BACKGROUND
human genome is a result of  <dig> years of evolution.
it is very complex and in some respects it is still evolving.
this contribution concerns evolution of the so-called alu elements, which are movable sequences of dna, very abundant in the human genome.
we present a mathematical random process, the griffiths-pakes discrete-time branching process with infinite-allele mutations, which is almost ideally suited for modeling of alu elements proliferation.
for the biologically important special case of the linear-fractional offspring distribution we derive semi-explicit expressions for the expected frequency spectra of classes of alleles existing in a given number of copies .
we compare the outcome with alu-element statistics data.
alu repeat sequences
background on alus
alu elements belong to the group of transposable or mobile elements, which occupy nearly 45% of the human genome  <cit> .
within this group of transposable and also highly repetitive elements, lines  and sines  form the two largest groups.
they occupy 21% and 13% of the human genome respectively  <cit> .
whereas the lines are dominated by l <dig> elements, the largest and hence most studied group of the sines is comprised of the alu elements.
while many transposable elements are present in all eukaryotic genomes, alu elements appear only in mammals.
a typical full-length alu sequence is approximately  <dig> bp long.
alu sequences amplify by retrotransposition, also known as "the copy and paste" mechanism.
at present it is estimated that more than one million copies of alu elements occupy about eleven percent of the human genome, and the number of elements seems to be growing  <cit> .
alu elements are non-autonomous and seem to have to use the l <dig> elements' tools for retrotransposition.
it has been hypothesized that l <dig> endonuclease causes a nick at the ttaaaa consensus site, after which alu anneals directly to the site of integration  <cit> ; then a second nick on the other strand completes the insertion.
these two staggered nicks introduce an identifiable characteristic of alu elements.
the newly inserted alu element is surrounded by an identical set of direct repeats, which are also called target site duplications .
these direct repeats range from  <dig> to  <dig> bp and are considered the prevalent feature of retrotranspositional insertion  <cit> .
this process of integration, also known as target-primed reverse transcription   <cit> , is responsible for the successful amplification of alu elements.
at present it is estimated that more than one million copies of alu elements occupy about eleven percent of the human genome, and the number of elements seems to be growing  <cit> .
based on diagnostic mutations, alu elements are divided into subfamilies.
the three major families of alu sequences are j, s and y. the letters are chosen in alphabetical order to convey the different ages of each family.
alu sequences in the j family are the oldest, while alu sequences in the y family are the youngest.
the most interesting family in the current research of alu elements is the y family, which contains the youngest and most active alu elements  <cit> .
due to their recent integration,  <dig> percent of their loci are still polymorphic  <cit> .
an alu locus is defined to be polymorphic if some individuals have an alu element at that particular location while others do not.
these polymorphic loci can be used as genetic markers for disease association studies.
unlike single-nucleotide polymorphisms  alu markers are small in numbers, but they are identical by descent and essentially homoplasy-free markers and their ancestral state, which is defined by their absence from a specific locus, is always known.
polymorphic alu loci have been used in genetic diversity studies, forensic studies and disease association studies  <cit> .
alu insertions have influenced the architecture of human genome by duplication, deletion, inversion, transduction and translocation  <cit> .
alu elements frequently appear in introns, 3' untranslated regions of genes, and intergenic genomic regions  <cit> .
alu insertions act as insertional mutagens and are responsible for  <dig>  percent of human genetic disorders  <cit> .
almost all these diseases are caused by alu elements from the youngest subfamilies  <cit> .
for a comprehensive list of aluy disease loci and their associated diseases, one can also consult  <cit> .
their summary of alu insertion induced diseases includes neurofibromatosis, hemophilia a and b, huntington disease and apert syndrome.
deiniger and batzer  <cit>  attribute diseases such as insulin-resistant diabetes type ii, lesch-nyhan syndrome, tay-sachs disease, familial hypercholesterolaemia and -thalassaemia to alu-mediated recombination.
additionally, several types of cancer, including ewing sarcoma, breast cancer and leukemia are shown to be caused by alu elements  <cit> .
alu sequence data used in this study
dr. jerzy jurka of the genetic information research institute  kindly provided alu sequence data for our analysis.
all alu subfamilies were extracted from the march  <dig> assembly of the uscs human genome database.
only recognizable full-length alu sequences were retained for analysis.
overall, alu sequences for nine different alu subfamilies were extracted from the uscs reference genome: aluya <dig>  aluya <dig>  aluya <dig>  aluya <dig>  aluyb <dig>  aluyc <dig>  aluyd <dig>  aluye <dig>  and aluye <dig>
the goal was to extract alu sequences that belonged to relatively large subfamilies , such as aluya <dig>  aluya <dig>  aluyb <dig>  aluyc <dig>  and aluye <dig>  for each subfamily, a consensus or reference alu sequence was used to screen the entire human genome for matching sequences.
a match occurred when stretches of nucleotides that include the main diagnostic mutations agreed with the alu subfamily consensus sequences.
since the insertion mechanism of an alu element introduces large differences in their poly-a tails, these need to be deleted from analysis.
dr. jurka provided the alu sequence data with poly-a tails already deleted.
alu sequences contain the middle a-stretch, another highly variable region similar to the poly-a tail, which lies between the two monomers that constitute an alu sequence, and can be considered the a-tail of the first monomer.
to accurately delete the middle-a stretch, it is necessary to align the alu sequences for each subfamily.
a consensus sequence for each subfamily was obtained from repbase  <cit> , a database of repetitive elements, which is maintained by giri.
in each subfamily, pairwise alignment of each alu sequence in the subfamily with the repbase consensus sequences, was performed using clustalw  <cit> .
mega <dig> software  <cit>  was used to display the alignments including the middle-a stretch.
after deleting the middle-a stretch, the average length of an alu sequence is about  <dig> base pairs.
following preparatory steps described above, we obtained the counts of alu sequences that had n identical copies in the sample, for n =  <dig>   <dig>   <dig> .... to obtain these counts for each alu subfamily, a program was written in r-language.
these counts or corresponding percentages represent final data, which were tested against the theoretical distribution based on the branching process model.
RESULTS
maximum-likelihood fits
to fit the branching process model to the alu sequence data, we use the maximum likelihood method.
the highest value of the likelihood determines the estimates for our parameters.
since the log-likelihood of the sample does not exist in a closed form, we evaluate it numerically.
for these runs we set the value of the probability of mutation at μ = 10- <dig>  sensitivity of the outcome to variation in parameter μ is very slight as long as this parameter is small .
figures  <dig>   <dig>   <dig>   <dig> depict the maximum-likelihood fits of the model to the data from aluya <dig>  aluya <dig>  aluyb <dig> and aluyc <dig> subfamilies, respectively.
they are presented in the semi-logarithmic scale, to amplify the tail probabilities.
the graphical comparison demonstrates that the data fit relatively well for allele classes  <dig> and  <dig> -  <dig>  notably, the allele class  <dig> shows the worst fit among the first seven allele classes.
these seven classes account for at least  <dig>  cumulative class frequency observed in the data.
simulation-based test
testing for significance of the differences between the theoretical and observed frequencies of allele classes is in our case made difficult by the dominance of class  <dig> frequency.
this causes that, with finite sample sizes, counts of alleles in the further classes are quite low .
therefore the usual restrictions for minimum number of observations and minimum number of classes in the χ <dig> test  are met only in two data sets.
for this reason, we resort to a simulation-based approach.
in this approach we use the χ <dig> statistic as our test statistic.
we start with randomly drawing a sample of size n from the uniform distribution u, where n is equal to the number of alu sequences in each subfamily.
based on the distribution {Ψj} produced by our fitted model we obtain the probabilities of alu sequences with j =  <dig>   <dig>   <dig> ... copies.
these probabilities determine the bin a random draw from the uniform distribution will be placed in.
repeating this n times results in a distribution of counts per bin.
from this simulated distribution of counts we compute the χ <dig> statistic by using the expected counts under our fitted model and the counts from the simulated  approach.
for each alu subfamily, this process was repeated  <dig>   <dig> times.
the χ <dig> values were sorted and then plotted to display their distribution.
when comparing the χ2statistic from the actual data to the simulated χ <dig> statistics for each subfamily , it becomes apparent that the data produce a very high χ <dig> value, which is highly unlikely under the proposed model.
the two alu subfamilies that have χ <dig> values less than the maximum simulated χ <dig> values are aluya <dig> and aluyd <dig>  a closer look, however, reveals that both of the χ <dig> values for the aluya <dig> and aluyd <dig> subfamilies are among the ten highest values in the sample of  <dig>   <dig> simulated χ <dig> values .
the large χ <dig> statistic for our data is mostly due to the difference between the observed and the expected counts in bin number  <dig>  and in the combined bins of the tail.
we notice it as a systematic departure.
for each subfamily  <dig>   <dig> simulations were performed.
CONCLUSIONS
the current study seems to constitute the first application of the griffiths-pakes process to biological data.
the outcome is interesting in the sense that a generally plausible fit is obtained to the alu element frequency distribution.
it is not quite clear, why the fit fails worst at the frequency class  <dig>  this may be influenced by initial steps of data preparation.
if a region containing a a relatively frequent variant were removed so that sequences could be aligned, some unique variants might migrate to class  <dig>  another possibility is that the difference is caused by a departure from neutrality in alu evolution.
we should notice that the current model does not involve genetic drift.
in reality, the genomes evolve within individuals and properly, the branching process should have been embedded in a population genetic model of wright-fisher or moran type.
however, this would lead to enormous complications.
as an additional exercise, we attempted to fit the alu class frequencies by the classical ewens sampling formula, using a test developed by slatkin  <cit> , but the fit is rather bad.

 human genome is a result of years of evolution. it is very complex and in some respects it is still evolving. this contribution concerns evolution of the so-called alu elements, which are movable sequences of dna, very abundant in the human genome. we present a mathematical random process, the griffiths-pakes discrete-time branching process with infinite-allele mutations, which is almost ideally suited for modeling of alu elements proliferation. for the biologically important special case of the linear-fractional offspring distribution we derive semi-explicit expressions for the expected frequency spectra of classes of alleles existing in a given number of copies. we compare the outcome with alu-element statistics data. alu repeat sequences background on alus alu elements belong to the group of transposable or mobile elements, which occupy nearly 45% of the human genome. within this group of transposable and also highly repetitive elements, lines and sines form the two largest groups. they occupy 21% and 13% of the human genome respectively. whereas the lines are dominated by l elements, the largest and hence most studied group of the sines is comprised of the alu elements. while many transposable elements are present in all eukaryotic genomes, alu elements appear only in mammals. a typical full-length alu sequence is approximately bp long. alu sequences amplify by retrotransposition, also known as "the copy and paste" mechanism. at present it is estimated that more than one million copies of alu elements occupy about eleven percent of the human genome, and the number of elements seems to be growing. alu elements are non-autonomous and seem to have to use the l elements' tools for retrotransposition. it has been hypothesized that l endonuclease causes a nick at the ttaaaa consensus site, after which alu anneals directly to the site of integration; then a second nick on the other strand completes the insertion. these two staggered nicks introduce an identifiable characteristic of alu elements. the newly inserted alu element is surrounded by an identical set of direct repeats, which are also called target site duplications. these direct repeats range from to bp and are considered the prevalent feature of retrotranspositional insertion. this process of integration, also known as target-primed reverse transcription, is responsible for the successful amplification of alu elements. at present it is estimated that more than one million copies of alu elements occupy about eleven percent of the human genome, and the number of elements seems to be growing. based on diagnostic mutations, alu elements are divided into subfamilies. the three major families of alu sequences are j, s and y. the letters are chosen in alphabetical order to convey the different ages of each family. alu sequences in the j family are the oldest, while alu sequences in the y family are the youngest. the most interesting family in the current research of alu elements is the y family, which contains the youngest and most active alu elements. due to their recent integration, percent of their loci are still polymorphic. an alu locus is defined to be polymorphic if some individuals have an alu element at that particular location while others do not. these polymorphic loci can be used as genetic markers for disease association studies. unlike single-nucleotide polymorphisms alu markers are small in numbers, but they are identical by descent and essentially homoplasy-free markers and their ancestral state, which is defined by their absence from a specific locus, is always known. polymorphic alu loci have been used in genetic diversity studies, forensic studies and disease association studies. alu insertions have influenced the architecture of human genome by duplication, deletion, inversion, transduction and translocation. alu elements frequently appear in introns, 3' untranslated regions of genes, and intergenic genomic regions. alu insertions act as insertional mutagens and are responsible for percent of human genetic disorders. almost all these diseases are caused by alu elements from the youngest subfamilies. for a comprehensive list of aluy disease loci and their associated diseases, one can also consult. their summary of alu insertion induced diseases includes neurofibromatosis, hemophilia a and b, huntington disease and apert syndrome. deiniger and batzer attribute diseases such as insulin-resistant diabetes type ii, lesch-nyhan syndrome, tay-sachs disease, familial hypercholesterolaemia and -thalassaemia to alu-mediated recombination. additionally, several types of cancer, including ewing sarcoma, breast cancer and leukemia are shown to be caused by alu elements. alu sequence data used in this study dr. jerzy jurka of the genetic information research institute kindly provided alu sequence data for our analysis. all alu subfamilies were extracted from the march assembly of the uscs human genome database. only recognizable full-length alu sequences were retained for analysis. overall, alu sequences for nine different alu subfamilies were extracted from the uscs reference genome: aluya aluya aluya aluya aluyb aluyc aluyd aluye and aluye the goal was to extract alu sequences that belonged to relatively large subfamilies, such as aluya aluya aluyb aluyc and aluye for each subfamily, a consensus or reference alu sequence was used to screen the entire human genome for matching sequences. a match occurred when stretches of nucleotides that include the main diagnostic mutations agreed with the alu subfamily consensus sequences. since the insertion mechanism of an alu element introduces large differences in their poly-a tails, these need to be deleted from analysis. dr. jurka provided the alu sequence data with poly-a tails already deleted. alu sequences contain the middle a-stretch, another highly variable region similar to the poly-a tail, which lies between the two monomers that constitute an alu sequence, and can be considered the a-tail of the first monomer. to accurately delete the middle-a stretch, it is necessary to align the alu sequences for each subfamily. a consensus sequence for each subfamily was obtained from repbase, a database of repetitive elements, which is maintained by giri. in each subfamily, pairwise alignment of each alu sequence in the subfamily with the repbase consensus sequences, was performed using clustalw. mega software was used to display the alignments including the middle-a stretch. after deleting the middle-a stretch, the average length of an alu sequence is about base pairs. following preparatory steps described above, we obtained the counts of alu sequences that had n identical copies in the sample, for n =.... to obtain these counts for each alu subfamily, a program was written in r-language. these counts or corresponding percentages represent final data, which were tested against the theoretical distribution based on the branching process model.  maximum-likelihood fits to fit the branching process model to the alu sequence data, we use the maximum likelihood method. the highest value of the likelihood determines the estimates for our parameters. since the log-likelihood of the sample does not exist in a closed form, we evaluate it numerically. for these runs we set the value of the probability of mutation at μ = 10- sensitivity of the outcome to variation in parameter μ is very slight as long as this parameter is small. figures depict the maximum-likelihood fits of the model to the data from aluya aluya aluyb and aluyc subfamilies, respectively. they are presented in the semi-logarithmic scale, to amplify the tail probabilities. the graphical comparison demonstrates that the data fit relatively well for allele classes and - notably, the allele class shows the worst fit among the first seven allele classes. these seven classes account for at least cumulative class frequency observed in the data. simulation-based test testing for significance of the differences between the theoretical and observed frequencies of allele classes is in our case made difficult by the dominance of class frequency. this causes that, with finite sample sizes, counts of alleles in the further classes are quite low. therefore the usual restrictions for minimum number of observations and minimum number of classes in the χ test are met only in two data sets. for this reason, we resort to a simulation-based approach. in this approach we use the χ statistic as our test statistic. we start with randomly drawing a sample of size n from the uniform distribution u, where n is equal to the number of alu sequences in each subfamily. based on the distribution {Ψj} produced by our fitted model we obtain the probabilities of alu sequences with j =... copies. these probabilities determine the bin a random draw from the uniform distribution will be placed in. repeating this n times results in a distribution of counts per bin. from this simulated distribution of counts we compute the χ statistic by using the expected counts under our fitted model and the counts from the simulated approach. for each alu subfamily, this process was repeated times. the χ values were sorted and then plotted to display their distribution. when comparing the χ2statistic from the actual data to the simulated χ statistics for each subfamily, it becomes apparent that the data produce a very high χ value, which is highly unlikely under the proposed model. the two alu subfamilies that have χ values less than the maximum simulated χ values are aluya and aluyd a closer look, however, reveals that both of the χ values for the aluya and aluyd subfamilies are among the ten highest values in the sample of simulated χ values. the large χ statistic for our data is mostly due to the difference between the observed and the expected counts in bin number and in the combined bins of the tail. we notice it as a systematic departure. for each subfamily simulations were performed.  the current study seems to constitute the first application of the griffiths-pakes process to biological data. the outcome is interesting in the sense that a generally plausible fit is obtained to the alu element frequency distribution. it is not quite clear, why the fit fails worst at the frequency class this may be influenced by initial steps of data preparation. if a region containing a a relatively frequent variant were removed so that sequences could be aligned, some unique variants might migrate to class another possibility is that the difference is caused by a departure from neutrality in alu evolution. we should notice that the current model does not involve genetic drift. in reality, the genomes evolve within individuals and properly, the branching process should have been embedded in a population genetic model of wright-fisher or moran type. however, this would lead to enormous complications. as an additional exercise, we attempted to fit the alu class frequencies by the classical ewens sampling formula, using a test developed by slatkin, but the fit is rather bad. 
alu elements occupy about eleven percent of the human genome and are still growing in copy numbers. a comparison of the alu sequence data, obtained by courtesy of dr. jerzy jurka, with our model shows that the distributions of alu sequences in the aluy family systematically deviate from the expected distribution derived from the branching process. this observation suggests that alu sequences do not evolve neutrally and might be under selection. since alu elements substantially impact the shape of our genome, there is a need for modeling the amplification, mutation and selection forces of these elements.
 alu elements occupy about eleven percent of the human genome and are still growing in copy numbers. since alu elements substantially impact the shape of our genome, there is a need for modeling the amplification, mutation and selection forces of these elements. methods our proposed theoretical neutral model follows a discrete-time branching process described by griffiths and pakes. from this model, we derive a limit frequency spectrum of the alu element distribution, which serves as the theoretical, neutral frequency to which real alu insertion data can be compared through statistical goodness of fit tests. departures from the neutral frequency spectrum may indicate selection.  a comparison of the alu sequence data, obtained by courtesy of dr. jerzy jurka, with our model shows that the distributions of alu sequences in the aluy family systematically deviate from the expected distribution derived from the branching process.  this observation suggests that alu sequences do not evolve neutrally and might be under selection. 20– november international workshop on computational systems biology approaches to analysis of genome complexity and regulatory gene networks singapore 