi at 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules. Here we can see that all three species are now at 10 molecules throughout the duration of the simulation.
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To strengthen the validation of the final C-element design, we provide two further plots for selected initial configurations. We include in Fig. 11c a deterministic plot with starting configuration of input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules. After 1 s the system converges to output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo at 10 molecules. Figure 11d shows an LNA simulation with starting configuration of input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{hi}$$\end{document}Xhi,Yhi at 10 molecules and output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo at 10 molecules. After 1 s we reach output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi with probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx$$\end{document}≈ 1. Both these simulations show that our output changes if both inputs change. From Fig. 11b we observe that the output does not change if both inputs are different.
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An issue to address is the use of dual-rail systems in which our chemical output is precisely a value of 0 molecules or K molecules, where K is the largest number of molecules achievable by a population in the system. In reality, a species may not reach its maximum population and, due to variance, we may have a situation, as seen in Fig. 9, where one species has moderate probability of being higher than the output species we wish to present. Fortunately, we can use our approximate majority circuit to boost species. Figure 10 shows an example where species \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi are weak, but the output is boosted by the C-element back to the maximum output of 10 molecules. The gates are also reusable: specifically, in Fig. 11d we can observe the inherent reusability of the C-element because the output reacts to the change in input.Fig. 10LNA simulation demonstrating two weak input signals, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}$$\end{document}Xhi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{hi}$$\end{document}Yhi at 6 and 4 molecules respectively, boosted by the C-element with an output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules Fig. 11 Simulation of the final dual-rail Muller C-element design on selected input configurations. a Dual-rail AM circuit which is our final C-element design. b Deterministic simulation with inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{hi}$$\end{document}Xlo,Yhi at 10 molecules and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules, which does not exhibit any change over time. c Deterministic simulation resulting in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo (seen in light blue) based on the initial configuration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi at 10 molecules. d LNA simulation resulting in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi (seen in yellow) based on the initial configuration of input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi}, Y_{hi}$$\end{document}Xhi,Yhi at 10 molecules and initial output \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo at 10 molecules. Note that some plots are overlayed but are either set to 0 or 10 molecules. (Color figure online)
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