dataesr/scientific-paragraphs-categorization huggingface dataset:
For instance, for the composition structures {2, 2} and {2, 3} that are the most restrictive composition structures for k = 4 and k = 5 inputs, the corresponding enrichment factors are 40. 37 and 45760. 08. To check the level of significance of this effect, we applied a standard statistical test (see "Methods"). In Supplementary Table  S8 , we list the enrichment factors for all non-trivial composition structures having k ≤ 5 inputs and we give the corresponding one-sided p values. These p values show that the enrichment effects are indeed statistically significant, providing evidence in biological systems of a selection pressure in favor of each of the non-trivial composition structures. Figure  4b  is a bar plot of the fractions in the reference biological dataset of the four biologically meaningful sub-types when focusing on the BFs satisfying a given composition structure.
In addition, the black dots give the corresponding fractions when using the random ensemble instead of the reference biological dataset. We call "relative enrichment" E R the ratio of these fractions that focuses on both a given composition structure and a given biologically meaningful sub-type of BF. The E R s are larger than 1 for all non-trivial composition structures with number of inputs k ≤ 5 , suggesting that the four biologically meaningful sub-types of composed BFs are enriched within any composition structure in the reference biological dataset. Table  3  gives the E R values for the four biologically meaningful sub-types in all non-trivial composition structures with number of inputs k ≤ 5. Furthermore, the computed relative enrichment values are statistically significant as determined by one-sided p values (see Supplementary Table  S9 ).
A previous analysis  38  showed that biologically meaningful BFs are enriched in our reference biological dataset. Notably, those enrichments are likely driven by complexity minimization, with NCFs and RoFs respectively minimizing two complexity measures namely, average sensitivity and Boolean complexity  38. An immediate question that then arises is whether the enrichments of composed BFs as found in Fig. 4  might just be driven by enrichments of NCFs and RoFs. To examine that possibility, let T C denote the set of BFs allowed by a composition structure C at a given number of inputs k, and let T NCF denote the set of NCFs with k inputs. We have determined the enrichment factors of three disjoint sets of BFs: composed BFs that are also NCFs (i. e. , T C ∩ T NCF ), composed BFs that are not NCFs (i.
e. , T C \ T NCF ), and NCFs that are not composed BFs (i. e. , T NCF \ T C ). Table  4  shows the enrichment factors for these three disjoint sets of BFs, for all non-trivial composition structures with k ≤ 5 inputs. We find that the BFs belonging to the set T C ∩ T NCF display a very high enrichment factor. Moreover, for composition structures {2, 2} , {2, 3} and {1, 2, 2} , we find that both the sets T C \ T NCF and T NCF \ T C are enriched in the biological datasets. However, the enrichment factor is much larger for the set T NCF \ T C. Finally, for the composition structures {1, 2} , {1, 3} , {1, 1, 2} , {1, 4} , {1, 1, 3} and {1, 1, 1, 2} that are a superset of the correspond- ing NCFs, we find that the set T C \ T NCF is either depleted or shows a lower enrichment factor compared to the set T C ∩ T NCF.
After repeating the above analysis for RoFs to estimate the enrichment factors for T C ∩ T RoF , T C \ T RoF and T RoF \ T C , we find that the results are similar to those for NCFs (Table  4 ). Furthermore, all these enrichment are statistically significant as determined by one-sided p values (Supplementary Table  S10 ). These results suggest that although composed BFs are subject to positive selection in real biological networks, the primary driving force for enrichment is the property of being an NCF or an RoF. Table  3. Relative enrichment of biologically meaningful BFs among composed BFs of different composition structures in the reference biological dataset. This table gives the relative enrichment values E R in the reference biological dataset for the four biologically meaningful sub-types within composed BFs for different non-trivial composition structures with number of inputs k ≤ 5.
Composition structure T C ∩ T NCF T C \ T NCF T NCF \ T C T C ∩ T RoF T C \ T RoF T RoF \ T C {1,2} 3 Our final set of analyses are a set of 3 statistical tests to identify whether composed BFs are enriched within biological logics and if so, what are the factors that drive their enrichment. First, we find that the composed BFs are indeed enriched in our reference biological dataset in comparison to the space of all BFs. Then by computing the relative enrichment of a biologically meaningful sub-type in non-trivial composition structures (for instance, the relative enrichment of NCFs when considering BFs compatible with the composition structure {2,2}), we find that these sub-types are enriched, though the cause of its enrichment could be attributed either to the property of being biologically meaningful or to the property of belonging to the composition structure.
Taking p 1 (x 1 ) = x 1 , p 2 (x 2 , x 3 ) = x 2 x 3 , and g(x, y) = x + y leads to the composed BF h(x 1 , x 2 , x 3 ) = x 1 + x 2 x 3. However the BFs obtained by permuting the labels of these variables, namely x 2 + x 1 x 3 and x 3 + x 1 x 2 , are just as relevant biologically; indeed, the labels point to genes and these are hardly ever equivalent. Thus, we count all three of the cases above as valid composed BFs. In contrast, Fink and Hannam 47 count them as one composed BF. A code to generate all the composed BFs for any given composition structure after accounting for all the permutations of the input variables is available from the associated GitHub repository (see https:// github. com/ asama llab/ CoSt). Note that this example shows that the two ways of counting are not generally related by the number of permutations (k!) of k labels because of possible symmetries within these expressions.
TF binding regions and active enhancers. We relied on two types of published datasets for estimating the prevalence of composition structures arising from cis-regulatory modules: (i) transcription factor binding regions and (ii) active enhancers. We focused on the two well-studied human cell lines HepG2 and K562 because there is ample published data for them. We obtained the DNA binding regions of the TFs as ChIP-seq narrow-Peak bed files for the two cell lines from the human ENCODE project  78. The active enhancers are obtained from data processed using the STARRPeaker peak-calling software  79. Employing these two datasets, we consider that a TF binds to an active enhancer if and only if both the midpoint and the summit of the ChIP-seq peaks for that TF fall within the active enhancer region.
Notably, there were no cases wherein the summit of the peaks were not provided in the ChIP-seq files obtained from the human ENCODE project. The ChIP-seq narrowPeak bed files for the HepG2 and K562 cell lines were last downloaded on April 28th 2022 and April 29th 2022, respectively, from the human ENCODE project website: https:// www. encod eproj ect. org. The processed datasets from human ENCODE used for this analysis and the associated codes are available at: https:// github. com/ asama llab/ CoSt. This study was carried out in accordance with relevant guidelines and regulations. Reference biological dataset of 2687 Boolean functions from reconstructed models. We utilized the empirical dataset compiled by Subbaroyan et al. 38  consisting of 2687 BFs extracted from 88 published discrete models of biological systems to quantify the abundance of composed BFs in biological networks.
This dataset of BFs, available for download at https:// github. com/ asama llab/ MCBF/ tree/ main/ biolo gical_ datas et, was compiled using information on published models in online repositories Cell Collective 31 (https:// cellc ollec tive. org/), GINSIM  28  (http:// ginsim. org/) or BioModels  29  (http:// www. ebi. ac. uk/ biomo dels/) and by manually retrieving information from published literature. The compiled collection of 88 published models spans a diverse range of biological processes from various kingdoms of life. Though most of the 88 models are Boolean, some models are not Boolean but nevertheless discrete, implying that each node may have more than 2 states in its discrete logic model. For such models, BFs alone were compiled into the dataset, i. e. , nodes which had a binary state and whose inputs were restricted to binary states were included in the 2687 BFs.
Statistical tests. Enrichments and relative enrichments. For a given number of inputs k, let T be the set of composed BFs allowed by the composition structure {t 1 , t 2 ,. , t r } , with t 1 + t 2 +. + t r = k. Let f 0 be the fraction occupied by the set T among the set of all k-input BFs. This fraction f 0 is equal to the probability of obtaining a BF belonging to the set T when drawing at random (uniformly) within all k-input BFs (black dots in Fig. 4a ). Let f 1 denote the fraction of BFs belonging to T among all k-input BFs that are present in the reference biological dataset (bar plots in Fig. 4a ). T's "enrichment factor", E, is equal to the fraction f 1 /f 0. An enrichment factor E > 1 implies that the composition structure {t 1 , t 2 ,. , t r } is enriched in the reference biological dataset, whereas E < 1 implies that the composition structure is depleted.
Consider now a refinement of the previous notion of enrichment to probe the roles of biologically meaningful types. Specifically, let s be one of the four types of biologically meaningful BFs (UFs, CFs, NCFs, or RoFs). Denote by T s the subset of BFs of type s in T. We then define the "relative enrichment" E R of type s in T as E R = f s,1 /f s,0. In that ratio, f s,0 (respectively f s,1 ) is the fraction of BFs in T that belong to T s (respectively the fraction of BFs 2 2 t 1 2 2 t 2. 2 2 tr 2 2 r. in the intersection of T and the reference biological dataset that belong to T s ). A relative enrichment close to 1 means that an enrichment of T s is driven by an enrichment of T (i. e. , by the composition structure) rather than by the biologically meaningful sub-type s. In contrast, a large relative enrichment suggests that sub-type s is driving enrichment of such BFs in the reference biological dataset even after accounting for enrichment of BFs compatible with a given composition structure.
Associated p values. We first describe the procedure that we employed to test the significance of an enrichment factor E for the set T of composed BFs in the composition structure {t 1 , t 2 ,. , t r }. A similar statistical test was presented and implemented by Subbaroyan et al. 38. First, we introduce the null hypothesis, denoted by H 0 , in which all the k-input BFs in the reference biological dataset are drawn from a random ensemble comprised of uniformly distributed k-input BFs. The p value associated with rejecting this null hypothesis H 0 is computed as follows. Recall that f 0 is the probability of choosing a k-input BF belonging to the set T from the random ensem- ble. Define M to be the number of k-input BFs in the reference biological dataset. Then, we draw M BFs from all k-input BFs in the random ensemble, and compute the probability of getting m BFs that also belong to the set T.
This probability is equivalent to getting m successes when tossing a biased coin M times, and is thus given by the binomial distribution M m f m 0 (1 -f 0 ) M-m. The p value is then given by m>=Mf 1 M m f m 0 (1 -f 0 ) M-m. Here, Mf 1 is the number of BFs belonging to the set T among all k-input BFs that are present in the reference biological dataset. We perform a similar statistical test to determine whether a relative enrichment E R for a given T and s is statistically significant. In this case, the null hypothesis H 0 hypothesizes that although there is a selection for T (as evident from a large value of E), the elements that are drawn within T have a uniform probability, that is the individual elements belonging to T s are not more probable than the other elements of T. In practice, we draw a sample of size M under H 0 where as above M is the number of k-input BFs in the reference biological dataset.
If this sample contains M T elements in T as does the reference biological dataset, the distribution of the number of elements in T s is then known. Specifically, the probability to have m elements in T s is M T m f m s,0 (1 -f s,0 ) M T -m where f s,0 is the ratio of sizes of T s and T in the random ensemble. The p value associated with rejecting H 0 is then just the sum of all such probabilities under the condition that m is larger or equal to the number of T s ele- ments in the reference biological dataset.
Acknowledgements We are thankful to  Satyanarayan Rao  for discussions and  Donghoon Lee  for sharing the list of active enhancers from  STARRPeaker. Areejit Samal  acknowledges research support from the  Department of Atomic Energy (DAE) ,  India  and  Max Planck Society, Germany  (through a  Max Planck Partner Group in Mathematical Biology ). IPS2 benefits from the support of  Saclay Plant Sciences-SPS  ( ANR-17-EUR-0007 ).
Data availability The list of human macromolecular complexes was abtained from the EBI Complex Portal database  57  , and the list of human TFs was obtained from http:// human tfs. ccbr. utoro nto. ca/  58. The families of human TFs are provided in the JASPAR database  64. The ChIP-seq peaks for TFs in HepG2 and K562 were obtained obtained the human ENCODE project  78. The active enhancer regions were obtained from data processed using STARRPeaker peakcalling software  79.
Generally that will require many different such complexes, but one cannot exclude that just a few complexes are involved in the control of many genes. From the data on macromolecular complexes in humans obtained from the EBI Complex Portal  57  , we find that for approximately 6. 5% of the complexes (86 out of 1325), all of their monomeric subunits are identified as TFs. (For such an identification, we imposed that they be present in the database of 1617 human TFs from Lambert et al. 58  and come with strong evidence for DNA binding as ascertained by manual curation of the literature. ) Furthermore, we find that 4. 57% of the human TFs belong to the bZIP and bHLH classes that are known to bind to DNA as homodimers or heterodimers. It is likely that the collection of complexes in the EBI Complex Portal are biased towards complexes which do not act as TRs given that the detection and characterization of heteromeric protein complexes which act as TRs is experimentally challenging.
By analyzing ChIP-seq and enhancer datasets in the two human cell lines HepG2 and K562, we find that 32. 68% and 44. 31% of their respective active enhancers bind to at least two TFs. Our result suggests that composition structures with cis-regulatory elements acting as transcriptional regulators are likely to be prevalent in bipartite gene regulatory networks. We remark that experimental limitations do not allow for the detection of all the enhancers for a given target gene in a given cell type, preventing the identification of exact composition structures from empirical data. Fink and Hannam  47  showed that composition structures can severely restrict the number of Boolean logics in the space of all BFs. In the present contribution, we address many questions from the perspective of providing a comprehensive comparison between the BFs of a composition structure and the BFs that are biologically meaningful.
The main questions we address are as follows: (i) How restrictive are composition structures compared to biologically meaningful logic rules? (ii) For a given number of inputs, do the BFs belonging to two different composition structures overlap? (iii) Do BFs in a given composition structure overlap with those having a biologically meaningful logic? (iv) Are composed BFs enriched in an empirical dataset of biological logic rules? First, we provide the corrected values for the number of BFs belonging to a given composition structure by Vol:. (  1234567890  www. nature. com/scientificreports/ independent of x j for j = i. Here, x i is the canalyzing input variable, a is the canalyzing input value, and b is the canalyzed output value.
5 Figure 4. 54 Figure 4. Abundance of composed BFs in reconstructed biological networks. (a) Bar plots give the fraction of BFs in the reference biological dataset that are compatible with each of the composition structures. The black dots indicate the fraction when considering all possible BFs instead of only the ones in the reference biological dataset. Note that since the sets of BFs allowed by different composition structures overlap with each other, the sum of the bar plot values may be larger than 1. (b) For all BFs of the reference biological dataset compatible with a given composition structure, the bars give the fraction of these BFs that belong to each of the four biologically meaningful sub-types: Unate functions (UFs), Canalyzing functions (CFs), Nested canalyzing functions (NCFs), and Read-once functions (RoFs).
We note that it closely resembles the experimental trace in Fig. 2e. Based on a dataset of such simulated traces (n =75), all laboratories inferred FRET efficiencies (Fig. 2c ) and rate constants (Fig. 2d  ) ,w h i c ha g r e ev e r yw e l l :t h eF R E Te f ficiencies deviate by less ??    than 17% from the GT (1% average deviation), and the inferred rate constants deviate with a maximum of 12% from the GT (5% average deviation), with a slight systematic underestimation in most cases, i. e. , the determined rate constants were slower. Pomegranate, FRETboard, and Step finding infer the most accurate rate constants under the tested conditions (Table  3 ). The equilibrium constants K = k 21 =k 12 vary generally less since systematic deviations balance each other in this case (Supplementary Fig.
1a ). In contrast, the reported uncertainty measures vary greatly, independent of the analysis type (0. 4% to 21% relative to the inferred rate constant). For comparison, we estimated the minimal uncertainty given the finite size of the dataset, by quantifying the standard deviation of the rate constants obtained from one million simulated samples (see Methods). This standard deviation is ≥3% of the rate constants for the provided dataset (gray and light gray bars in Fig. 2bshownfor1σ  and 2σ, respectively). Thus, most analysis tools reported reasonable uncertainty estimates, while some tools reported uncertainties that are smaller than this lower limit (Tracy, Correlation, STaSI) or provided no uncertainty measures (Edge finding). FRETboard version 0. 0. 2 reported consistently very large uncertainties, which was solved in their latest software version 0.
g. camera blurring) of the FRET efficiencies can lead to mid-FRET observations that are indistinguishable from those caused by a bona fide biomolecular conformation. While, at the single datapoint level this discretization artefact cannot be prevented, the inference accuracy may intervals. Sample size as in (b). Uncertainty measures (CI, SD) as listed in Table  1. Analysis tools are numbered as in (e). e The inferred cyclic flow in the counterclockwise direction determined by calculating ∆GfromEq. (1)andcomparedwith the GT value (solid vertical line). The uncertainty intervals (dark and light gray) are plotted as in (d). Sample size as in (b). Uncertainty measures (CI, SD) as listed in Table  1. Additional simulations to validate the dataset are shown in Supplementary Fig. 3. ‡ denotes results that were submitted after the GT was known.
Edge finding did not report uncertainties. § denotes that the misassignment of start and end states was corrected after the GT was known. Source data are provided as a Source Data file for panels (c, d,ande). be improved by treating discretization-induced averaging explicitly in the analysis  33, 40  ; or using pulsed illumination to reduce blurring  41, 42. Overall, postFRET and Tracy inferred the most accurate rate constants with average GT deviations of 9% and 14%, respectively. As shown in Fig. 3e , qualitatively, the net flow was correctly resolved (most accurately by postFRET, Tracy,andFRETboard), while quantitatively it was mostly underestimated, which we attribute to the aforementioned systematic misallocation of transitions between states 1 and 3. For this simulated dataset, the theoretical lower limit of the uncertainty (as introduced above for the 2-state system) is smaller because the dataset is larger.
4d , detailed in Methods) to compare models with the correct number of FRET states but differences in the kinetic model, such as the connectivity of states or the number of hidden states (rate constants of all inferred models are reported in the Supplementary Table  1 , and in the Supplementary Datafiles). Out of the seven independently inferred   1. The intrinsic uncertainty of the dataset is shown as dark gray (1σ) and light gray (2σ) intervals. Beyond the six displayed rate constants, these additional rate constants were inferred: for Hidden Markury k 31 = 0. 045 and k 34 = 0. 003, for SMACKS k 13 =0. 0001,k 31 = 0. 0055, k 34 = 0. 0034, for MASH-FRET (prob. ) k 31 = 0. 033. All inferred values of all models are reported in the Supplementary Tables 1 and in the Supplementary Datafiles. ‡ denotes results that were submitted after the GT was known.
Full complexity of a black-box experiment Encouraged by the previous results, we tested all tools vis-à-vis the full experimental complexity to see if they perform similarly as in the simulated case (Fig. 4 ). Three experimental datasets of the same biological system (protein binding to a fluorescently labelled DNA, see Methods), under different experimental conditions and thus different kinetic behaviour, served as a test case. However, the analysts had no prior information on the molecular system causing the dynamics. This means that all the effects discussed so far could potentially be present in these experimental datasets: multiple FRET states, diverse noise sources, fluorophore blinking, directional steady-state flow and kinetic heterogeneity. In addition, the fluorescence intensity variation between single molecules was particularly high in these datasets (see Fig.
5a, d, g ), which complicated the inference of the number of states and rates involved (subsequently referred to as model selection). Under these complex conditions, the inferred number of FRET states (Fig. 5b, e, h ) varied more than in the simulated case (discussed in Fig. 4 ). Most tools found two FRET states (Fig. 5b , e, h, some of them including kinetic heterogeneity), but also three, four, or more different FRET states were reported (Supplementary Fig. 4 ), and the kinetic rate constants varied accordingly. Given the inherent lack of GT information in experimental data, we cannot quantitatively assess the accuracy in this comparison. To balance this fact, we qualitatively compare the inferred results for all three datasets. The 6-7 models with two FRET states (and possibly more hidden states) are compared in (Fig.
5c, f, i ). Other models with three, four, or more FRET states are compared in Supplementary Figs. 4 5 6. (All inferred rate constants are given in Supplementary Tables  2 3 4 and Supplementary Datafiles). Again, we Fig. 5  | Increased experimental complexity. Results inferred from three experimental datasets where naturally no GT exists. a, d, g Experimental traces, offset in time and separated by dashed vertical lines, with donor and acceptor fluorescence intensity (green, red) and FRET efficiency (FRET, black), representative for the datasets used in (b, c), (e, f), (h, i), respectively, with n(traces): 134, 163, 118; and n(datapoints): 36,604, 37,067, 43,512; sampling rate = 33 Hz, time per datapoint = 30 ms. All three datasets were kindly provided by M. Schlierf. b, e, h FRET efficiency histograms and FRET efficiencies inferred by the analysis tools numbered as in (j).