Co-occurrence pattern mining based on a biological approximation scoring matrix

Abstract

Mining co-occurrence frequency patterns from multiple sequences is a hot topic in bioinformatics. Many seemingly disorganized constituents repetitively appear under different biological matrices, such as PAM250 and BLOSUM62, which are considered hidden frequent patterns (FPs). A hidden FP with both gap and flexible approximation operations (replacement, deletion or insertion) deepens the difficulty in discovering its true occurrences. To effectively discover co-occurrence FPs (Co-FPs) under these conditions, we design a mining algorithm (co-fp-miner) using the following steps: (1) a biological approximation scoring matrix is designed to discover various deformations of a single FP pattern; (2) a data-driven intersection tactic is used to generate candidate Co-FPs; (3) a deterministic Apriori-like rule is proposed to prune unnecessary Co-FPs; and (4) finally, we employ a backtracking matching scheme to validate true Co-FPs. The co-fp-miner algorithm is an unified framework for both exact and approximate mining on multiple sequences. Experiments on DNA and protein sequences demonstrate that co-fp-miner is more efficient on solutions, time and memory consumption than that of other peers.

Appendices

Appendix 1: Measurement of the large number $N_{l,op_{max}}$

While $op_{max}=0$, Zhang et al. [22] inferred the upper limit number of exact occurrences $N_{l,op_{max}}=N_l$ as follow:

$$\begin{aligned} N_l=\left[ L-(l-1)\left(\frac{M+N}{2}+1\right)\right] W^{l-1} \end{aligned}$$

(5)

where l is the length of pattern P, L is the length of subject sequence S, and $W=M-N+1$ is the total length of gap $\phi ^M_N$. Formula (5) must meet the condition that the maximal span of pattern $L_{P_{Max}}=l+(l-1)*M<L$. While $L_{P_{Max}}+op_{max} \ll L$, we extend $N_l$ to the upper limit of the number of approximate occurrences $N_{l,op_{max}}$, where $op_{max}$ is the operation threshold defined in Definition 3 in Sect. 3.

For the same gap $\phi ^M_N$, we consider three situations as follows:

• (1) Number by exact matching and replacement $(N_{E/R})$:

On the basis of conclusion of Zhang et al. in [22], $N_l$ describes the number of distinct length-l offset queues with gap $\phi ^M_N$ in sequence S. The length of each occurrence by exact matching and replacement is still l with the same gap $\phi ^M_N$. Hence, $N_{E/R}=N_l$.

• (2) Incremental number by insertion ($N_I$):

By Definition 3 in Sect. 3, the insertion position does not appear in any occurrence, and the length of the real occurrence is still l; however, the gap will be changed. Some new occurrences by insertion operations overlap $N_{E/R}$ and some do not. It is equivalent to count the incremental number of length-l offset matching position queues that exceed the gap’s upper limit.

Supposing the number of insertion operations is i and the number of inserted positions in an occ is j, then $j\le i \le op_{max}$.

First, to insert i wildcards into j positions in an occ, there are $\sum ^i_{j=1}Y_{i,j}\cdot C^j_{l-1}$ ways, where $Y_{i,j}$ is the number of ways to divide i wildcards into j groups, and $C^j_{l-1}$ is the number of ways to select j gaps from P’s l-1 gaps. We discover that $Y_{i,j}$ is the jth coefficient in the ith row in Pascal’s triangle. By Pascal’s rule, $Y_{i,j}=C^{j-1}_{i-1}$.

Second, with the j positions fixed, we need to confirm how many occurrences can be selected to extend new occurrences. The selected occurrences must satisfy the least one, and more gaps upper limit is M. P’s j gaps are fixed with M, and the other l-j gaps are flexible. Similar to Zhang et al.’s conclusion [22], with flexible l-j gaps, the number of the selected occurrences is no more than $N_{l-j}$. Thus, under the number of insertion operations i, there is $\sum ^{i}_{j=1}Y_{i,j}\cdot C^j_{l-1}\cdot N_{l-j}$ as an incremental number by insertion operations.

Therefore, under the virtual operation threshold $op_{max}$ ($i\le op_{max}$), there is $N_{I}=\sum ^{op_{max}}_{i=1}\sum ^i_{j=1}\left[ Y_{i,j}\cdot C^j_{l-1}\cdot N_{l-j}\right] =\sum ^{op_{max}}_{i=1}\sum ^i_{j=1}\left[ C_{i-1}^{j-1}\cdot C^j_{l-1}\cdot N_{l-j}\right]$.

• (3) Incremental number by deletion ($N_D$);

The deletion operation is divided into two situations as follows:

$\textcircled {\small {1}}$ Delete sub-patterns. Under $op_{max}$, the length of incremental occurrences of pattern P could be $l-op_{max}$, $l-op_{max}+1$ and $l-1$ with the same gap $\phi ^{M}_N$. Similar to Zhang et al.’s conclusion [22], with flexible $l-op_{max}$, $l-op_{max}+1$, ..., and $l-1$ gaps, the number of new occurrences is no more than $\sum ^{l-1}_{l'=l-op_{max}}N_{l'}=\sum ^{op_{max}}_{i=1}N_{l-i}$.

$\textcircled {\small {2}}$ Delete wildcards between gaps. If a deletion operation happens on a gap, the length of occurrences is still l, but the gap is changed. This case is similar to the incremental number by insertion, but the gaps are lessening. The selected occurrences must satisfy the least one, and more gaps’ lower limit is N. The total number of new occurrences is no more than $\sum ^{op_{max}}_{i=1}\sum ^{i}_{j=1}\left[ C^{j-1}_{i-1}\cdot C^j_{l-1}\cdot N_{l-j}\right]$.

• Conclusion:

The unified large number $N_{l,op_{max}}=N_{E/R}+N_I+N_D$. When $op_{max}$=0, there is exact mining, and we force that $N_I=N_D =0$. Therefore, the formula of $N_{l,op_{max}}$ is

$$\begin{aligned}&N_{l.op_{max}}=\left\{\begin{array}{ll}N_l,&if(op_{max}=0) \\N_l+\sum \limits^{op_{max}}_{i=1}N_{l-i}+2\sum \limits ^{op_{max}}_{i=1}\sum \limits^i_{j=1}\left[ C^{j-1}_{i-1}\cdot C^{j}_{l-1} \cdot N_{l-j}\right],&else\qquad \\ \end{array} \right. \nonumber \\&s.t. \left\{\begin{aligned}&j\le i \le op_{max}\\&{\text {while}}(l-i\le 0),N_{l-i}=0 \end{aligned} \right. \end{aligned}$$

(6)

We stipulate the rule that when $l-i\le 0$, then $N_{l-i}=0$.

For different gaps {$\phi ^{M_i}_{N_i} (1\le i<l)$}:

If we set $W=max\{M_i-N_{i}+1\}(1\le i\le l-1)$, $N_{l,op_{max}}$ still satisfies formula (6). We express it more accurately in formula (7):

$$\begin{aligned}&N_{l.op_{max}}=\left\{ \begin{array}{ll}N_l,&if(op_{max}=0) \\ N_l+\sum \limits ^{op_{max}}_{i=1}N_{l-i}+2\sum \limits ^{op_{max}}_{i=1}\sum \limits ^i_{j=1}\left[ C^{j-1}_{i-1}\cdot C^{j}_{l-1} \cdot N_{l-j}\right] ,&else\qquad \\ \end{array} \right. \nonumber \\ &s.t. \left\{ \begin{array}{ll}N_l=[L-(l-1)\left( \frac{M_{max}+N_{max}}{2}+1\right) ]W^{l-1}\\&j\le i \le op_{max}\\&{\text {while }}(l-i\le 0),N_{l-i}=0 \end{array} \right. \end{aligned}$$

(7)

where $M_{max}$ and $N_{max}$ denote the maximum and minimum local length of gaps, respectively.

Appendix 2: Theorem proof

Lemma 1

Given a length- n pattern $P=p_1\phi ^{M_1}_{N_1}p_2\ldots \phi ^{M_{n-1}}_{N_{n-1}}p_n$ and its any length- l sub-pattern $Q=p_k\phi ^{M_{k}}_{N_{k}}p_{k+1}\phi ^{M_{k+1}}_{N_{k+1}}\ldots p_{k+l-1}$ $(1\le k\le n-l+1,0<l<n)$ , here is Sup(Q) $\ge Sup(P)/w^*$ , where

$$w^*=\left\{ \begin{array}{ll}\prod _{i\in [1,k]\cup [k+l,n-1]}(M_i-N_i+1),&if (op_{max})=0\\ \prod _{i\in [1,k]\cup [k+l,n-1]}(M_i-N_i+2\times op_{max}+3),&else.\qquad \end{array} \right.$$

(8)

Proof

As shown in Fig. 5 in Sect. 4, ${\mathcal {M}}_{i,j}$ can be traced back to ${\mathcal {M}}_{i-1, 0}$, ${\mathcal {M}}_{i-1,s}$, ${\mathcal {M}}_{i-1,s'}$, ${\mathcal {M}}_{i-1,s''}$, $M_{i,j-1}$, and $M_{i-1,j}$.

• (1) The paths from ${\mathcal {M}}_{i-1,0}$, ${\mathcal {M}}_{i-1,s}$, ${\mathcal {M}}_{i-1,s'}$, and ${\mathcal {M}}_{i-1,s''}$ are replacement paths. Occs (occ, an occurrence of pattern P in sequence S) by the replacement paths do not overlap each other.

• (2) The insertion path from ${\mathcal {M}}_{i,j-1}$ to ${\mathcal {M}}_{i,j}$ must be neglected because the insertion operation after $p_i$ is a virtual operation. By Definition 3 in Section 3, any occ from the path (${\mathcal {M}}_{i,j-1}$ to ${\mathcal {M}}_{i,j}$) can overlap an occ at the (j-1)th position of sequence S.

• (3) Finally, the path from ${\mathcal {M}}_{i-1,j}$ to ${\mathcal {M}}_{i,j}$ is a deletion path. The deletion is also a virtual operation; however, its generated occs’ lengths are changed. Hence, its generated occs also do not overlap any occ.

Therefore, under the virtual operation threshold, there is no more than $w'=\left[ (j-N_{i-1}-1+op_{max})-(j-M_{i-1}-1-op_{max})+1\right] +1+1=M_i-N_i+2op_{max}+3$ times to expand sub-pattern $P_{k\ldots i-1}$ to sub-pattern $P_{k\ldots i}$, where $P_{k\ldots i}= p_k\phi ^{M_{k}}_{N_{k}}p_{k+1}\phi ^{M_{k+1}}_{N_{k+1}}\ldots p_i$. $\square$

By the analogy method, it needs to expand sub-patterns $\{ p_1, p_2, \ldots , p_k \}$ and $\{ p_{k+l}, \ldots , p_n \}$ to convert from Q to P. Therefore, $sup(Q)\ge sup(P)/w^*$.

Note that for exact mining, $op_{max}=0$, the deletion path and insertion path are discarded. There is $w'= \left[ (j-N_{i-1}-1+op_{max})-(j-M_{i-1}-1-op_{max})+1\right] =M_i-N_i+1$.

Lemma 1’

Lemma 1 for the same gap $\phi ^M_N$ here is $sup(Q) \ge sup(P)/w^{n-l}:$

$$w=\left\{ \begin{array}{ll}W=M-N+1,&if (op_{max})=0\\ M-N=2\times op_{max}+3,&else\qquad \end{array} \right.$$

(9)

Proof

For the same gap $\phi ^M_N$, $w^*=w^{n-l}$. Lemma 1’ is a special situation of Lemma 1. $\square$

Theorem 1

(Apriori-like). Given the same length- n pattern P, length- l sub-pattern Q and variable $w^*$ in Lemma 1 , if Q satisfies $Freq(Q)<\eta _{n-l,op_{max}}\times \rho$ , then P is not a frequent pattern, where $\eta _{n-l,op_{max}}=\displaystyle \frac{N_{n,op_{max}}}{N_{l,op_{max}\times w^*}}.$

Proof

We use the method of reduction to absurdity.

Suppose P is a frequent pattern, by Lemma 1, there is $\rho \le Freq(P)=\displaystyle \frac{sup(P)}{N_{n,op_{max}}} \le \displaystyle \frac{sup(Q)\times w^*}{N_{n,op_{max}}} .$ Thus, we have $sup(Q)\ge N_{n,op_{max}}\times \rho /w^*$.

Then, $Freq(Q)=\displaystyle \frac{sup(Q)}{N_{l,op_{max}}}\ge \displaystyle \frac{N_{n,op_{max}}\times \rho }{N_{l,op_{max}}\times w^*}=\eta _{n-l,op_{max}}\times \rho$. However, it is contrary to the hypothesis that $Freq(Q)<\eta _{n-l,op_{max}\times \rho }$.

Hence, the proof is given. $\square$