[Paper Review] Multiple pattern matching: A Markov chain approach

Manuel E. Lladser, Meredith D. Betterton+1 more|ArXiv.org|Apr 24, 2007

RNA and protein synthesis mechanisms62 references3 citations

TL;DR

This paper presents a unified Markov chain embedding framework using finite automata to analyze the probability of multiple patterns—especially correlated RNA motifs—in random strings generated by memoryless sources. By modeling pattern matching as a Markov process on an automaton, it enables exact computation of occurrence probabilities, first-passage times, and frequency distributions, with asymptotic results derived via generating functions and partial fraction decomposition.

ABSTRACT

RNA motifs typically consist of short, modular patterns that include base pairs formed within and between modules. Estimating the abundance of these patterns is of fundamental importance for assessing the statistical significance of matches in genomewide searches, and for predicting whether a given function has evolved many times in different species or arose from a single common ancestor. In this manuscript, we review in an integrated and self-contained manner some basic concepts of automata theory, generating functions and transfer matrix methods that are relevant to pattern analysis in biological sequences. We formalize, in a general framework, the concept of Markov chain embedding to analyze patterns in random strings produced by a memoryless source. This conceptualization, together with the capability of automata to recognize complicated patterns, allows a systematic analysis of problems related to the occurrence and frequency of patterns in random strings. The applications we present focus on the concept of synchronization of automata, as well as automata used to search for a finite number of keywords (including sets of patterns generated according to base pairing rules) in a general text.

Motivation & Objective

• To develop a general mathematical framework for computing the probability of occurrence of multiple patterns in random biological sequences.
• To address the statistical significance of RNA motifs, particularly those with structural correlations such as base-paired modules.
• To unify disparate approaches from automata theory, generating functions, and Markov chains into a coherent method for pattern analysis.
• To provide exact analytical solutions for first-passage times and frequency distributions of patterns, avoiding computationally expensive Monte Carlo simulations.
• To extend the framework to handle both independent and correlated patterns, including modular RNA motifs with long-range dependencies.

Proposed method

• Construct deterministic finite automata (DFAs) to recognize target patterns, including compound and modular patterns with base-pairing constraints.
• Embed the random text generation process into the automaton by defining a Markov chain where state transitions follow the i.i.d. symbol probabilities of the memoryless source.
• Use transfer matrix methods and generating functions to compute the probability generating function of pattern occurrence times.
• Apply partial fraction decomposition to extract asymptotic expressions for the first-passage time distribution, such as $\text{Prob}[T=n] \sim c_2 \cdot n^2 / 2^n$.
• Leverage automaton synchronization and state aggregation to model multiple keyword searches and correlated modular patterns.
• Derive explicit formulae for the expected number of pattern occurrences and the probability of first occurrence after $n$ characters using recursive state transitions.

Experimental results

Research questions

• RQ1 What is the exact probability that a given set of patterns first appears after $n$ characters in a random string generated by a memoryless source?
• RQ2 How can the frequency of multiple patterns in a random string of length $n$ be computed analytically, without simulation?
• RQ3 What is the asymptotic distribution of the first-passage time for a compound pattern composed of two keywords?
• RQ4 How can correlated modular patterns—where sub-patterns are linked by base-pairing rules—be modeled and their occurrence probabilities computed?
• RQ5 Can the Markov chain embedding framework be extended to handle non-i.i.d. models such as Markovian or hidden Markov models?

Key findings

• The first-passage time distribution for a pattern in a random string exhibits asymptotic behavior $\text{Prob}[T=n] \sim c_2 \cdot n^2 / 2^n$, where $c_2$ is a computable constant derived from the partial fraction decomposition of the generating function.
• The probability that a compound pattern (e.g., two keywords) appears for the first time after $n$ characters can be computed exactly using the transition matrix of the embedded Markov chain on the automaton.
• The expected number of occurrences of a pattern in a random string of length $n$ can be derived from the generating function of the pattern's occurrence process.
• For modular patterns with base-pairing constraints, the framework allows computation of the asymptotic frequency distribution of such patterns in random sequences.
• The method provides a significant improvement over Monte Carlo simulations by enabling exact, analytical computation of low $p$-values for rare events.
• The framework unifies the analysis of various pattern-matching problems in bioinformatics, including motif discovery, splicing regulation, and ribozyme motif detection, under a single mathematical formalism.