Patent Application: US-48032504-A

Abstract:
the present invention relates to a method for gene mapping from chromosome and phenotype data , which utilizes linkage disequilibrium between genetic markers m i , which are polymorphic nucleic acid or protein sequences or strings of single - nucleotide polymorphisms deriving from a chromosomal region . the method according to the invention is based on discovering and assessing tree - like patterns in genetic marker data . it extracts , essentially in the form of substrings and prefix trees , information about the historical recombinations in the population . this infor - mation is used to locate fragments potentially inherited from a common diseased founder , and to map the disease gene into the most likely such fragment . the method measures for each chromosomal location the disequilibrium of the prefix tree of marker strings starting from the location , to assess the distribution of disease - associated chromosomes .

Description:
it is an object of the present invention to provide a method for gene mapping aiming to discover a gene region affecting a certain trait using chromosome data . empirical evaluation on a realistic , simulated data shows that the method according to the invention is competitive with other recent data mining based methods , and clearly outperforms more traditional methods . our experiments , explained later , show that the method according to the invention , treedt , is effective in extreme conditions typical for current mapping problems : with lots of noise ( only 10 - 20 % of affected chromosomes carry the mutation , lots of missing data ) and with small sample sizes ( 200 affected and 200 control chromosomes ). however , the highest potential of the method according to the invention lies in the data intensive tasks of future — such as genome scanning with larger samples and larger number of markers — due to its low computational complexity . in comparison to state of the art methods , treedt is most competitive . in terms of gene localization accuracy , it gave best results in the case of multiple founders and demonstrated good robustness with respect to missing data . unlike the compared methods , treedt can be used to predict whether a gene is present at all or not . finally , in comparison to its closest competitor , hpm , treedt has much smaller computational cost . an additional advantage of treedt is that it has only one input parameter , the ( maximum ) number of deviant subtrees , whereas for hpm one has to set several more or less arbitrary thresholds . for any pair of chromosomes in the sample there has been a common origin in the population history , an ancestral chromosome at which their paths have diverged . due to recombinations different parts of chromosomes have different histories . at any given location the chromosomes in the sample and their most recent common origins form a coalescence tree . in the coalescence tree for the ds gene location , all the chromosomes in one or more subtrees carry the ds mutation , and we should observe excess of disease - associated haplotypes as the leaves of these subtrees . the closer the tree is located to the ds gene , the more and larger subtrees are identical to those in the tree at the ds gene location . based on the observed haplotypes , the method of the invention defines a prefix tree estimating the most likely coalescence tree at a number of locations along the analyzed chromosome , and then assesses the subtree clustering of disease - associated haplotypes in these trees . it is a further object of this invention to provide a novel tree disequilibrium test , intended for predicting ds gene locations in the method of the invention . the vicinity of the location for which the test gives the lowest p value is the most likely candidate area for the ds gene location . the method also calculates the corrected overall p value for the best finding . this p value can be used for predicting whether the chromosome carries a ds gene . further , a method for the estimation of statistical significance of individual findings as well as the whole process , based on multiple permutations but carried out at the cost of a single permutation , is provided . the subsumption relation of the substrings overlapping a given location forms a directed acyclic graph ( dag ). the tree structures obtainable by pruning the dag may be considered as possible coalescence trees at the location , as shown in fig3 , with the following exceptions : 1 ) the order of nodes may differ from that in the true coalescence tree , e . g . - 34 -- might actually be a more recent node than -- 1234 --. however , because the expected length of the shared region of two chromosomes decreases monotonically as the time from their divergence increases , it is easy to see that the order given by subsumption is the most likely one . 2 ) because haplotypes may also share a substring by chance , the internal nodes may represent a combination of nodes in true coalescence tree . the upper nodes of the coalescence tree must be very old and the corresponding shared chromosomal regions extremely short , and therefore it is very likely that a large number of coalescence nodes is contained in the empty substring root . on the other hand the younger coalescence nodes with shared regions spanning over several markers are more likely to have one - to - one correspondence with observed recurrent substrings . instead of considering alternative coalescence trees leading to the same observed haplotypes , the method of the invention uses the unique haplotype prefix tree as a canonical representation of such set of coalescence trees . an example of a prefix tree is shown in fig4 . the method of the invention builds the prefix trees between each pair of consecutive markers and tests their disequilibrium . according to one embodiment of the invention , the prefix tree t is tested by the tree disequilibrium test ( treedt ) testing the alternative hypothesis the distribution of the disease - association statuses deviates in some subtrees of t from the overall distribution of statuses against the null hypothesis the disease - association statuses are randomly distributed in the leaves of t . treedt identifies the subtree set in which the observed status distribution deviates most from the expectation under the null hypothesis , and returns the significance of the deviation as a p value . treedt takes the maximum number of deviant subtrees as a parameter . in principle there is no need to set an upper limit for the subtree count , but whenever ld - mapping is applicable , the majority of the mutation carriers is concentrated in a only few such subtrees in which the shared region is long enough to identify a deviant substring . in the experiments for this paper we use an upper limit of 6 subtrees . for measuring the disequilibrium of a tree , we use a variant of the z test . the test statistic z k for a tree with k deviant subtrees t 1 , . . . , t k is z k = ∑ i = 1 k ⁢ a i - n i ⁢ p n i ⁢ p ⁡ ( 1 - p ) , where a i is the number of disease - associated haplotypes and n i the total number of haplotypes in subtree t i εs , and p is the proportion of disease - associated haplotypes in the sample . the score measures the distance of the observed number of disease - associated chromosomes ( a i ) from the expectation ( n i p ) in standard deviations ( the square root of n i p ( 1 − p )), under the assumption of binomial distribution with parameters n i and p . we use a one - tailed test , since we are interested only in subtrees in which the proportion of disease - associated haplotypes is greater than expected . we could use a 2 ×( k + 1 ) χ 2 - statistic as a measure of deviation for a given subtree set s . the χ 2 - statistic , however , is not easily maximized in the space of all possible subtree sets and is therefore not a very practical choice . z k can be efficiently maximized simultaneously for all k using a recursive algorithm , as shown in the algorithms section . treedt takes the maximum number of deviant subtrees as a parameter . in principle there is no need to set an upper limit for the subtree count , but whenever ld - mapping is applicable , the majority of the mutation carriers is concentrated in only few such subtrees in which the shared region is long enough to identify a deviant substring . in the experiments for this paper we use an upper limit of 6 subtrees . z k is a measure for the disequilibrium of a given tree , corresponding to a certain location in the chromosome , with given k deviant subtrees . given a tree , treedt finds for each k the set s of subtrees that maximizes z k . in order to find the best k for the given tree , simple maximization is not possible . since the statistics for different degrees of freedom k are not comparable , treedt estimates the p value for each maximized z k ( under the null hypothesis of random distribution of disease status ). because the distribution of the maximized z k is very complex and dependent on the tree structure , p values are estimated by a permutation test . in order to get a single p value for the disequilibrium at a given location , we need to combine the information from the trees to the left and to the right of the location . as a combined measure we use the product of the lowest p value over all k from each side . again , since the measures are not necessarily directly comparable , a new p value for the combination is estimated . the results are now comparable between different locations . the output of treedt is essentially the p value ranked list of locations . a point prediction for the gene location is obtained by taking the best location ; a ( potentially fragmented ) region of length l is obtained by taking best locations until a length of l is covered . since multiple locations are tested for a p value , and also since the p values at nearby locations are not independent , a direct link between the p value and the probability that the gene is indeed close to the location can not be established . the p values are used simply as a method of ranking the locations . however , a single corrected p value for the best finding can be obtained with a third test using the lowest local p value as the test statistic . this p value can also be used to answer the question whether there is a gene in the investigated are in the first place or not . all these three nested p value tests ( for each tree and k , for each location , for the best location ) can be carried out efficiently at the cost of a single test . table 1 summarizes the three levels of the nested test . the haplotype prefix - trees to the left and right from each analyzed location can be efficiently identified using a string - sorting algorithm . the algorithm produces as intermediate results for each marker the sorted list of the partial haplotypes to the right from the marker . all the right - side trees can be easily derived from these intermediate lists , because the haplotypes belonging to a single node form a continuous block in the sorted list . the left - side trees can be identified similarly by sorting the inverted haplotypes . the computational cost of constructing the trees is negligible compared to the cost of the permutation test procedure . the same process can also be used to enumerate all the recurrent substrings , or all the closed substrings . a substring s is closed , if and only if none of its superstrings match all the same haplotypes than s . the nodes in the right - side prefix trees have one - to - one correspondence to recurrent substrings starting at the same marker . nodes that are to be split in the next step of the sort algorithm correspond to closed substrings . it is essential that the time - complexity of the algorithm for maximizing the z - values is as low as possible , because it must be executed for each tree location and permutation in turn . the key observation is that if s is the set of k deviant subtrees of t with the greatest value of z k , t ′ is a subtree of t , and s ′ ⊂ s is a set of m subtrees in t ′, then s ′ has the maximum value of z m in t ′. also , if s = s 1 ∪ . . . ∪ s n , and k is the subtree count in s , and k i is the subtree count in s i , then z k ⁡ ( s ) = ∑ i ⁢ z k i ⁡ ( s i ) . these observations lead us to the following recursive algorithm that propagates the locally maximized z - values upwards in the tree : input : a haplotype prefix tree t output : maximum values of z k in the tree t for each k call maximize ( t ) maximize ( t ): if t is not a leaf : 1 . for each immediate subtree t i of t : recursively call maximize ( t i ). 2 . for each k : calculate the maximum value z max , k ( t ) for z k ( s , t ) over all s that can be obtained by combining subtree sets from each subtree t i of t . 3 . calculate z 1 for t . if z 1 & gt ; z max , 1 ( t ) then set z max , 1 ( t ): = z 1 . if t is a leaf , then set z max , 1 ( t ): = 0 . step 2 can be further refined : 2 . 1 set y k : = 0 and z max , k ( t ): = 0 for all k , 1 ≦ k ≦ n , where n is the number of leaves in t . 2 . 2 for each subtree t ′ of t : 2 . 2 . 1 for each pair ( i , j ), 1 ≦ i ≦ p and 1 ≦ j ≦ q , where p is the number of leaves in t ′ and q is the total number of leaves in all the subtrees processed prior to t ′: if z max , i ( t ′)+ y j & gt ; z max , i + j ( t ), then set z max , i + j ( t ): = z max , i ( t ′)+ y j . if z max , k ( t ′)& gt ; z max , k ( t ), then set z max , k ( t ): = z max , k ( t ′). if z max , k ( t )& gt ; y k ( t ), then set y k ( t )= z max , k ( t ) the time complexity of the algorithm is o ( n 2 ) ( proof omitted ), where n is the number of leaves in the tree i . e . the number of haplotypes in the data set . by setting an upper limit k for the size of the subtree sets , the average time complexity can be reduced to o ( n ) with a constant coefficient proportional to k 2 , k being typically small , ≦ 10 . the straightforward algorithm for a three - level nested permutation test using nested loops would have time complexity of o ( n 3 qr ), where n is the number of permutations at each level , q is the time complexity of maximizing the z k - statistic for all k , and r is the number of tested locations in the chromosome . the test would be intractable already with rather small permutation counts . however , the time complexity can be drastically reduced using the same set of permutations at each level of the test and thus only maximizing the z k - values n instead of n 3 times for each location . 1 . compute z max , k ( t )= max z k ( t , s ) for each subtree count k and each coalescence tree t over all sεsubtreesets ( t ). 2 . randomly generate n + 1 permutations of disease - association statuses for the haplotypes and for each permutation i and ( t , k ): compute z max , k ( i , t )= max z k ( i , t , s ) over all sεsubtreesets ( t ). 3 . for each ( t , k ): 3 . 1 calculate a p value p ( t , k ) by comparing z max , k ( t ) to z max , k ( i , t ), 1 ≦ i ≦ n . 3 . 2 for each permutation i : calculate a p value p ( i , t , k ) by comparing z max , k ( i , t ) to all z max , k ( j , t ), j ≠ i . 4 . for each pair of opposed trees rooted at the same location t =( t 1 , t 2 ): 4 . 1 choose p min ( t )= min p ( t 1 , k 1 ) p ( t 2 , k 2 ) over all k 1 , k 2 4 . 2 for each permutation i : choose p min ( i , t )= min p ( i , t 1 , k 1 ) p ( i , t 2 , k 2 ) over all k 1 , k 2 . 4 . 3 calculate a p value p ( t ) by comparing p min ( t ) to p min ( i , t ), 1 ≦ i ≦ n . 4 . 4 for each permutation i : calculate a p value p ( i , t ) by comparing p min ( i , t ) to all p min ( i , t ), j ≠ i . 5 . choose p min = min p ( t ) over all t . 6 . for each permutation i : choose p min ( i )= min p ( i , t ) over all t . 7 . calculate the overall corrected p value by comparing p min to p min ( i ), 1 ≦ i ≦ n . the time complexity of steps 3 . 2 and 4 . 4 is o ( n log n ) using an algorithm which first sorts the values of the test statistic for all the permutations . step 2 predominates the time complexity of the algorithm , o ( nqr ), where s is the upper limit for the number of subtrees allowed in a set , q is the time complexity of maximizing the z k - statistic for all k , and r is the number of tested locations in the chromosome . due to the finite number of permutations , the precision of the p values given by a permutation tests may not be sufficient for accurate localization . in some situations even a very large number of permutations does not produce any values for the test statistic more extreme than the observed values for several consecutive tree locations . for this purpose the p values returned by the first and second level permutation tests are determined slightly unconventionally : at level 1 we use a slightly modified version of algorithm 2 to obtain an upper bound of z k for all k . at level 2 the smallest possible value for the test statistic is zero . these values correspond to p values of 1 / 2 ( n + 1 ). the returned p value is interpolated between the p values corresponding to the next lower and higher values for the test statistic obtained by permutations . the top - level test returning the overall p value is implemented in the usual conservative manner . certain embodiments and results of the present invention are described in the following non - limiting examples . we compare treedt empirically to tdt , an established mapping method , and to hpm , our recent proposal based on pattern discovery . we evaluate the methods on a difficult data collection carefully simulated to resemble a realistic population isolate . we designed several different test settings , with variation in the fraction ( a ) of mutation carriers in the disease - associated chromosomes , in the number of founders who introduced the mutation to the population , and in the amount of missing information . for statistical analyses , we created 100 independent artificial data sets in each test setting . great care was taken to generate realistic data by a simulation procedure that included four steps : pedigree generation , simulation of inheritance , diagnosing , and sampling . the population pedigree was set to grow from 100 to 100 , 000 individuals in a period of 20 generations . in each generation , the selection of parents for each child was random , but once a couple was formed , all subsequent children allocated to either of the parents were set to be common children of the couple . the inheritance of chromosomes within the population pedigree was simulated by first allocating a continuous chromosomal segment of 100 centimorgans to each founder individual in generation 1 . morgan is a unit of genetic length . 1 cm is the distance at which recombination occurs 1 out of every 100 times , about 10 6 base pairs . human chromosomes are roughly of 50 - 300 cm . next , the entire pedigree was traversed top - down , and , in each inheritance event , gametes were created by simulating meiosis under the assumption that the number of chiasmata in the pair of homologous chromosomes was taken from poisson distribution with parameter one ( corresponding to the genetic length of 100 cm ), and their locations selected randomly . a related approach was originally presented in ( terwilliger et al ., 1993 ). for a baseline test setting we selected a challenging disease model where only a small proportion ( a = 10 %) of the disease - associated chromosomes carries the disease - predisposing mutation , a complication that often is encountered in the analysis of common diseases . in the baseline setting there is one founder , and on average 3 . 7 % of alleles are missing , making the mapping task more difficult but also more realistic . the location of the mutation was selected randomly and independently for each of the 100 data sets produced in every setting . each data set was in turn collected from 100 affected individuals . the length of the region to be analyzed was 100 cm . allelic data were created using a map of 101 equidistantly spaced markers , each having 5 alleles . both chromosomes of each affected individual in each sample were labeled disease - associated whereas the control chromosomes were constructed from the non - transmitted alleles in the parental chromosomes . each data set thus consisted of 200 disease - associated and 200 control chromosomes first we assess the prediction accuracy of treedt with different values of a , the proportion of disease - associated chromosomes that actually carry the mutation ( fig5 a ). the results are reported as curves that show the percentage of 100 data sets where the gene is within the predicted region , as a function of the length of the predicted region . or , in other words , the x coordinate tells the cost a geneticist is willing to pay , in terms of the length of the region to be further analyzed , and the y coordinate gives the probability that the gene is within the region . for a = 20 % or 15 % the accuracy is very good , and with lower values of a the accuracy decreases until with a = 5 % only in 20 - 30 % of data sets can the gene be localized within a reasonable accuracy of 10 - 20 cm . we remind the reader that the test settings have been designed to be challenging , and to test the limits of the approach . next we evaluate the effect of the only parameter of treedt , the number of deviant subtrees that are searched for in each tree . an upper limit of 6 subtrees , used in the previous test , is evaluated against fixed amounts of 1 , 2 , or 3 subtrees , with a varying number of founders that introduced the mutation ( fig5 b ). as we increase the number of founders , evidence about the gene location becomes more fragmented , and accordingly the performace degrades . while the differences between different numbers of subtrees are not large , it is interesting to note that for each number of founders , the same number of subtrees gives marginally the best result . the upper limit of 6 subtrees gives consistently competitive results , so we continue using it in the following experiments . gene mapping studies like the ones imitated in the above tests assume , based on some other analyses , that a disease susceptibility gene is indeed present in the analyzed area . treedt has the important advantage over plain gene localization methods that it can also be used to predict whether the analyzed region contains a disease susceptibility gene at all or not . the overall p value treedt produces indicates the corrected significance of the best single finding , and by setting an upper limit for its value treedt can be used to classify data sets to ones that do or do not contain a gene . for data sets with no gene , treedt correctly produces overall p values that are uniformly distributed in [ 0 , 1 ]. so , smaller thresholds for p result in less false positives , but also in less true positives . fig5 c shows the experimental relationships between power ( ratio true positivites / all positives ) and overall p ( ratio false positives / all negatives ). for higher values of a the classification accuracy is extremely good . for a = 5 % it is comparable to random guessing , although treedt is still able to locate an existing gene adequately in 20 - 30 % of the cases ( fig5 a ). treedt , hpm , and m - tdt have practically identical performance in localizing the ds gene in the baseline setting ( fig6 a ). tdt is clearly inferior compared to the other methods . tests with other values of a give similar results . in a test setting with three founders who introduced the mutation to the population , differences between the three best methods start to appear ( fig6 b ). treedt has an edge over hpm , which in turn has an edge over m - tdt . tdt barely beats random guessing . finally , we compare the methods with a large amount of missing data ( fig6 c ). expectedly , hpm is most robust with respect to missing data since it allows gaps in its haplotype patterns . surprisingly , treedt is not much weaker than hpm , although no actions have been taken in it to account for missing or erroneus data . performance of m - tdt degrades much more clearly . method to method comparisons ( not shown ) indicate that the prediction errors are mostly caused by random effects in population history — since different methods tend to make mistakes in the same data sets — rather than by systematic differences between the methods . however , those cases where one method succeeds and another fails will give useful input for further improvements of the methods . the execution time of treedt for a single dataset is about ten minutes using 1 , 000 permutations on a 450 mhz pentium ii . the respective time for hpm with permutations is over 20 minutes . r . agrawal , t . imielinski , and a . swami . mining association rules between sets of items in large databases . in p . buneman and s . jajodia , editors , proceedings of 1993 acm sigmod conference on management of data , pp 207 - 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