A Linear Programming Based Algorithm for Multiple Sequence Alignment by Using Markov Decision Process

Transcription

1 inear rogramming Based lgorithm for ultiple equence lignment by Using arkov ecision rocess epartment of omputer cience University of aryland, ollege ark hirin ehraban dvisors: r. ern unt and r. hau-en seng ay,

2 able of ontents. bstract ntroduction Background.... lgorithm and mplementation esults elated orks onclusion.... eferences...

3 . bstract Bioinformatics is a scientific field that combines biology and information technology for the purpose of developing tools that identify genes and proteins for medical and biotechnology applications. n the last decade, there has been an enormous increase in the study of biological sequences. nalyzing biological sequences requires matching a set of protein sequences composed of unknown properties with the protein sequences whose properties are well known. his project proposes a promising algorithm that offers an alternative approach to the problematic process of aligning and matching longer biological sequences.. ntroduction Bioinformatics is the field of science in which biology, computer science, and information technology merge to form a single discipline. he ultimate goal of the field is to enable the discovery of new biological insights as well as to create a global perspective from which unifying principles in biology can be discerned. [] here are several objects of interest in bioinformatics such as the enome sequences ( and protein sequences) and the acromolecular structures (a structure in which all of the atoms are linked by chemical bonds as a unit). One of the important tools in bioinformatics is sequence alignments. variation of the sequence alignments is called the multiple sequence alignments. he multiple sequence alignments identify the conserved regions of the biological sequence that can have major medical or biological importance. he theory of evolution shows that all living creatures are descended from a common ancestor by a process known as mutation. he principal non-experimental method of tracing biological history is an example of multiple sequence alignment. equence alignment is a method of 3

4 writing one sequence on top of another in a way that shows how mutation occurs during evolution. hese changes are known as insertions (inserting amino acids to the second sequence) and deletions (the process in which nucleotide pairs are removed from a gene which is denoted with a gap). n our work different combinations of insertions and deletions are called actions. 3. Background he methods used to align biological sequences are based on solving a minimization problem (cost) or a maximization problem (score). n this project, a coring arkov decision model is used to calculate the probability matrix of the alignment. he resulting matrix is used to solve a minimization or a maximization problem. arkov chain is a random process consisting of states that can change with time. One of the arkov hain s properties is that it is memory-less, meaning at time n+ it has no memory of the previous states at time,,..., n-. n a arkov chain, the probability of traveling from one state to another is important. hus, a matrix that consists of these probabilities is called arkov chain matrix where each row and column is labeled with state numbers. wo properties are extremely important in building the arkov chain. he first is that the probability of traveling from one state to another should be greater or equal to zero. he second is that the summation of probabilities of traveling from one state to all other states should be equal to one. hese properties can be formulated as shown in below.

5 arkov hain properties: () n = i =,..., m j= he method of aligning sequences used in the algorithm uses inear rogramming. inear rogramming is a method to minimize or maximize a linear function of one or more variables. he solution must satisfy certain equations or inequalities called constraints. e obtain alignments by solving a linear programming problem derived from the arkov chain decision model. he arkov decision model is a process used to define the next state s action in a particular time. fter the observation of the state of the process, an action must be chosen for the next state. oreover, if the process is in state i at time n and action a is chosen, then the next state of the process is determined by the probability of the present state and subsequent action as shown in equation (). ( a) = r ob{ X = j given X = i and a = a } () n + n n urthermore for each action a, the two properties stated in equation (3) and () should hold. ( a ) (3) i j m j= ( a) = () 5

6 . lgorithm and mplementation he algorithm proposed in this paper is based on the arkov decision model which creates probability matrices used to set up the linear programming problem. Because we focus on aligning three sequences, there are eight different possibilities of combining gaps and letters. gap is represented as and letter is represented as. oreover, these different possibilities are numbered for ease of reference to the actions as shown in igure. a = igure ach column of the aligned sequence represents a state and is defined by a number as shown in igure. tate umber : igure or example, assume that we want to align the sequences stated in figure 3. igure 3 6

7 7 he first step is to assign the state number to each column of our sequence. igure shows the assigned states for the sequences in igure igure ach state represents an action, which is defined by the algorithm. or the example mentioned in igure 3 the algorithm defines the actions of each state as described in igure 5. igure 5 he following formula calculates the frequency of transition from a particular state to other actions. ), ( ),, ( ) ( a i n a j i n a =, where a denotes the action, i represents the initial state, and j is defined as the final state.

8 n case of the example shown in igure 3 above, the algorithm creates a matrix shown in igure 6, where includes i, j, a,, and (a) which is calculated using the above formula.. i j ( a) a igure 6 hese frequencies are stored in a matrix and will be used to solve a maximization linear function as shown in equations (5) and (6) below. maximize y j a ja ( j, a) onstraints :.. j j y a ja y ja = = b j α + α y i a ia ( a) (6) y ja, all j, a y ja measures how often from state j one should choose action a when there is a maximum total score b j the initial distribution of states in the arkov hain is set to

9 /m where m is the number of states in the sequence. (a) is the calculated frequency in arkov decision model associated with action a. ( j, a ) represents the score of transition from state j to another state with action a. s observed in the equations (5) and (6) above, the linear programming function consists of two constraints. owever, the first constraint can be derived from the second constraint. hus, it can be assumed that the problem has only one independent constraint. o solve this linear programming, we are required to produce a matrix for (a) his matrix can also be denoted as accumulation of different frequencies for each action as shown in igure 7. ( a ) = () () (3) () (5) (6) (7) () igure 7 herefore, in the above example, the algorithm creates a (a) matrix with the result shown in figure 6. igure represents part of the ) (a for the previous example. 9

10 =.5.5 () igure s stated above, b j the initial distribution of states in the arkov hain is set to /m where m is the number of states in the sequence. n the above example the number of states is. igure 9 shows the initial distribution. = b j igure 9 here are two different types of sequences used for testing the algorithm. he first sequences uses letters and gaps as shown in igure. igure

11 ccording to the type of sequence that is being processed, different score values are calculated. he score of a state is not permanent; however, it depends on a transition to different actions. n the set of data shown in igure, a pair of sequences is selected and then the score from Blosum6 (which is a generally accepted table for scoring pairs of amino acids) is selected. ccording to the action, if there is a deletion, a deletion penalty, d would be deducted, and if there is an extension with gaps, an extension penalty, e, would be deducted. he following procedure is repeated for each sequence, and the result of the following calculation is divided by the number of sequences that are being aligned as shown in equation (7) and (). xample: lignment of three sequences (, ) = [Bl (-, - ) e e + Bl (-, ) e d + Bl ( -, ) e d ]/3 (7) (, ) = [ Bl (, ) d + Bl (, ) + Bl (, ) d]/3 () he second set of data contains letters, gaps, and question marks as shown in igure. he question mark symbol in a sequence represents the effect of an insertion or deletion at the beginning of a sequence. his effects the translation into protein at those positions.

12 igure he score for data with a question mark is calculated with equations (9), (), and (), where f is the penalty of having question mark. ( letter, ) = Bl ( letter, - ) f ( -, ) = ( f + d) (9) (, ) = f xample: = ), ( [ Bl (, - ) f e + Bl (, ) + Bl (-, ) f e ]/3 () = ), ( [ (, ) d d + ( -, ) d e + (-, ) d e ]/3 () his algorithm uses equations (7), and () to calculate the scores of each state. he score results for action using the sequences in igure 3 are represented in igure below.

13 ( j, ) = igure ince atlab s solves minimization problems, we insert a negative sign in front of the maximization function to solve for minimization problem. he goal to be achieved is to determine what insertion/deletion pattern should be chosen for the next column given state i. herefore, the probability of choosing action a in state i is calculated as shown in equation (), and is denoted as β (a). β i ( a) = yia y a ia i () where { y ia } are solution of or example igure 3 shows a section part of the y i, which is a part of the linear programming solutions for action of sequences shown in igure 3. 3

14 y i = e 6 5.6e.75e igure 3 y is the summation of inear rogramming solutions on state i over all a ia possible actions a. he set of all β (a) is called a policy. i By using theses policies, the suggested action from this algorithm is chosen. he results of these policies are stored in a matrix where the columns represent the actions and the rows represent the states. urthermore, for each state if β (a) = for some action a, then action a would be chosen as the suggested action. f < β (a) <, then an interval method would be used to select action a with probability (a). random number would be chosen in the interval method. urthermore, the action whose number is greater and is closest to the random number would be selected for the next action. igure demonstrates the policy result from the algorithm with figure 3 sample set. i β i i

15 tate olicy igure ubsequently, the alignments that result from the suggested actions were compared with the alignments obtained by using the U method. hese comparisons are done on the same set of sequences so this is just a test of how consistent the algorithm is with U. he purpose of this project is to discover a new algorithm which consists of the same result as U to be used in a longer set. 5

16 5. esult he algorithm has been tested with onghui an data set on the first sequences. he sequences have been aligned with U with a length of. his set of data is derived from protein cytochrome p5. he range of extension penalty, e, for the test was as to insteps of. he deletion penalty, d, had the same range and α.9 in steps of.. few of examples are described below. he following is the result observed from the data set with e = (extension penalty), d = (deletion penalty), and α =.5. he final result of this algorithm is a matrix that includes the state number with the corresponding suggested and actual action for each of the states. igure 5 shows the result of this algorithm for sample data shown in igure 3. tate uggested ction ctual ction igure 5 he results of the algorithm are represented with symbols for ease of reference. n igure 6, represents the discrepancies and represents that the suggested and actual sequences matched each other. his means that the result from the algorithm was 6

17 7 consistent with the result achieved with U. n this example the discrepancy is equal to since one of the states have been repeated two times. igure 6 nother example of the sample data with the same extension and deletion penalties is shown below. he discrepancies is also in this example. he second set of data that the algorithm has been tested upon is the data supplied by r. rlin toltzfuss of the enter for dvanced esearch in Biotechnology at the ational nstitute of tandards and echnology. his set of consists of aligned proteins of species used in a study of protein evolution. he data set is aligned according to U. he range of extension penalty, e, for the test were to with insteps of

18 . he deletion penalty, d, had the same range and α.9 in steps of. as describe in the sample data above. he result of three of the sequences from the data set with e =, d = 6, andα =. with discrepancies = 3 is shown below. nother result from the same data set with e =, d = 6, and α =. with discrepancy of is shown below.

19 9 6. elated orks s mentioned in the background section on page [], the methods used to align biological sequences are based on solving a minimization problem (cost) or a maximization problem (score). he propose algorithm is based on solving the maximization problem, also known as dual. owever, there exists another method which is called primal where the linear function solves a minimization (cost) problem as shown in (3) below. inimization... onstrain ( ) (, ) n n i j j i a i a b u b u b u u u u α >= >= (3) he result of multiple sequence alignments using the dual method on the sequences from the onghui an set, each with a length of, shows improvement. or example, igure 7 shows the result of the discrepancies of the algorithm for the data set. he discrepancy is. owever, with the same set of data, the result of discrepancies from primal problem is 3, as demonstrated in igure. igure 7

20 igure igure 9 demonstrates another example of the result of our algorithm and igure demonstrates the result of the primal problem. igure 9 igure

21 7. onclusion equence alignment is based on biological evaluation. lignment is a tool for discovering the biological function of genes and proteins, and is important for discovering genetic and related diseases. he proposed algorithm is a new approach in the multiple sequence alignment that uses arkov decision model to solve a linear maximization problem. his algorithm establishes more improved results than primal problem since it generates fewer discrepancies. he algorithm can still be improved to produce more improved result which are consistent with U.

22 . eferences [] [] [3] ubrin., ddy., rogh., itchison. Biological equence nalysis: robabilistic odels of roteins and ucleic cids. ambridge University ress, 999. [] unt., earsley., O gallagher. inear rogramming Based lgorithm for ultiple equence lignment, roceedings of the 3 Bioinformatics onference B 3, omputer ociety 3. [5] unt., earsley. n Optimization pproach to ultiple equences lignment, ational nstitution of tandard and echnology, arch. [6] Jeanmougin,., hompson J., ouy., iggins., and ibson. ultiple sequence alignment with clustalx. rends in Biochemical ience, October 99. [7] oss, heldon. ntro to robability odel, 7 th edition. arcourt, cademic ress, Burlington,. [] oss, heldon. pplied robability odels with Optimization pplications. olden ay, an rancisco, 97.