Parametric Sequence Alignment
4.7 Notes and Comments
Parametric issues in optimization, especially linear programming, have been studied since the 1950s. Parametric linear programming, where the coefficients of the objective function are variable, was initially formulated by Gass and Saaty [24]. In the terminology of this chapter, Gass and Saaty presented a simplex-based algorithm for one-dimensional sensitivity analysis of parametric linear programs. The method can be used for construction and search as well. The combinatorial complexity of parametric linear programming was studied by Murty [39], who showed that there exists a parametric linear program with n variables and 2n constraints where there are 2n basic feasible solutions, each of which is a unique optimal solution for some suitably chosen value of the parameter. Parametric versions of various optimization problems have been studied and bounds for various problems have been established. A sampling of the parametric optimization problems considered in the literature includes network flows [23], stable marriage [30], matroid optimization [12], and scheduling [32].
Many of the approaches discussed in this chapter are specializations of more general techniques to sequence comparison. The geometric definitions and results of Section 4.2.1 are adapted from Agarwal and Sharir’s text [2]. Megiddo’s method of parametric search technique originally appeared in [36] as a method for solving optimization problems with ra-tional objective functions. An improvement based on simulating parallel algorithms instead of sequential ones is also due to Megiddo [37]. The application of Megiddo’s method to sensitivity analysis was first investigated by Gusfield [26, 27]. Ray shooting is an important problem in computer graphics, where it is used to detect and remove hidden surfaces and in computing the intersection of polyhedra; Agarwal and Matouˇsek describe these and other geometric applications of parametric search in [1] (see also Salowe’s survey [48]). Newton’s zero-finding algorithm and the gradient descent method are classical algorithms that can be traced back to Newton and Cauchy respectively. Radzik [47] describes the application of Newton’s method to solve fractional combinatorial optimization problems. The gradient descent method for optimization is well known and discussed in many textbooks [44, 40].
Polyak [43] was among the first to study the subgradient method’s theoretical aspects. Held and Karp [33] were the first to apply the method to mathematical programming problems.
4-28 Handbook of Computational Molecular Biology Parametric sequence comparison was first considered by Fitch and Smith [22], who studied the effect of varying the gap penalty on the optimum alignment of two sample sequences. By careful analysis, they showed that there are 7 and 11 different optimal alignments (optimal-ity regions) for their sample pair when end gaps are weighted and unweighted, respectively.
Waterman et al. [52] proposed a systematic way of finding the optimality regions. Vin-gron and Waterman [51] studied the implications of parameter choice through a series of case studies. Independently of Waterman et al.’s work, Gusfield et al. [29] formally defined parametric alignment and gave the first bounds on the number of regions. Among their results is the O(n2/3) on the number of optimality regions for global alignment with zero gap penalty presented in 4.3.1. Fern´andez-Baca et al. [18] prove that this bound is tight when the alphabet size is unbounded [18]; in fact, it is the only combinatorial complexity bound for parametric sequence comparison known to be exact. The best known lower bound when the alphabet size is bounded is Ω(√
n) [18]. The properties of parametric alignment problems with feature-based scoring schemes were first investigated by Fern´andez-Baca et al. [19], who obtained combinatorial bounds for several problems, such as multiple sequence alignment and phylogeny construction, by observing that they all have a similar integer parametric nature. Tighter bounds (Theorem 4.2) are due to Pachter and Sturmfels [42]
and Fern´andez-Baca and Venkatachalam [21].
Gusfield [26] attributes the one-parameter construction algorithm of 4.5.1 to Eisner and Severance [16]. The two-parameter construction algorithm presented in 4.5.2 is due to Fern´andez-Baca and Srinivasan [20]. Zimmer and Lengauer [53] used this algorithm in their parametric sequence alignment software. The techniques for reconstructing multi-dimensional convex geometric objects through probing, upon which Theorem 4.7 is based, were developed by Dobkin et al. [13, 14], who extended the work on two-dimensional probing by Cole and Yap [10].
Gusfield and Stelling’s publicly-available XPARAL system [31] implements the ray-shooting approach (using Newton’s method) for two-parameter sensitivity analysis described in 4.4.6 and applies it to construct the maximization diagram for two-parameter alignment prob-lems under a wide variety of scoring functions. While, in principle, ray shooting can be used for sensitivity analysis (and, hence, construction) for any number of parameters, there appears to be no reference to this in the literature. One way to achieve this generalization is to use the probing idea mentioned above; the probes here are ray-shooting queries, instead of evaluations. Each probe returns a supporting hyperplane of the region being generated.
The results of [13, 14] imply a number of queries proportional to the number of vertices and facets of the region.
Pachter and Sturmfels [42, 41] describe an implementation of the lifting algorithm for the construction problem mentioned at the beginning of 4.5. Their software relies on Gawrilow and Joswig’s polymake tool [25].
XPARAL solves the inverse alignment problem using the gradient descent method de-scribed in 4.4.4. Sun et al. [50] give efficient algorithms for inverse sequence alignment with and without gaps that exploit the properties of feature-based scoring. Other inverse parametric optimization problems are studied by Eppstein [17].
For general background on hidden Markov models, a good starting point is Rabiner’s survey article [45]. HMMs were first used for sequence alignment by Borodovsky et al. [6, 7, 8]. Durbin et al. [15] and Baldi and Brunak [4] give good introductions to the application of HMMs to sequence alignment and bioinformatics in general. Pachter and Sturmfels [42, 41] build a mathematical theory of statistical models for biological applications and show connections between parametric analysis and statistical models.
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