Exact asymptotic results for the Bernoulli matching model of sequence alignment

Abstract

Finding analytically the statistics of the longest common subsequence (LCS) of a pair of random sequences drawn from c alphabets is a challenging problem in computational evolutionary biology. We present exact asymptotic results for the distribution of the LCS in a simpler, yet nontrivial, variant of the original model called the Bernoulli matching (BM) model which reduces to the original model in the c → ∞ limit. We show that in the BM model, for all c, the distribution of the asymptotic length of the LCS, suitably scaled, is identical to the Tracy-Widom distribution of the largest eigenvalue of a random matrix whose entries are drawn from a Gaussian unitary ensemble. In particular, in the c → ∞ limit, this provides an exact expression for the asymptotic length distribution in the original LCS problem.

Universal randomness

“…, has been derived in terms of the optimal fluctuation approach [50][51][52], where it has been demonstrated that both asymptotics (left and right) of the function P * (f ) are consistent with the Tracy-Widom distribution [1] which was known to describe the statistical properties of many other systems [2][3][4][5][6][7].…”

“…Nowadays we have got rather comprehensive list of various systems (both purely mathematical and physical) whose macroscopic statistical properties are described by the same universal Tracy-Widom (TW) distribution function. These systems are: the longest increasing subsequences (LIS) model [2] (Section I.A) zero-temperature lattice directed polymers with geometric disorder [3] the polynuclear growth (PNG) system [4], (Section I.B) the oriented digital boiling model [5], the ballistic decomposition model [6], the longest common subsequences (LCS) [7], the onepoint distribution of the solutions of the KPZ equation [8] (which describes the motion of an interface separating two homogeneous bulk phases) in the long time limit [9,10], and finally finite temperature directed polymers in random potentials with short-range correlations [11][12][13][14]. It should be noted that directed polymers in a quenched random potential have been the subject of intense investigations during the past three decades (see e.g.…”

Large Deviations of Extreme Eigenvalues of Random Matrices

“…Amazingly, the Tracy-Widom distribution has since emerged in a number of seemingly unrelated problems such as the longest increasing subsequence problem [4], directed polymers in (1 + 1)-dimensions [5], various (1 + 1)-dimensional growth models [6], a class of sequence alignment problems [7] and in finance [8]. Recently, it has been shown that the statistics of the largest eigenvalue is also of importance in population growth of organisms in fluctuating environments [9].…”

Unzipping of two random heteropolymers: Ground-state energy and finite-size effects

“…The Bernoulli Matching model for this problem has been considered in details in [35]. An example of the random matrix with the optimal path is outlined by the bold line in Fig.4 (only filled circles, i.e.…”

“…The ground state energy, E m,n (a = 0), has a meaning of the LIS length of "1" (see [35]). The mean value E m,n in the thermodynamic limit n = m → ∞ equals to…”