dataesr/scientific-paragraphs-categorization huggingface dataset:
Population Genetic Structure, Genetic Distance, and Isolation-by-distance Results from the PCA (Supplementary Material Figure  S3 ), the DACP (Supplementary Material Figure  S4 ), and from ADMIXTURE (Supplementary Material Figure  S5-S6 ) were congruent and all revealed a single genetic cluster (K=1). When ADMIXTURE was conducted separately on each dataset of yearling owls versus the adult owls, both also revealed a single genetic cluster (K=1). Pairwise F ST between wintering areas averaged 7e -5 ± 3e -4 (Figure  1 ). However, pairwise F ST analysis on adults from W and E, the two most distant wintering areas, revealed a small, but significant genetic differentiation (F ST = 0. 002; p = 0. 017; Figure  1  and Supplementary Table  S2 ), which also held for the adult females between W and E (F ST = 0.
SUPPLEMENTARY MATERIAL Population genetic structure, genetic distance and Isolation-by-distance Unless mentioned, all analyses were in R (3. 6. 2. ; R Core Team 2019). We first computed a Principal Component Analysis (PCA) with the function glPCA using the package Adegenet*2. 1. 1  (Jombart 2008 ) using the four wintering regions (W, C, GL and E), retaining 100 PCA axes according to the eigenvalues plot. To infer the number of clusters of genetically related individuals to best fit our data, we used a Discriminant Analysis of Principal Components (DAPC;  Jombart et al. 2010 ) also implemented in the package Adegenet  (Jombart 2008). We followed the recommendation from  Miller et al. (2020)  to report the methods and results. Without prior information (de novo, unsupervised DAPC), we computed the function find.
cluster() to determine the optimal number of clusters. Precisely, we ran successive K-means clustering with an increase in the number of clusters (K=1-10). We used the smallest Bayesian information criterion (BIC) value associated with K for defining the number of clusters. We explored grouping individuals by sex or age (adult only), but we did not find any pattern. We also used  ADMIXTURE v 1. 3 (Alexander et al. 2009)  to estimate the number of clusters in our dataset. We ran 10 replicates of ADMIXTURE (without any prior) allowing for the number of clusters (K) in the model to vary from 1 to 10. We generated random seeds for each run (beta < 0. 0001). We chose the K value that minimized cross-validation error and hence best fit the data  (Alexander et al. 2009). The output generated from 10 independent runs across each K was summarized and graphically represented using the main pipeline implemented in CLUMPAK with default options  (Kopelman et al.
2015). We performed 10 independent runs of ADMIXTURE with adults and first years separated. Additionally, we computed pairwise F ST among wintering areas (W vs. C vs. GL vs. E) with the gl. fst. pop in the dartR package  (Gruber et al. 2018). We calculated pairwise F ST by gender (male vs. female) and age group (adult vs. first year) among wintering areas. We computed global and within wintering regions F IS using gl. basics. stats from DartR and tested if mean F IS were significantly different from zero using Wilcoxon signed rank tests with continuity correction. We computed a beta estimator genetic distance matrix (i. e. a relatedness estimation matrix; according to  Weir and Goudet 2017)  between each pair of individuals with the function snpgdsIndvBeta in the package SNPRelate  (Zheng et al.
The properties of random Boolean networks as models of gene regulation have been investigated extensively by the statistical physics community. In the past two decades, there has been a dramatic increase in the reconstruction and analysis of Boolean models of biological networks. In such models, neither network topology nor Boolean functions (or logical update rules) should be expected to be random. In this contribution, we focus on biologically meaningful types of Boolean functions, and perform a systematic study of their preponderance in gene regulatory networks. By applying the k[P ] classification based on number of inputs k and bias P of functions, we find that most Boolean functions astonishingly have odd bias in a reference biological dataset of 2687 functions compiled from published models.
Subsequently, we are able to explain this observation along with the enrichment of read-once functions (RoFs) and its subset, nested canalyzing functions (NCFs), in the reference dataset in terms of two complexity measures: Boolean complexity based on string lengths in formal logic which is yet unexplored in the biological context, and the average sensitivity. Minimizing the Boolean complexity naturally sifts out a subset of odd-biased Boolean functions which happen to be the RoFs. Finally, we provide an analytical proof that NCFs minimize not only the Boolean complexity, but also the average sensitivity in their k[P ] set.
Due to possible overlap among these types of BFs, a solitary exploration of any type is insufficient. Hence, we systematically study the different types of biologically meaningful functions and their relationships in both the complete space of 2 2 k BFs and a reference biological dataset of 2687 BFs compiled from 88 published models. Furthermore, we look at these different types of biologically meaningful functions through the lens of k[P ] classification  [28]. This led to multiple theoretical conjectures on relevant properties of these BF types for which we are able to provide analytical proofs. ture. We borrow concepts from the computer science literature to quantify the notion of simplicity (or complexity) of a BF, and thereafter, perform a thorough evaluation of the biologically meaningful types of functions from the perspective of complexity.
Computing the Boolean complexity of a function As shown above, the definition of Boolean complexity of a BF necessitates its reduction to a minimal (equivalent) formula. Obtaining a minimal formula for a given BF or expression is a computationally hard problem  [41]. In practice, for more than a handful of variables, one has to resort to heuristic algorithms such as the QMV proposed by Vigo  [42]  for reducing expressions. Thus, barring exceptions, one can only obtain an upper bound on the Boolean complexity for BFs with several inputs. In this work, to obtain the factorized minimal expression of a BF, we employ the logic synthesis software "ABC"  [43, 44]. To improve the estimated Boolean complexity of a BF, we give as input to the ABC software four types of Boolean expressions, namely the full DNF, the full CNF, the Quine-McCluskey minimized DNF expression  [45, 46]  and the Quine-McCluskey minimized CNF expression, corresponding to the same BF.
To retain only the representative RoFs, we iteratively walk through each RoF in a k[P ] set given by the above scheme, generate its list of permutations, and remove any RoFs in the k[P ] set which belong to that list. We separately store the RoF with which we started out, essentially retaining just one of the permutations. S2  for the exact numbers). (b) The in-degree distribution (or number of inputs) for nodes in the reference biological dataset of 2687 BFs from 88 models of living systems. The number of BFs decreases when the in-degree grows. Further, the number of BFs with odd bias (indicated in 'green') are significantly greater than those with even bias (indicated in 'red') in the dataset for k < 10. (c) The plots show that the biologically meaningful types, EF, UF, CF, RoF, NCF and non-NCF RoF, for a given number of inputs k ≤ 8 are over-represented in the reference dataset (except for EFs with k = 4, 5).
VI. ENRICHMENTS IN THE BIOLOGICAL DATA In this section, we report our observations on the relative abundance and associated statistical significance of the different types of BFs in a compiled dataset of BFs from reconstructed models. In Appendix C, we provide details on the compilation and curation of this reference biological dataset of 2687 BFs from 88 published models. The in-degree distribution for these 2687 BFs comprising the dataset analyzed here is shown in Fig. 3(b). As the number of inputs k to a BF increases, the number of nodes or functions with k inputs decreases very rapidly (Fig. 3(b) ). The key methodology hereafter consists in focusing on the relative abundances of the different types of BFs (and the statistical significance thereof) when comparing the ensemble of all BFs to the ensemble composed of our reference dataset.
An enrichment, if it is statistically significant, is suggestive of some level of selection pressure on the BFs present in the biological networks. For details about the statistical tests, see Appendix D. A. Enrichment in types when comparing to the ensemble of random Boolean functions Fig. 3 (b) shows the abundances of the BFs as a function of in-degree in the reference biological dataset of 2687 functions. From Fig. 3 (b), we observe that for each input 1 ≤ k ≤ 8, the odd bias BFs are dominant and statistically enriched in the reference dataset. It is not immediately apparent why BFs with odd bias should be preferred over BFs with even bias as in theory biologically meaningful BFs with even bias also exist (see Fig. 3(a) ). Fig. 3 (c) shows the relative abundances in that dataset for different BF types, namely EFs, UFs, CFs, RoFs, NCFs, and non-NCF RoFs.
Also shown in the figure are the values of these relative abundances in the ensemble of random BFs. We have performed statistical tests to determine whether the relative abundances in the reference biological dataset are larger than expected (one-sided p-values) under the null hypothesis that the reference BFs are drawn from the ensemble of random BFs. We find that most of these types are indeed more abundant than expected (see stars above the bars in Fig. 3(c) ). The main exception is the EF type of function; that feature can be understood by the fact that random functions are typically effective (see Fig. 2 ). For the detailed p-values, refer to Table  S8. From the enrichment ratios shown in Table  1 , it is clear that the RoF, NCF and also the non-NCF RoF types are all strongly enriched.
B. Relative enrichment in sub-types when comparing to random Boolean functions Comparing the enrichments of the different types of biologically meaningful BFs can provide signatures of causes of enrichment. For instance, if selection operated only in favor of unateness, each sub-type therein (NCF, RoF or non-NCF RoF) would be expected to have a relative abundance (proportion within UF) that is the same whether one considers the reference biological dataset or the ensemble of random BFs. In effect, the proportions of different sub-types of BFs in the two ensembles points to which factors drive the different enrichments. We thus developed a way to test the hypothesis that a sub-type enrichment is solely due to the enrichment in one of its englobing types. First, we consider the enrichment ratios of NCFs and RoFs in the three types of BFs: odd bias, EFs and UFs.
From the Table  S9 , it is clear that for k > 2, the relative enrichment ratios (when comparing the observed to the expected under the null hypothesis) of both the NCFs and RoFs are much greater than 1, implying that the enrichment of these sub-types does not follow from the enrichment of their super-sets: biological selection solely in favor of being odd biased, effective or unate is not consistent with the enrichments found of the NCFs or RoFs in the reference dataset, some other factors are also at work. Since NCFs are a subset of CFs, we can also ask whether canalyzation is the factor driving the enrichment of NCFs. Since the relative enrichment ratios are high and the p-values low (see Table  S10 ), we can conclude that selection for canalyzation alone does not explain the enrichment observed for NCFs.
VII. ENRICHED FUNCTIONS IN BIOLOGICAL DATA HAVE MINIMUM COMPLEXITY A plausible explanation for the enrichment of the RoFs and NCFs in the dataset is the low complexity of these functions. We have examined the different types of BFs using two notions of complexity (see section III). In terms of the first notion (Boolean complexity), the RoFs, of which NCFs are a subset, have in fact the minimum Boolean complexity among all EFs (see Property IV. 7). RoFs and NCFs have the same Boolean complexity but interestingly they differ for the second measure of complexity, namely average sensitivity. Therefore the relative enrichment of NCFs and relative depletion of non-NCF RoFs within RoFs in the reference dataset can be possibly explained by using two measures of complexities which have relevance to regulatory logic in biological systems.
VIII. DISCUSSION AND CONCLUSIONS Ever since Kauffman  [10, 11]  proposed the Boolean modeling of gene regulatory networks, there has been considerable interest in this framework for studying the dynamics of biological networks. Subsequently, Kauffman  [2]  and others surmised that the logical rules or BFs determining the dynamics in such models of living systems are likely to have specific properties of biological relevance. Over the years, different types of "biologically meaningful" BFs have been introduced in the literature including effective (EFs)  [29] , unate (UFs)  [30] , canalyzing (CFs)  [2]  and nested canalyzing (NCFs)  [21]. In particular, we introduced for the first time in a biological setting, the read-once functions (RoFs)  [51]  from the computer science literature. In this study, we have systematically investigated the occurrences and relationships among these different types of BFs in: (a) the space of all 2 2 k BFs, and (b) a reference dataset of 2687 BFs compiled from published Boolean networks  [25, 58, 59 ] of biological systems.
Firstly, these biologically meaningful types of BFs represent a tiny fraction of the space of all BFs (see Fig. 2 ), and yet we find that they cover nearly all BFs found in our reference biological dataset (Fig. 3 ). In particular, we find that the odd bias BFs, UFs, CFs, RoFs and NCFs are highly over-represented in the reference biological dataset. Secondly, by studying the corresponding enrichments, we concluded that the enrichment of RoFs and NCFs in the reference dataset cannot be due only to odd bias, effectiveness or unateness. We remark that while the abundance of CFs and NCFs in biological networks has been previously reported in several publications  [2, 21, 22, 31, 50, [53] [54] [55] , we believe this is the first systematic study of 7 different types of BFs in a large curated reference biological dataset.
Further, previous studies neither carried out statistical tests nor assessed the relative enrichments in sub-types, e. g. NCFs within CFs or RoFs, and in this respect, our study is able to shed light on possible driving factors. Subsequently, we systematically examined two measures of complexity namely, Boolean complexity  [28, 38]  and average sensitivity  [26, 34, 39] , and found that their minimization largely accounted for the enrichments of RoFs and NCFs in the reference dataset. Inspired by the work of Feldman  [28] , we classified BFs into their respective k[P ] sets, and thereafter examined abundances as a function of the three quantities: bias P , Boolean complexity and average sensitivity (Fig. 4 ). Based on this investigation, we find that RoFs minimize the Boolean complexity among EFs while NCFs minimize both the Boolean complexity and the average sensitivity for a given odd bias P.
When instead P is even, the minimum corresponds to BF that are not effective, though once the ineffective inputs are removed one recovers again precisely NCFs with fewer variables. As a result, NCFs have the surprising property that they minimize both Boolean complexity and average sensitivity. The framework we used both supports and formalizes Kauffman's  [2]  qualitative view in which "simplicity" should be a driver of the regulatory logic in biological systems. Kauffman argued that canalyzing functions were simpler than random functions, and therefore should be expected to arise quite frequently in biological systems  [2, 53]. Our use of an extensive curated dataset generated from published Boolean models of biological networks enabled us to compare different notions of simplicity, and thereby confront Kauffman's view to real data in a well defined quantitative framework.
In Appendix E, we describe our program to check if a userspecified BF is a RoF. Appendix C: Biological dataset compiling Boolean functions from reconstructed discrete models of living systems In order to assess the abundance of different types of biologically meaningful BFs in reconstructed discrete models of living systems, we first compiled a large dataset of 88 models which have been published to date. These 88 models were either downloaded from databases such as Cell Collective  [25]  (https://cellcollective. org/), GIN-SIM  [58]  (http://ginsim. org/) or BioModels  [59]  (http: //www. ebi. ac. uk/biomodels/), or directly obtained from the corresponding published article. Notably, most of these 88 models were downloaded from the Cell collective database  [25]. Further, this compilation of 88 models spans the overwhelming majority of Boolean models of biological systems reconstructed and published to date.
While compiling the BFs from these 88 models, we have also gathered the information on the signs of interactions between regulators (input nodes) and target gene (output node). Such information is typically obtained from associated experimental literature. Across the 88 models in this compilation, the number of nodes in a model varies between 4 and 128. From these 88 models, we have compiled 2687 BFs pertaining to 2687 nodes that have number of inputs k ≥ 1 (Fig. 3(b) ). The BFs assigned to each node in these 88 models are the result of many authors manually identifying appropriate input-output relations during network reconstruction. In other words, the 2687 BFs in the reference biological dataset were chosen during model reconstruction process such as to capture the known regulatory information.
Enrichments and relative enrichments Consider a given type of BF (say unate with k inputs) which we refer to as T. Denote by f 0 the fraction of functions that are of type T in the random ensemble and by f 1 the corresponding fraction in our reference biological dataset. The enrichment ratio E is simply f 1 /f 0. If E > 1, then the T is enriched while if E < 1, T is depleted. If E = 1, there is neither enrichment nor depletion. In our study we are also interested in relative enrichments to probe for possible causes of enrichments. For that we consider a type T and one of its subtypes, say T s. For instance in our comparison of the two measures of complexity we examined the case where T =RoF and T s =NCF. In direct analogy with what was done for enrichments, we define the relative enrichment E R = (f s,1 /f 1 )/(f s,0 /f 0 ) where the subscript s refers to type T s.
Associated p-values We developed a first statistical test to determine whether an observed enrichment E was statistically significant. The underlying statistical distribution of the random variable E is obtained by formalizing an underlying hypothesis referred to as H 0. Here H 0 corresponds to hypothesizing that the functions in the reference biological dataset are drawn from the random ensemble where all 2 2 k BFs with k inputs are equiprobable. The (rightsided) p-value is then just the probability that such a drawing leads to a value of E as large as the one actually observed. This probability is computed as follows. The fraction f 0 is first determined (see Table  S2 ). Then we consider drawing M BFs from the random ensemble and count the number m of these functions that belong to type T (M is the number of BFs in the reference biological dataset).
The probability of having a given value m is given by the binomial distribution: M m f m 0 (1 -f 0 ) M -m. The desired p-value is then just the sum of all such probabilities under the condition that m is larger or equal to M f 1. This sum is computed numerically. The second type of test we perform concerns the statistical significance of a relative enrichment E R deviating from 1. Again we formalize this by introducing an H 0 hypothesis. Using the notation of the previous subsection, H 0 corresponds to assuming that although there is a selection for T evident from a large value of E), the elements that are drawn within T have a uniform probathat is members of T s are not more probable than the other elements of T. Consider then drawing a sample of size M under H 0. If it leads to M T elements in T as in the reference biological dataset, the distribution of the number of elements in T s is known.
Specifically, the probability to have m elements in T s is M T m f m 0 (1 -f 0 ) M T -m where now f 0 is the ratio of sizes of T s and T in the random ensemble of T s (see Table  S2 ). The desired p-value is then just the sum of all such probabilities under the condition that m is larger or equal to the number of T s elements in the reference biological dataset. Again, this sum is computed numerically. The number of functions belonging to a particular type of BF was obtained from both computation and theory. The number of CFs for k = 6, 7, 8 and NCFs for k = 7, 8 were obtained from  [73]  and  [74]. In certain cases (EFs and UFs having 6, 7 or 8 inputs), it was computationally unfeasible to obtain the exact number of functions in these types and there was no data in the literature as well, hence we used sampling to estimate the probability of a BF to belong to these types, for the specified number of inputs.
If all the representative RoFs are tested without any success, then the input BF is not a RoF. The catalog of RoFs along with the python code to check for RoFs is available via the GiHub repository: https://github. com/asamallab/MCBF. TABLE  S3. Parity distribution of biologically meaningful types of Boolean functions (BFs) for a given number of inputs k. The fraction of even parity functions of a particular type of BF is calculated with respect to the total number of functions of that BF type. Here, EF corresponds to effective functions, UF to unate functions (all sign combinations), CF to canalyzing functions, EUF to effective and unate functions, ECF to effective and canalyzing functions, UCF to unate and canalyzing functions, EUCF to effective, unate and canalyzing functions. Note that entries labeled '-' are those which could not be computed due to inadequate computational resources.
Fractions of functions that are RoFs, non-NCF RoFs or NCFs, in the space of all 2 2 k BFs (f0) or in the reference biological dataset (f1). E(= f1/f0) is the enrichment ratio; it indicates the extent of the over-representation of such functions in the reference dataset. Over-representation is highest for NCFs but clearly non-NCF RoFs are also highly over-represented. Computations are reported for functions with k ≤ 8 inputs.
Number of different types of biologically meaningful Boolean functions (BFs) in the reference biological dataset. Here, k is the number of inputs, 'All' is the total number of BFs for a given number of inputs, EF corresponds to effective functions, UF to unate functions (all sign combinations), CF to canalyzing functions, EUF to effective and unate functions, ECF to effective and canalyzing functions, UCF to unate and canalyzing functions, EUCF to effective, unate and canalyzing functions, NCF to Nested Canalyzing functions and RoF to Read-once functions.
Fraction of different types of biologically meaningful Boolean functions (BFs) in the reference biological dataset. Here, k is the number of inputs, EF corresponds to effective functions, UF to unate functions (all sign combinations), CF to canalyzing functions, EUF to effective and unate functions, ECF to effective and canalyzing functions, UCF to unate and canalyzing functions, EUCF to effective, unate and canalyzing functions, NCF to Nested Canalyzing functions and RoF to Read-once functions.