**Assessing the significance of repeats**

Let’s calculate the mean repeats number (l,k) in a random sequence consisting of N nucleotides occurring with frequencies p_{1},p_{2},…,p_{m}, where m=4 – is the number of various nucleotide types.

The probability of two nucleotides being identical is:

(1)

The probability of two random sequence segments of length l differ by k nucleotides is

where p is from (1), Cl – is the permutation number of k differences in l positions

The number of possible mutual locations of two non-overlapping segments of length l nucleotides could be calculated as follows. The first segment could be located in N-2*l+1 positions of the sequence. If the frist segment starts from i-th position, then the second one could be located in N-i-2*l+2 positions non-overlapping. Thus the overall number of nonoverlapping pairs of segments is

Mean number of repeats in a random sequence of length N is calculated as:

Let’s consider only repeats (l,k) which occur for 1 time in a sequence of length N on the average. In this case we can use the binomial distribution to assess the probability of finding exactly n repeats (l,k) in a target sequence. This probability equals:

Then we find threshhold n_{o} given a significance level q that satisfies:

(4)

If the repeat number exceeds the threshold n(l,k) > n_{o}(l,k), then we consider the number n as significantly deviating from the mean number En(l,k) for a significance level q= 0.05, and the repeat found in n copies we call the nonrandom one.

The formulae considered are valid for the direct, complementary, symmetrical and inverse repeats. In case of complementary repeats and inverse ones we use the probability of nucleotides to complement each other instead of (1), that is:

**References for statistical basis**

Solovyov V.V., Zharkikh A.A., Kolchanov N.A., Ratner V.A. – FEBS Lett., 1984, v.165, p.72-78.

**Program description**

The program is destined to search for non-perfect repeats of various types in a nucleotide sequences.

**Definitions**

- Non perfect repeat is a pair of DNA segments of length l differing one from another by k nucleotides. This pair of segments is called
**(l,k) repeat**. - The overall set of (l,k) repeats is called
**(l,k) group**.

**Types of repeats definition**

Depending on type of correspondence we consider the following types of repeats:

**Direct repeat**– two (l,k) segments are located in the same orientation in a single DNA strand.**Symmetrical**– two (l,k) segments are located in opposite orientation in a single DNA strand.**Direct complementary repeats**– two segments are located in the same orientation in different DNA strands.**Inverse repeats**– two segments are located in opposite orientation in the different DNA strings.

**Program features**

The program performs the repeats search by group clusters (l,k). After the search the optimisation procedure follows consisting of the following steps:

- The concatenation of the groups (l,k) in one set is performed. During that:
- The repeats flanks are extended if there are coinciding nucleotides
- All abundant repeats are deleted

**Program Parameters**

**Repeats types to search parameters:**

*Direct Repeats*– turns on the search of direct repeats.*Direct Complementary Repeats*– turns on the search of direct complementary repeats.*Symmetric Repeats*– turns on the search of symmetric repeats.*Invert Repeats*– turns on the search of invert repeats.

**Parameters specifying the features of group (l,k) to search**

**min length** (l_{1}), **max length** (l_{n}), **Limit type**, **Limit** are the parameters to define the set of groups (l,k) that are subject to search:

(l

_{1},0),…,(l_{1},k_{1}),…

(l

_{n},0),…,(l_{n},k_{n}).

Number of mismatches k_{i} depends on statistical threshold given by n_{o}(l,k) which is described above. The maximal number of mismatches is limited by max mismatch number parameter.

**min length** – the minimal length of the repeated segment

**max length** – the maximal length of the repeated segment

**max mismatch number** -the maximal mismatch number

**Limit type** – Statistical threshold type. Can be of two values:

- In case
**Limit type=P**, accepted are the groups (l,k), that satsify P <=**Limit**, where P – from (2) - In case
**Limit type= En**, accepted are groups (l,k), that satisfy En <=**Limit**, where En – the expected repeats number for in a sequence of length N (see (3))

**Selection of statistically significant (l,k) groups**

**Save Mode ** – Statistical significance switch.

- If
**Save Mode**= NOT RANDOM, accepted are groups (l,k), having nonrandom number of repeats - If
**Save Mode**= ALL, accepted are all groups (l,k)