Optimization and Performance Analysis of CAT Method for DNA Sequence Similarity Searching and Alignment

1. Introduction

2. Materials and Methods

2.2. Method for DNA Sequence Alignmment Based on Trilateration

Finding a beginning point, or benchmark, against which the other data in the database can be compared is at the core of the concept of a favorite sequence. Alternatively, if we were to approach the problem mathematically, sequence favorite could be represented as a function of N unknowns (in the context of DNA, the unknowns being the 4 bases adenine, thymine, guanine, and cytosine), and the remaining database entries could then be represented once more as functions of the same variables. In this scenario, the distance between each individual sequence and the preferred sequence would be represented by the similarity comparison. In other words, determine the relationship between a point defined by the favorite sequence function and a point described by the sequence function.

A point is generated somewhere in the center of the cloud of points that is used as a reference (sequence favorite) when comparing to a set of points (the database entries) because there is no coordinate system. However, if a coordinate system or three or more reference points are found, it would be possible to use trilateration or elementary analytical geometry to determine the positions of the points relative to one another, which would reflect the degree to which the database records match one another. Moreover, to do away with the requirement for favorite computation sequence.

A novel approach, known as the CAT method, has been introduced for aligning DNA sequences, utilizing the trilateration technique [30]. This method establishes three consistent reference points for trilateration application, resulting in a steadfast reference sequence. This sequence, comprising C-benchmark, A-benchmark, and T-benchmark, remains constant regardless of alterations in the database content.

Once the three constant benchmarks for applying trilateration are established, issue (1) is eliminated. Constant sequence favorite – i.e. independent of the records in the database and to remain the same when the data set is changed.

Since the constant benchmark sequences are determined (i.e. they do not depend on either the data or their count), this allows comparisons to be made at the very beginning - when the sequences are uploaded into the database and this is metadata information accompanying each sequence. This way, sequences won’t need to be compared during lookup (which is the slowest operation), but instead only the metadata information generated during the upload will be compared. By establishing the benchmark sequences, issue (3) unification of favorite sequences for all databases is also eliminated. There are now unified sequences that are standardized for all databases using the described alignment algorithm. When two sequences have the same profiles, this means that they have regions with the same alignment, and one hundred percent complete matching of one sequence on the other can be expected.

For the evaluation of two sequences, it is necessary to calculate the distance of the segment S1S2 in Figure 1.

$$S_1{S}_2=\sqrt{{\left|A{D}_1-A{D}_2\right|}^{2 }+{\left|h_1-h_2\right|}^{2 }}$$

(1)

For now, ∆AS1T is considered, then analogous calculations and reasoning are per- formed for AS2T. What is known about ∆AS1T is the sides AT = |1|, AS1 = distance from S1 to A-benchmark (it is known), S1T = distance from S1 to T-benchmark. Use the cosine theorem to find ∡TAC, then side AD1:

$$S_1{T}^2=A{S_1}^2+A{T}^2-2.A{S}_1.AT.\mathit{cos}\left({\alpha }_1\right)$$

(2)

$$\mathit{cos}\left({\alpha }_1\right)=\frac{A{S_1}^2+A{T}^2-S_1{T}^2}{2.A{S}_1.AT}$$

(3)

$$A{D}_1=A{S}_1.\mathit{cos}\left({\alpha }_1\right)$$

(4)

$$h_1=\sqrt{A{S}_1^2-{(A{S}_1.cos\left({\alpha }_1\right))}^2}$$

(5)

The calculations for triangle AS2T are analogous. After substituting the values found, a value for S1S2 is obtained.

$$S_1{S}_2=\sqrt{{\left|A{D}_1-A{D}_2\right|}^{2 }+{\left|h_1-h_2\right|}^{2 }}$$

(6)

The smaller value obtained for the intercept, the greater the probability of a complete match, expressed as a percentage. This allows the database to be quickly searched for sequences with a certain percentage of similarity, which can later be aligned and compared with more accurate algorithms such as Needelman-Wunsch or Smith-Waterman.

It is possible to occur collisions in such proposed DNA sequence alignment method based on trilateration, i.e.:

• More than one sequence of the same length to get the same values for AD and h:
  • Due to the nature of benchmark sequences and the fact that the real sequence projects at most a quarter of the bases onto the benchmark, i.e. with benchmark ACGT and projection of G at the second position, there are no matches and no value is accumulated for match rate.
• During S1S2 calculation, the same values are obtained:
  • Because of statistical errors accumulated when calculating AD and h.
  • Because of rounding in calculations due to the range of data types, this cannot be avoided even with the use of more precise types.

To minimize collisions, the precision in calculations for AD and h should be increased by adding the dependency on neighboring bases. Like in local alignment, when the current base of the benchmark sequence does not match the current base of the real sequence, additional points can be added or subtracted, depending on whether a neighboring base match is. After Needelman-Wunsch alignment, the places where the bases do not match appear as gaps “_” positions and are given different points accordingly. A similar principle can be applied in the proposed method. If a base and index match is given value of 1. If there is a mismatch, a neighboring base from the benchmark sequence is checked and given a value of 0.6 or 0.4 depending on how far the neighbor is, i.e. when there is match with the left or right base from the benchmark then give 0.6, when is the far – 0.4. If we define baseDistance array for near matches it will looks like: [0, 0.6, 0.4, 0.6]. Left and right neighbors are with indexes 1 and 3, 0 index is for exact match and index 2 is the far neighbor.

Example of benchmark ACGT base at position 2 G

ACGT

XGXX

G at position 2 corresponds to C from the benchmark sequence and instead of 0 a match value of 0.6 can be given because it is adjacent to the right and in Needelman- Wunsch ordering it has the following alignment:

ACG_T

X_GXX

In this way, the precision of AD and h calculation is increased, the accumulation of statistical errors is reduced 1., thus reducing collisions 2. After finding a suitable sequence in the base, one with a minimum value for S1S2, a more accurate alignment algorithm can be applied. The comparison calculations in the direction calculate the CAT of two sections proposed in the presented method with a constant complexity that makes it to be applied as a first step suitable in the FASTA algorithm as well as for multiple alignments such as ClustalW.

Add dependency with a previous match and current position to increase uniqueness of the CAT profile. Like in positional numeral system contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. It’s need something similar to be done here, but considering the average length of the sequence exact same approach cannot be used. Instead, when a previous exact or near match is detected, a sum of current maximum theoretical sequence of matches is tracked, will be increased. The ratio of current maximum theoretical sequence of matches to current index will be added to the sum of previous matches, and this will be considered as kind of a bonus points. All given bonuses from previous matches must be tracked as well, since they are need in a later stage to correct the distances, so that a tringle can be formed. To be able to apply method of trilateration calculated distances should ensure that in any conditions a triangle can be formed. I.e. the sum of any two sides must be greater than the third one.

3. Results

4. Discussion

References

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