Are RNA networks scale-free?
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Abstract
A network is scale-free if its connectivity density function is proportional to a power-law distribution. It has been suggested that scale-free networks may provide an explanation for the robustness observed in certain physical and biological phenomena, since the presence of a few highly connected hub nodes and a large number of small-degree nodes may provide alternate paths between any two nodes on average—such robustness has been suggested in studies of metabolic networks, gene interaction networks and protein folding. A theoretical justification for why many networks appear to be scale-free has been provided by Barabási and Albert, who argue that expanding networks, in which new nodes are preferentially attached to highly connected nodes, tend to be scale-free. In this paper, we provide the first efficient algorithm to compute the connectivity density function for the ensemble of all homopolymer secondary structures of a user-specified length—a highly nontrivial result, since the exponential size of such networks precludes their enumeration. Since existent power-law fitting software, such as powerlaw, cannot be used to determine a power-law fit for our exponentially large RNA connectivity data, we also implement efficient code to compute the maximum likelihood estimate for the power-law scaling factor and associated Kolmogorov–Smirnov p value. Hypothesis tests strongly indicate that homopolymer RNA secondary structure networks are not scale-free; moreover, this appears to be the case for real (non-homopolymer) RNA networks.
Keywords
RNA secondary structure Scale-free network Small-world network Dynamic programmingMathematics Subject Classification
62G32 62G07 68W99 05C82 68R051 Introduction
The connectivity (or degree) of a node v in a network (or undirected graph) is the number of nodes (or neighbors) of s, connected to v by an edge. A network is said to be scale-free if its connectivity function N(k), which represents the number of nodes having degree k, satisfies the property that \(N(a\cdot k) = b\cdot N(x)\), the unique solution of which is a power-law distribution, which by definition satisfies \(N(k) \propto {k^{-\alpha }}\) for some scaling factor \(\alpha >1\) (Newman 2006). Scale-free networks contain a few nodes of high degree and a large number of nodes of small degree, hence may provide a reasonable model to explain the robustness^{1} often manifested in biological networks—such robustness or resilience must, of course, be present for life to exist.
Barabasi and Albert (1999) analyzed the emergence of scaling in random networks, and showed that two properties, previously not considered in graph theory, were responsible for the power-law scaling observed in real networks: (1) networks are not static, but grow over time, (2) during network growth, a highly connected node tends to acquire even more connections—the latter concept is known as preferential attachment. In Barabasi and Albert (1999), it was argued that preferential attachment of new nodes implies that the degree N(k) with which a node in the network interacts with k other nodes decays as a power-law, following \(N(k) \propto k^{-\alpha }\), for \(\alpha >1\). This argument provides a plausible explanation for why diverse biological and physical networks appear to be scale-free. Indeed, various publications have suggested that the the following biological networks are scale-free: protein–protein interaction networks (Ito et al. 2000; Schwikowski et al. 2000), metabolic networks (Ma and Zeng 2003), gene interaction networks (Tong et al. 2004), yeast co-expression networks (Van Noort et al. 2004), and protein folding networks (Bowman and Pande 2010).
How scale-free are biological networks?
The validity of a power-law fit for previously studied biological networks was first called into question in Khanin and Wit (2006), where 10 published data sets of biological interaction networks were shown not to be fit by a power-law distribution, despite published claims to the contrary. Estimating an optimal power-law scaling factor by maximum likelihood and using \(\chi ^2\) goodness-of-fit tests, it was shown in Khanin and Wit (2006) that not a single one of the 10 interaction networks had a nonzero probability of being drawn from a power-law distribution; nevertheless, some of the interaction networks could be fit by a truncated power-law distribution. The data analyzed by the authors included data from protein–protein interaction networks (Ito et al. 2000; Schwikowski et al. 2000), gene interaction networks determined by synthetic lethal interactions (Tong et al. 2004), metabolic interaction networks (Ma and Zeng 2003), etc.
In Clauset et al. (2009), 24 real-world data sets were analyzed from a variety of disciplines, each of which had been conjectured to follow a power-law distribution. Estimating an optimal power-law scaling factor by maximum likelihood and using goodness-of-fit tests based on likelihood ratios and on the Kolmogorov–Smirnov statistic for non-normal data, it was shown in Clauset et al. (2009) that some of the conjectured power-law distributions were consistent with claims in the literature, while others were not. For instance, Clauset et al. (2009) found sufficient statistical evidence to reject claims of scale-free behavior for earthquake intensity and metabolic degree networks, while there was insufficient evidence to reject such claims for networks of protein interaction, Internet, and species per genus.
It is possible to come to opposite conclusions, depending on whether \(\chi ^2\) or Kolmogorov–Smirnov (KS) statistics are used to test the hypothesis whether a network is scale-free, i.e. follows a (possibly truncated) power-law distribution. Indeed, Khanin and Wit (2006) obtained a p value of \(<10^{-4}\) for \(\chi ^2\) goodness-of-fit for a truncated power-law distribution for the protein–protein interaction data from Ito et al. (2000), while Clauset et al. (2009) obtained a p value of 0.31 for KS goodness-of-fit for a truncated power-law for the same data.
In this paper, we introduce the first efficient algorithm to compute the exact number of homopolymer RNA secondary structures having k neighboring structures, for each value of k, that can be reached by adding or deleting one base pair. Since there are exponentially many secondary structures, our \(O(n^5)\) time and \(O(n^3)\) space algorithm uses dynamic programming. By applying the Kolmogorov–Smirnov test, we then show that homopolymer RNA secondary structure networks are not scale-free. We also provide evidence that the same is true for real RNA networks. Prior to this paper, only fragmentary results were possible by exhaustively enumerating all secondary structures having free energy within a certain range obove the minimum free energy (Wuchty 2003).
Our work investigates properties of the ensemble of RNA secondary structures, considered as a network, and so extends results of Clote (2015), which described a cubic time dynamic programming algorithm to compute the expected network degree. The RNA connectivity algorithm described in Sect. 2.3 is completely unrelated to that of Clote (2015), and allows one to compute all finite moments, including mean, variance, skew, etc.
The plan of the remaining paper is as follows. Section 2 presents a brief summary of basic definitions, followed by a description of an efficient dynamic programming algorithm to determine the absolute [resp. relative] frequencies N(k) [resp. p(k)] for secondary structure connectivity of a given homopolymer, which allows non-canonical base pairs. Section 3 presents the statistical methods used to both fit RNA connnectivity data to a power-law distribution and to perform a goodness-of-fit test using Kolmogorov–Smirnov distance. Section 4 shows that RNA networks are not scale-free, by performing (computationally efficient) Kolmogorov–Smirnov bootstrapping tests. Section 5 presents concluding remarks, while the Appendix presents data that suggests that RNA networks satisfy a type of preferential attachment. The rigorous proof that RNA networks satisfy modified form of preferential attachment is suppressed for reasons of space, but is available in the preprint (Clote 2018).
2 Computing degree frequency
Section 2.1 presents basic definitions and notation used; Sect. 2.2 presents an algorithm to compute the frequency of each degree less than K in the ensemble of all secondary structures with run time \(O(K^2 n^4)\) and memory requirements \(O(K n^3)\). Section 2.3 presents a more efficient algorithm, with run time \(O(K^2 n^3)\) and memory requirements \(O(K n^2)\), for the special case of a homopolymer, in which all possible non-canonical base pairs are permitted. We implemented both algorithms in Python, cross-checked for identical results, and call the resulting code RNAdensity. Since this paper is a theoretical contribution on network properties, we focus only on homopolymers and do not present the details necessary to extend the algorithm of Sect. 2.2 to non-homopolymer RNA, where base pairs are required to be Watson–Crick or GU wobble pairs—such an algorithm is possible to develop, using ideas of Sect. 2.2; however, since the resulting complexity is formidible, with \(O(n^9)\) time and \(O(n^7)\) space requirements, and since there are no obvious applications, we do not pursue such an extension.
2.1 Preliminaries
A secondary structure for a length nhomopolymer is a set s of base pairs (i, j), such that (1) there exist at least \(\theta \) unpaired bases in every hairpin, where \(\theta \) is usually taken to be 3, though sometimes 1 in the literature, (2) there are no basd triples, so for \((i,j), (k,\ell ) \in s\), if \(\{ i,j \} \cap \{k,\ell \} \ne \emptyset \), then \(i=k\) and \(j=\ell \), (3) there do not exist base pairs \((i,j), (k,\ell ) \in s\), such that \(i<k<j<\ell \); i.e. a secondary structure is a type of outerplanar graph, where each base pair \((i,j) \in s\) satisfies \(j-i>\theta \). The free energy of a homopolymer secondary structure s is defined to be \(-1\) times the number |s| of base pairs in s [Nussinov–Jacobson energy model (Nussinov and Jacobson 1980)]. Since entropic effects are ignored, this is not a real free energy; however it allows us to use the standard notation “MFE” for ‘minimum free energy’. Note that the MFE structure for a length n homopolymer has \(\lfloor \frac{n-\theta }{2} \rfloor \) many base pairs.
For a given RNA sequence, consider the exponentially large network of all its secondary structures, where an undirected edge exists between any two structures s and t, whose base-pair distance equals one—in other words, for which t is obtained from s by either removing or adding one base pair. The connectivity (or degree) of a node, or structure, s is defined to be the number of secondary structures obtained by deleting or adding one base pair to s—this corresponds to the so-called \(MS_1\) move set (Flamm et al. 2000). At the end of the paper, we briefly consider the \(MS_2\) move set, where the degree of a structure s is defined to be the number of secondary structures obtained by adding, deleting or shifting one base pair (Bayegan and Clote 2015). The \(MS_1\) [resp. \(MS_2\)] connectivity of the MFE structure for a homopolymer of length n is \(\lfloor \frac{n-\theta }{2} \rfloor \) [resp. \(\lceil \frac{n-\theta }{2} \rceil \)]. ConnectivityN(k) is defined to be the absolute frequency of degree k, i.e. the number of secondary structures having exactly k neighbors, that can be obtained by either adding or removing a single base pair. The degree densityp(k) is defined to be the probability density function (PDF) or relative frequency of k, i.e. the proportion \(p(k) = \frac{N(k)}{Z}\) of all secondary structures having k neighbors, where Z denotes the total number of secondary structures for a given homopolymer. A network is defined to be scale-free, provided its degree frequency N(k) is proportional to a power-law, i.e. \(N(k) \propto k^{-\alpha }\) where \(\alpha >1\) is the scaling factor.
2.2 Computing the degree density
Suppose that every hairpin loop is required to have at least \(\theta \ge 1\) unpaired positions; i.e. if (i, j) is a base pair, then \(i+\theta +1 \le j\). As described in the Eqs. (5–12) for Base Cases A–D, let Z(i, j, k, h, v) denote the number of secondary structures on the interval [i, j], for \(1 \le i \le j \le n\) for the homopolymer model, that have exactly k neighbors, and for which there are exactly h unpaired positions (or holes) in \([i,j-\theta -1]\), and for which there are exactly \(v \in [0,\theta +1]\) visible positions among \(j-\theta ,j-\theta +1,\ldots ,j\). Concretely, this means that either(i)\(v=\theta +1\) and the rightmost \(\theta +1\) positions in the interval [i, j] are all unpaired, or(ii) that \(0 \le v < \theta +1\), and that position \(j-v\) is paired to some \(r \in [i,j-v-\theta -1]\). In base case D and inductive case D below, we will treat the two possible subcases of (i), in which the rightmost \(\theta +1\) positions are unpaired – namely, the subcase (i)\(_a\) in which \(j-\theta \) is unpaired (hence the rightmost \(j-\theta +1\) positions are unpaired), and the subcase (i)\(_b\) in which position \(j-\theta \) is paired with some \(r \in [i,j-v-\theta -1]\).
In line 1, arrays \(Z,Z^*\) are initialized to zero, then Base Cases A–D are applied; lines 2–9 then treat the Inductive Cases A–D. In this dynamic programming (DP) algorithm, the idea is to define \(Z,Z^*\) for all intervals [i, j] where \(d=j-i\), after having computed and stored the values for \(Z,Z^*\) for all intervals [i, j] where \(j-i=d-1\). All secondary structures of the interval [i, j] can be partitioned into structures having exactly degree k (i.e. k\(MS_1\) neighbors, in which k structures that can be obtained by either adding or removing a single base pair). To support an inductive argument, in proceeding from interval [i, j] to \([i,j+1]\), we need additionally to determine the number of structures having degree k, which have a certain number h of positions that are visible (external to every base pair), which can be paired with the last position \(j+1\). Note that the position \(j-\theta \) can not be base-paired with j in [i, j]; however, \(j-\theta \)can be base-paired with j in \([i,j+1]\). Thus in addition to keeping track of the number h of holes (positions in \(i,\ldots ,j-\theta -1\) that are external to all base pairs, hence can be paired with j), we introduce the variable v to keep track of the number of visible positions in \(j-\theta ,\ldots ,j\). This explains our need for the function Z(i, j, k, h, v) as defined in the Eqs. (5–12) for Base Cases A–D. We now proceed to the details, where for ease of the reader, some definitions are repeated.
Let \(\theta =3\) denote the minimum number of unpaired positions required to be present in a hairpin loop. For a length n homopolymer, let \(1 \le i \le j \le n\), \(0 \le k \le {{n-\theta } \atopwithdelims ()2}\), \(0 \le h \le j-i-\theta \), \(0 \le v \le \theta +1\). Recall that Z(i, j, k, h, v) denotes the number of secondary structures on the interval [i, j], for \(1 \le i \le j \le n\) for the homopolymer model, that have exactly k neighbors, and for which there are exactly h unpaired positions (or holes) in \([i,j-\theta -1]\), and for which there are exactly \(v \in [0,\theta +1]\) visible positions among \(j-\theta ,j-\theta +1,\ldots ,j\). For \(0 \le v \le \theta \), this means that position \(j-v\) is base-paired to some \(r \in [i,j-v-\theta -1]\) while positions \(j-v,j-v+1,\ldots ,j\) are not base-paired to any position in [i, j]. When \(v=\theta +1\), this means simply that the rightmost \(\theta +1\) positions in the interval [i, j] are all unpaired.
Recall as well our definition that \(Z^*(i,j,k) = \sum _h \sum _v Z(i,j,k,h,v)\). We begin by initializing \(Z(i,j,k,h,v)=0\) for all values in corresponding ranges. Letting N(i, j) denote the number of secondary structures on [i, j] for the homopolymer model, as computed by Eq. (2), the following recurrences describe an algorithm that requires \(O(K \cdot n^3)\) storage and \(O(K^2 \cdot n^4)\) time to compute the probability \(Prob[ \mathrm{deg}(s) = k] = \frac{Z^*(1,n,k)}{N(1,n)}\) that a (uniformly chosen) random secondary structure has degree k for \(0 \le k \le K\), where K is a user-defined constant bounded above by \(\text{ maxDegree }(n) = \frac{(n-\theta )(n-\theta -1)}{2}\).
Base Case A considers all structures on [i, j], as depicted in Fig. 1, that are too small to have any base pairs, hence which have degree zero.
Base Case B considers all structures on [i, j], as depicted in Fig. 2, that have only base pair (i, j), since other potential base pairs would contain fewer than \(\theta \) unpaired bases. The degree of such structures is 1, since only one base pair can be removed, and no base pairs can be added. Moreover, no position in [i, j] is external to the base pair (i, j), so visibility parameters \(h=0,v=0\). The arrow in Fig. 2 indicates that the sole neighbor is the empty structure, obtained by removing the base pair (i, j).
Base Case C considers the converse situation, consisting of the empty structure on [i, j] where \(j-i = \theta +1\), whose sole neighbor is the structure consisting of base pair (i, j). The arrow is meant to indicate that the structure on the right is the only neighbor of that on the left, as depicted in Fig. 3. Since the size of the empty structure on [i, j] is \(\theta +2\) and every position in [i, j] is visible (external to every base pair), \(h=1\) and \(v=\theta +1\). the dotted rectangle in Fig. 3 indicates the \(\theta +1\) unpaired positions at the right extremity as counted by \(v=\theta +1\).
Base Case D considers the empty structure on [i, j] where \(j-i>\theta +1\). The empty structure is the only structure having degree maxDegree\((i,j) = \frac{(j-i-\theta +1)(j-i-\theta )}{2}\), since maxDegree(i, j) many base pairs can be added to the empty structure. In Fig. 4, the dotted rectangle indicates the \(\theta +1\) rightmost unpaired positions, corresponding to visibility parameter \(v=\theta +1\), while dotted circles indicate the \(h = j-i-\theta \) holes, i.e. unpaired positions that could be paired with the rightmost position j.
Inductive Case A considers the case where left and right extremities i, j form the base pair (i, j), where \(j-i>\theta +1\). No position in [i, j] is visible (external to all base pairs), so visibility parameters \(h=0=v\). Recalling the definition of \(Z^*(i,j,k)\) from Eq. 4, we have the following.
Inductive Case B considers the case where last position j base-pairs with the r, where \(i<r<j-\theta \). The value \(r=i\) has already been considered in Inductive Case A, and values \(r=j-\theta +1,\ldots ,j-1\) cannot base-pair to j, since the corresponding hairpin loop would constain less than \(\theta \) unpaired positions. This situation is depicted in Fig. 6, where there are h holes (positions in \([i,j-\theta -1]\) that are external to all base pairs) and no visible positions in \([j-\theta ,j]\).
For each value \(v \in \{1,\ldots ,\theta +1\}\), inductive Case C(v) considers the case where position \(r \in [i,j-v-\theta -1]\) forms a base pair with position \(j-v\). The value \(v=0\) is not considered here, since it was already considered in Inductive Cases A,B. Note that a structure s of the format has k neighbors, provided the restriction of s to \([i,r-1]\) has \(k_1\) neighbors, and the restriction of s to \([r+1,j-1]\) has \(k_2\) neighbors, where \(k_1+k_2+vh+1=k\). The term vh is due to the fact that since base pair \((r,j-v)\) ensures that all holes are located in \([i,r-1]\), hence located at more than \(\theta +1\) distance from all visible positions in \([j-v+1,j]\), a neighbor of s can be obtained by adding a base pair from any hole to any visible suffix position—there are vh many such possible base pairs that can be added. Finally, the last term \(+1\) is present, since one neighbor of s can obtained by removing base pair \((r,j-v)\). This explains the summation indices and summation terms in Eq. (11). Figure 7 depicts a typical structure considered in case C(v).
Case D considers the case where there are h holes, and positions \(j-\theta -1,\ldots ,j\) are unpaired, so that \(v=\theta +1\). Note that \(v=\theta +1\) implies only that \(j-\theta ,\ldots ,j\) are unpaired, so Case D includes the addition requirement that position \(j-\theta -1\) is unpaired. Structures s satisfying Case D can be partitioned into subcases where the restriction of s to \([i,j-\theta -1]\) has \(h-w\) holes in \([i,(j-\theta -1)-(\theta +1)] = [i,j-2\theta -2]\), and \(1 \le w \le \theta +1\) visible positions in \([j-2\theta -1,j-\theta -1]\). Note that \((h-w)+w=h\), accounting for the h holes in structure s in \([i,j-\theta -1]\), and that it is essential that \(w\ge 1\), since the case \(w=0\) was considered in Case \(C(\theta +1)\).
The term \(\frac{w(w+1)}{2}\) is due to the fact that the rightmost position \(j-\theta -1\) in the restriction of s to \([i,j-\theta -1]\) can base-pair with position j, but not with \(j-1\), etc. since this would violate the requirement of at least \(\theta \) unpaired bases in a hairpin loop. Similarly, the second rightmost position \(j-\theta -2\) in the restriction of s to \([i,j-\theta -1]\) can base-pair with positions j and \(j-1\), but not with \(j-2\), etc.; as well, the third rightmost position \(j-\theta -3\) can base-pair with positions j, \(j-1\) and \(j-2\), but not with \(j-3\), etc. The number of neighbors of s produced in this fashion is thus \(\sum _{i=1}^w i = \frac{w(w+1)}{2}\). Finally, the term \((\theta +1)(h-w)\) is due to the fact that each of the \(h-w\) holes in the restriction of s to \([i,j-\theta -1]\) can base-pair to each of the \((\theta +1)\) positions in \([j-\theta ,j]\).
The argument just given shows the following. Let s be a structure that satisfies conditions of Case D with h holes and \(v=\theta +1\) visible positions, and suppose that the restriction of s to \([i,j-\theta -1]\) has \(h-w\) holes and w visible positions. Then s has k neighbors provided that the restriction of s to \([i,j-\theta -1]\) has \(k-\frac{w(w+1)}{2} - (\theta +1)(h-w)\) neighbors on interval \([i,j-\theta -1]\). The Eq. (12) now follows.
As in Case C(v), when implemented, this requires a test that \(h-w\ge 0\) (see Fig. 8).
Our implementation of Eqs. (5–12) has been cross-checked with exhaustive enumeration; moreover, we always have that \(\sum _{k} Z^*(i,j,k) = N(i,j)\), so the degree density is correctly computed.
2.3 Faster algorithm in the homopolymer case
3 Statistical methods
Current software for probability distribution fitting of connectivity data, such as Matlab™, Mathematica™, R and powerlaw (Alstott et al. 2014), appear to require an input file containing the connectivity of each node in the network. In the case of RNA secondary structures, this is only possible for very small sequence length. To analyze connectivity data computed by the algorithm of Sect. 2.3, we had to implement code to compute the maximum likelihood estimation for scaling factor \(\alpha \) in a power-law fit, the optimal degree \(k_{min}\) beyond which connectivity data is fit by a power-law, and the associated p value for Kolmogorov–Smirnov goodness-of-fit, as described in Clauset et al. (2009). We call the resulting code RNApowerlaw. This section explains those details.
Table comparing goodness-of-fit computations for software powerlaw (Alstott et al. 2014) and RNApowerlaw for homopolymer lengths less than 30 nt
n | \(S_n\) | \(k_{min}\) | \(\alpha \) (PL) | \(\alpha \) (RNAPL) | KSdist (PL) | KSdist (RNAPL) | \(\langle \text{ KSdist } \rangle \) | log odds ratio R (PL) | p-val for R (PL) | p-val (RNAPL) |
---|---|---|---|---|---|---|---|---|---|---|
10 | 65 | 3 | 3.13752 | 3.13753 | 0.05576 | 0.05576 | 0.02721 | \(-\)0.15 | 0.765 | 0.813 |
12 | 274 | 4 | 3.23011 | 3.23011 | 0.03650 | 0.03650 | 0.01277 | \(-\)0.81 | 0.482 | 0.746 |
14 | 1184 | 5 | 3.38933 | 3.38935 | 0.02021 | 0.02021 | 0.00669 | \(-\)1.70 | 0.270 | 0.699 |
16 | 5223 | 6 | 3.51285 | 3.51289 | 0.02252 | 0.02253 | 0.00603 | \(-\)6.78 | 0.029 | 0.051 |
18 | 23,434 | 9 | 3.79069 | 3.79073 | 0.02333 | 0.02333 | 0.00624 | \(-\)16.00 | 0.001 | 0.001 |
20 | 106,633 | 10 | 3.87168 | 3.87165 | 0.02116 | 0.02116 | 0.00581 | \(-\)82.12 | 0.000 | 0.000 |
22 | 490,999 | 10 | 3.85806 | 3.85809 | 0.02304 | 0.02304 | 0.00523 | \(-\)670.64 | 0.000 | 0.000 |
24 | 2,283,701 | 14 | 4.16480 | 4.16477 | 0.02242 | 0.02242 | 0.00484 | \(-\)1452.24 | 0.000 | 0.000 |
26 | 10,713,941 | 15 | 4.24485 | 4.24486 | 0.02298 | 0.02298 | 0.00417 | \(-\)7129.42 | 0.000 | 0.000 |
28 | 50,642,017 | 16 | 4.33086 | 4.33089 | 0.02167 | 0.02168 | 0.00347 | \(-\)33,020.89 | 0.000 | 0.000 |
30 | 240,944,076 | – | – | 4.33681 | – | 0.02393 | 0.00298 | – | – | 0.000 |
4 Results
Below, we use the algorithms described in previous sections to compute RNA secondary structure connectivity, determine optimal scaling factor \(\alpha \) and minimum degree \(k_{min}\) for a power-law fit, then perform Kolmogorov–Smirnov bootstrapping to determine the goodness-of-fit for parameters \(\alpha ,k_{min}\). In Appendix A, we show that preferential attachment appears to hold for the network of RNA structures, at least for our definition of preferential attachment.
4.1 Analysis of RNA networks using RNAdensity and RNApowerlaw
Table showing that approximate [resp. exact] scaling factor \(\alpha _0\) [resp. \(\alpha \)] and minimum degree \(k_{min}\) for optimal power-law fit of homopolymer connectivity data can not be reliably computed by using software powerlaw (Alstott et al. 2014) on data sampled from relative frequencies
N | \(10^2\) | \(10^3\) | \(10^4\) | \(10^5\) | \(10^6\) | \(10^7\) | RNAPL | \(S_n\) |
---|---|---|---|---|---|---|---|---|
\(\alpha _0\), \(n=20\) | 6.58318 | 3.66505 | 3.93389 | 3.86017 | 3.84749 | 3.84657 | 3.84648 | \(106633\approx 1.1 \cdot 10^5\) |
\(k_{min}\) | 10 | 7 | 10 | 10 | 10 | 10 | 10 | – |
\(\alpha _0\), \(n=30\) | 5.27581 | 4.42183 | 4.46307 | 4.35008 | 4.32651 | 4.32272 | 4.32213 | \(240944076 \approx 2.4 \cdot 10^{8}\) |
\(k_{min}\) | 12 | 13 | 16 | 16 | 16 | 16 | 16 | – |
\(\alpha _0\), \(n=40\) | 5.15978 | 5.09714 | 5.03719 | 5.24488 | 5.16985 | 5.70916 | 5.94561 | \(633180247373\approx 6.3 \cdot 10^{11}\) |
\(k_{min}\) | 15 | 19 | 23 | 29 | 29 | 42 | 49 | – |
\(\alpha \), \(n=20\) | 6.76575 | 3.70988 | 3.96139 | 3.88570 | 3.87271 | 3.87180 | 3.87165 | \(106633\approx 1.1 \cdot 10^5\) |
\(k_{min}\) | 10 | 7 | 10 | 10 | 10 | 10 | 10 | – |
\(\alpha \), \(n=30\) | 5.33162 | 4.44651 | 4.47963 | 4.36511 | 4.34122 | 4.33739 | 4.33681 | \(240944076 \approx 2.4 \cdot 10^{8}\) |
\(k_{min}\) | 12 | 13 | 16 | 16 | 16 | 16 | 16 | – |
\(\alpha \), \(n=40\) | 5.19197 | 5.11604 | 5.049419 | 5.25365 | 5.17824 | 5.65206 | 5.95033 | \(633180247373\approx 6.3 \cdot 10^{11}\) |
\(k_{min}\) | 15 | 19 | 23 | 29 | 29 | 41 | 49 | – |
Table showing maximum likelihood scaling factors \(\alpha \) with associated p values for optimal power-law fits of RNA secondary structure connectivity data for homopolymers of length \(n=30\) to 150
n | \(k_{max}\) | \(\%\) of \(S_n\) | \(k_{{peak}}\) | \(k_{{mfe}}\) | \(k_{min}\) | \(\alpha (k_{min}, k_{max})\) | \(KS (k_{min}, k_{max})\) | \(\langle KS \rangle \) | p val |
---|---|---|---|---|---|---|---|---|---|
30 | 60 | 0.998861 | 10 | 13 | 16 | 4.223674 | 0.014393 | 0.004877 | 0.000000 |
35 | 70 | 0.999174 | 12 | 16 | 18 | 4.395936 | 0.015391 | 0.005284 | 0.000000 |
40 | 80 | 0.999404 | 14 | 18 | 23 | 4.736679 | 0.015298 | 0.004859 | 0.000000 |
45 | 90 | 0.999563 | 16 | 21 | 30 | 5.146670 | 0.012075 | 0.004291 | 0.000000 |
50 | 100 | 0.999681 | 18 | 23 | 32 | 5.310801 | 0.012421 | 0.004345 | 0.000000 |
55 | 110 | 0.999762 | 20 | 26 | 39 | 5.674231 | 0.011649 | 0.003979 | 0.000000 |
60 | 120 | 0.999823 | 22 | 28 | 41 | 5.829310 | 0.012328 | 0.003979 | 0.000000 |
65 | 130 | 0.999866 | 24 | 31 | 49 | 6.200720 | 0.010772 | 0.003572 | 0.000000 |
70 | 140 | 0.999899 | 26 | 33 | 52 | 6.386452 | 0.010836 | 0.003464 | 0.000000 |
75 | 150 | 0.999923 | 28 | 36 | 60 | 6.721588 | 0.009719 | 0.003151 | 0.000000 |
80 | 160 | 0.999941 | 31 | 38 | 63 | 6.897103 | 0.009818 | 0.003067 | 0.000000 |
85 | 170 | 0.999955 | 33 | 41 | 67 | 7.097544 | 0.009737 | 0.002940 | 0.000000 |
90 | 180 | 0.999965 | 35 | 43 | 74 | 7.373569 | 0.008916 | 0.002726 | 0.000000 |
95 | 190 | 0.999973 | 37 | 46 | 78 | 7.564208 | 0.008755 | 0.002615 | 0.000000 |
100 | 200 | 0.999979 | 40 | 48 | 83 | 7.775022 | 0.008444 | 0.002476 | 0.000000 |
105 | 135 | 0.999388 | 42 | 51 | 67 | 7.204937 | 0.010712 | 0.003853 | 0.000000 |
110 | 140 | 0.999432 | 44 | 53 | 70 | 7.360192 | 0.010810 | 0.003854 | 0.000000 |
115 | 145 | 0.999474 | 46 | 56 | 73 | 7.513728 | 0.010889 | 0.003852 | 0.000000 |
120 | 150 | 0.999512 | 49 | 58 | 77 | 7.706717 | 0.010405 | 0.003703 | 0.000000 |
125 | 155 | 0.999549 | 51 | 61 | 80 | 7.856962 | 0.010504 | 0.003696 | 0.000000 |
130 | 160 | 0.999582 | 53 | 63 | 84 | 8.045458 | 0.010102 | 0.003556 | 0.000000 |
135 | 165 | 0.999614 | 55 | 66 | 88 | 8.231267 | 0.009724 | 0.003425 | 0.000000 |
140 | 170 | 0.999643 | 58 | 70 | 91 | 8.377410 | 0.009809 | 0.003418 | 0.000000 |
145 | 175 | 0.999670 | 60 | 71 | 94 | 8.522170 | 0.009884 | 0.003413 | 0.000000 |
150 | 180 | 0.999695 | 62 | 75 | 98 | 8.703041 | 0.009515 | 0.003289 | 0.000000 |
Figure 10a shows a scatter plot with regression line for the cut-off values \(x_c\), defined to be the least value such that the probability that a secondary structure for length n homopolymer has degree greater that \(x_c\) is at most 0.01. From this figure, we determined that for homopolymer length \(n>100\), it more than suffices to take degree upper bound \(K=n+30\). Figure 10b shows the connectivity degree distribution for a homopolymer of length 20, where degree dg(s) is redefined to be the number of structures t that can be obtained from s by adding, removing, or shifting a base pair in s. The so-called \(MS_2\) move set, consisting of an addition, removal or shift of a base pair is the default move set used in RNA kinetics software kinfold (Lorenz et al. 2011). Although a dynamic programming algorithm was described in Clote and Bayegan (2015) to compute the average \(MS_2\) network degree, the methods of this paper do not easily generalize to \(MS_2\) connectivity densities. Figure 11 shows a least-squares regression line for the log-log density plot for \(MS_2\) connectivity (computed by brute-force) for a homopolymer of length 20, together with an optimal power-law fit computed by RNApowerlaw. Since there are only 106.633 secondary structures for the 20-mer with \(\theta =3\), we ran powerlaw on \(MS_2\) connectivity data, which determined \(\alpha = 6.84\), \(k_{{xmin}}=36\), and a log odds ratio \(R= -2.06\) with p value of 0.248. Since RNApowerlaw determined \(\alpha = 6.84\), \(k_{{xmin}}=36\), and a Kolmogorov–Smirnov p value of 0.219, we can not reject the null hypothesis that a power-law distribution fits the tail of \(MS_2\) connectivity data for a 20-mer.
5 Conclusion
Table showing secondary structure preferential attachment probabilities
n | n+1 | \(S_{n}\) | \(S_{n+1}\) | Succ | Fail | Succ/(Succ+Fail) | \(\langle p(s',t'|s,t) \rangle \) |
---|---|---|---|---|---|---|---|
5 | 6 | 2 | 4 | 5 | 1 | 83.33% | \(0.8333 \pm 0.1667\) |
6 | 7 | 4 | 8 | 18 | 8 | 69.23% | \(0.7222 \pm 0.4157\) |
7 | 8 | 8 | 16 | 90 | 37 | 70.87% | \(0.7748 \pm 0.3260\) |
8 | 9 | 16 | 32 | 419 | 131 | 76.18% | \(0.8105 \pm 0.2941\) |
9 | 10 | 32 | 65 | 1891 | 575 | 76.68% | \(0.8122 \pm 0.2887\) |
10 | 11 | 65 | 133 | 7883 | 2498 | 75.94% | \(0.8125 \pm 0.2891\) |
11 | 12 | 133 | 274 | 33,069 | 9763 | 77.21% | \(0.8300 \pm 0.2730\) |
12 | 13 | 274 | 568 | 142,968 | 40,797 | 77.80% | \(0.8322 \pm 0.2709\) |
13 | 14 | 568 | 1184 | 621,884 | 171,384 | 78.40% | \(0.8366 \pm 0.2646\) |
14 | 15 | 1184 | 2481 | 2,723,993 | 723,887 | 79.00% | \(0.8428 \pm 0.2587\) |
15 | 16 | 2481 | 5223 | 12,041,929 | 3,108,978 | 79.48% | \(0.8478 \pm 0.2556\) |
16 | 17 | 5223 | 11,042 | 53,730,451 | 13,544,005 | 79.87% | \(0.8518 \pm 0.2523\) |
17 | 18 | 11,042 | 23,434 | 241,738,083 | 59,258,399 | 80.31% | \(0.8561 \pm 0.2485\) |
18 | 19 | 23,434 | 49,908 | 1,096,087,115 | 261,730,198 | 80.72% | \(0.8598 \pm 0.2455\) |
Given the current interest in whether certain naturally occurring networks are scale-free, we have introduced a novel algorithm to compute the connectivity density function for a given RNA homopolymer. Our algorithm requires \(O(K^2n^4)\) run time and \(O(Kn^3)\) storage, where K is a user-specified degree bound \(K \le \frac{(n-\theta )(n-\theta -1)}{2}\). Short of exhaustively listing secondary structures by brute-force, no such algorithm existed prior to our work. Since existent software appears unable to perform power-law fitting for exponentially large RNA connectivity data, we have also implemented code to compute and statistically evaluate the maximum likelihood power-law fit for an input histogram, using a very fast method to the density function of a sampled power-law distribution with given scaling parameter. Using the resulting code, called RNAdensity and RNApowerlaw, we have computed the connectivity density function for RNA secondary structure networks for homopolymers of length up to 150. Statistical analysis categorically shows that there is no statistically significant power-law fit for homopolymer RNA secondary structure network connectivity, despite the seemingly good visual fit shown in Fig. 9. Figure 12 shows that secondary structure network connectivity is not scale-free for the (real) 32 nt selenocysteine insertion sequence fruA. Figure 13 shows that the \(MS_1\) and \(MS_2\) degree distributions for other naturally occurring RNAs are not scale-free, in particular for the 65 nt RNA thermometer from Klebsiella pneumoniae subsp. pneumoniae with EMBL accession code CP000647.1/1773227-1773291 and the 76 nt alanine transfer RNA from Mycoplasma mycoides with accession code tdbR00000006 from tRNAdb Juhling et al. (2009) (accession code RA1180 from the Sprinzl tRNA database). While the density plot in Fig. 12 was produced by exhaustively enumerating all 971,299 secondary structures of the 32 nt fruA, Figure 13 was produced by enumerating all secondary structures having free energy within 13 kcal/mol of the minimum free energy, as computed by RNAsubopt from the Vienna RNA Package (Lorenz et al. 2011); this procedure generated 1,079,102 secondary structures (out of a total of \(\approx 1.99546 \times 10^{15}\) structures) for the 65 nt fourU RNA, and 408,414 secondary structures (out of a total of \(\approx 8.77347 \times 10^{17}\) structures) for the 76 nt tRNA.
Since (Day et al. 2003; Kihara and Skolnick 2003) have presented data that suggests that existent protein structures can be explained using only a small number of protein folds, we presented data in Table 4 that suggests that RNA secondary structures may satisfy a type of preferential attachment—a rigorous combinatorial argument establishes this fact for a modified notion of preferential attachment [data not shown, but available in the Appendix of Clote (2018)]. Finally, Python implementations of the algorithms from this paper are publicly available at http://bioinformatics.bc.edu/clotelab/RNAnetworks.
As an afternote, our personal opinion is that it doesn’t much matter whether a naturally occurring network arising from physical phenomena is precisely scale-free or not. If network connectivity appears to follow a power-law distribution, even approximately, then by results of Barabasi and Albert (1999), this suggests that preferential attachment could play a role in how the network may have been constructed by nature. Preferential attachment might well have been a factor in how protein and RNA structures have been formed by evolutionary forces—even in the emergence of stable folds in prebiotic times (Abkevich et al. 1997). It is noteworthy that only a small number of protein folds suffice to explain the diversity of all protein folds found in the Protein Data Bank (PDB) (Kihara and Skolnick 2003): “The number of proteins required to cover a target protein is very small, e.g. the top ten hit proteins can give 90% coverage below a RMSD of 3.5 Å for proteins up to 320 residues long.” As well, the 30 most populated metafolds represent “about half of a nonredundant subset of the PDB” (Day et al. 2003). However, other evolutionary factors seem to be present in the evolution of protein folds, such as kinetic accessibility (Cossio et al. 2010), as well as the ability to switch between alternate conformations (Porter and Looger 2018).
Footnotes
- 1.
A network is said to be robust, or resilient, if its connectivity is (relatively) unaltered in the event that random nodes have been removed; i.e. alternate pathways exist to connect nodes, even if a (random) node has been removed. Since functionality remains in the case of random node failure, network robustness is of obvious importance in massively parallel computers, in the World Wide Web, in metabolic pathways, signaling pathways, etc. This topic is discussed in detail in Chapter 16, “Percolation and Network Resilience”, in Newman (2010).
Notes
Acknowledgements
We would like to thank Amir H. Bayegan for providing the figures in Sect. 2.2 and Jenny Baglivo for a reference for the conditional method to sample from the multinomial distribution. This work was partially supported by National Science Foundation grant DBI-1262439. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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