Estimating Effective Population Size or Mutation Rate With Microsatellites

Corresponding author: Yun-Xin Fu, School of Public Health, University of Texas, 1200 Herman Pressler, Houston, TX 77030., yunxin.fu{at}uth.tmc.edu (E-mail)

Communicating editor: J. B. WALSH

	ABSTRACT

TOP ABSTRACT METHODS AND RESULTS APPLICATION DISCUSSION APPENDIX LITERATURE CITED

Microsatellites are short tandem repeats that are widely dispersed among eukaryotic genomes. Many of them are highly polymorphic; they have been used widely in genetic studies. Statistical properties of all measures of genetic variation at microsatellites critically depend upon the composite parameter = 4Nµ, where N is the effective population size and µ is mutation rate per locus per generation. Since mutation leads to expansion or contraction of a repeat number in a stepwise fashion, the stepwise mutation model has been widely used to study the dynamics of these loci. We developed an estimator of , _F, on the basis of sample homozygosity under the single-step stepwise mutation model. The estimator is unbiased and is much more efficient than the variance-based estimator under the single-step stepwise mutation model. It also has smaller bias and mean square error (MSE) than the variance-based estimator when the mutation follows the multistep generalized stepwise mutation model. Compared with the maximum-likelihood estimator _L by NIELSEN 1997 , _F has less bias and smaller MSE in general. _L has a slight advantage when is small, but in such a situation the bias in _L may be more of a concern.

MICROSATELLITE loci, also known as short tandem repeats, are tandem repeat loci with repeat motifs of two to six nucleotides in length (TAUTZ 1993 ). Microsatellites are highly informative as polymorphic markers. Variations at microsatellite loci have been used to study the history and genetic structure of individual populations, such as DNA fingerprinting, paternity and relatedness testing, reconstruction of evolutionary trees, and genetic distance. In addition, they are useful for inferring migration histories, for identifying individuals of unknown origin, and for detecting the hidden population substructure. Microsatellites are also widely used in linkage mapping.

Statistical properties of all measures of genetic variation critically depend upon the composite parameter = 4Nµ, where N is the effective population size and µ is the mutation rate per locus per generation. An accurate estimate of will greatly facilitate the inference on the basis of variation at microsatellite loci. While the variation at microsatellites is extremely useful, little has been done to estimate using microsatellite data. This is partly due to the unknown mutation mechanism at such loci. Microsatellite loci are hypervariable and the mechanisms that produce new variation at such loci are unusual in comparison with those of classical loci. While the exact mechanism of mutations at such loci is still not well characterized at a molecular level (JEFFREYS et al. 1994 ), it is generally believed that the processes and the patterns of mutations at different loci may differ from locus to locus, depending on the motif as well as the size of alleles at each locus. Empirical and theoretical studies indicate that for most microsatellite loci, mutations lead to stepwise changes of the repeat size of alleles although the rate of mutation leading to expansion may not be equal to that of contraction of allele size (CHAKRABORTY et al. 1997 ; DEKA et al. 1999 ). The stepwise mutation model, originally proposed for the study of protein charge changes (OHTA and KIMURA 1973 ), in a more generalized form may be more suitable for the study of most microsatellite loci (KIMMEL et al. 1996 ).

Although a number of estimators of (WEHRHAHN 1975 ; NIELSEN 1997 ; FU and CHAKRABORTY 1998 ) use microsatellite data, each has its limitations, in part being either too complicated or too simple. There is need for a relatively simple yet robust estimator and the purpose of this article is to develop one such estimator of using microsatellite data. Here we assume the neutral Wright-Fisher model without population substructure. The estimation of becomes the estimation of effective population size, N, when the mutation rate, µ, is known or the estimation of mutation rate, µ, when the effective population size, N, is known.

	METHODS AND RESULTS

TOP ABSTRACT METHODS AND RESULTS APPLICATION DISCUSSION APPENDIX LITERATURE CITED

Existing estimators:
Assuming the single-step stepwise mutation model, in which each mutation produces either one-step contraction or expansion in allele size, for a population without substructure and a neutral locus, the variance in allele size from a sample, V_s, has a mean equal to /2 (WEHRHAHN 1975 ). Then a convenient unbiased moment estimator is given by

(1)

The estimator _V is rather simple, but the price of its simplicity is a large variance. The variance of allele size variance, V_s, was given by ZHIVOTOVSKY and FELDMAN 1995 as

(2)

Consequently, the variance of _V is given by

(3)

Several examples of the value of _V are shown in Table 1. In general the standard deviation is >.

View this table: In this window In a new window   Table 1. Large sample variance of estimator V

An even better known quantity is heterozygosity, denoted as H and defined as the probability that two randomly chosen sequences are of different allelic type; it is a measure of genetic variation at a microsatellite locus. The complement of heterozygosity, F = 1 - H, is called homozygosity. Since F contains the information of both number of alleles and allele frequency, an estimator based on F may be a possible solution.

Under the single-step stepwise mutation model, for a population without substructure and a neutral locus, the expected homozygosity (OHTA and KIMURA 1973 ) is given by

(4)

Supposing a sample is taken from a population and letting k be the number of alleles in the sample, the homozygosity F can be estimated by

(5)

where p_i is the allele frequency of the ith allele in the sample. Then a moment estimator of can be derived from Equation 4, replacing F with :

(6)

Since the transformation is not linear, the estimator _F is usually biased, particularly when is large. Simple correction based on the infinite allele model was proposed before (ZOUROS 1979 ; CHAKRABORTY and WEISS 1991 ), which is based on an analytical relationship between the expected value of the estimator and real value of . Unfortunately, such an analytical formula is not yet known for genetic loci evolving under the stepwise mutation model.

Besides the two estimators of using microsatellite data, NIELSEN 1997 proposed an estimator using the maximum-likelihood approach. In addition to being restricted to the single-step stepwise mutation model, the estimator is rather demanding computationally and can handle only modest sample size. FU and CHAKRABORTY 1998 proposed an approach to simultaneously estimate all the parameters in a generalized stepwise mutation model, including . They use a minimum chi-square method to perform a grid search of all the possible values in the multidimensional parameter space, which makes it a challenge to analyze a large amount of data. To date, many population studies using microsatellites involve larger and larger samples and multiple loci. A relatively simple yet efficient estimator is highly desirable. In many ways, such an estimator can serve a role similar to that of Watterson's or Tajima's estimator of for DNA sequence data, despite the fact that several sophisticated estimators of for DNA sequence data have been available.

New estimator:
The approach we take uses a combination of computer simulation and statistical regression, trying to find the relationship between the expectation of _F and the real value of . On the basis of the relationship, we try to develop a new unbiased estimator of . Computer simulation is an efficient way to study the properties of the homozygosity-based estimator _F . For each combination of value and sample size, n, a large number of samples are simulated according to coalescent theory. For each sample, the homozygosity is estimated through Equation 5. Then the homozygosity-based estimate is obtained through Equation 6. Some of the results are shown in Fig 1, where each point in the figure is the mean of _F over 50,000 simulated samples. Fig 1 shows that _F on average overestimates . The magnitude of overestimation is a function of sample size n and , and, in many cases, the biases are severe.

View larger version (33K): In this window In a new window Download PPT slide   Figure 1. Relationship and regression of , sample size n, and mean of F. Each dot is the mean of F over 50,000 simulated samples and curves are the regression equations. The number on the right side of each curve is the value for simulating the samples upon which the mean F is taken.

To summarize the relationship among , n, and the mean of _F, a regression approach can be used. The challenge is to find the simplest equation that is sufficiently accurate for describing the relationship. From Fig 1, it seems that mean of _F is reversely related to sample size and positively proportional to . We include the terms 1/n and in the regression formula. We started to consider equations that incorporate 1/n and in various ways. Choosing as the basic unit was partly inspired by Equation 4. The most complex equation we consider is a polynomial including all combinations of 1/n, , and (1/n)².

The regression analysis shows that two regression equations summarize remarkably well (R² = 99.99%) the relationship of , n, and mean of _F (see Fig 1). For 10,

(7)

For > 10,

(8)

The regression equations have two nice properties. First, when _F = 0, we have = 0. Second, when sample size n , _F has a limit value, which does not depend on n. Actually, when n > 200, the effect of sample size is very small.

On the basis of the above regression equations, we propose the following new estimator _F:

The threshold value 15 is based on the observation that 90% of the value of _F is <15.0 with = 10. However, we found that the choice is not critical, because choosing 10 as the threshold value does not make much difference. This is because when is 10, Equation 7 and Equation 8 give very similar results.

The performance of _F was investigated through simulation. For a given combination of and sample size n, 50,000 samples were simulated and for each sample _F was estimated by Equation 6 and then corrected through Equation 7 or Equation 8. Some of the results are summarized in Table 2. Table 2 shows that the estimator is unbiased (or nearly so). The small bias is likely due to fluctuation in simulation and is insignificant compared to the variance.

View this table: In this window In a new window   Table 2. Properties of F

Next we compare the performance of our estimator _F with that of the estimator based on allele size variance, _V. There are two ways to compute the variance of _V. The theoretical value of the large sample variance can be computed through Equation 3 and the variance can also be estimated through computer simulation. We computed it in both ways because on the one hand the validity of our simulation program can be checked and on the other hand the results can corroborate each other. The results are summarized in Table 3. Table 3 shows that the theoretical value of the variance of _V agrees well with the simulation value, which indicates that our simulation is accurate. More importantly, Table 3 shows that while both estimators are unbiased, our homozygosity-based estimator _F is better than the size variance-based estimator _V in that the variance of is smaller than that of _V. The relative efficiency of _F against _V, defined as the ratio of the variance of _V and variance of _F, is also given in Table 3. The relative efficiency increases as increases, which means that _F becomes more and more efficient with increasing value. Note that since microsatellite loci have a relatively high mutation rate, the value can easily be of the range of 10–100, which makes _F superior to _V for most microsatellite loci.

View this table: In this window In a new window   Table 3. Comparison of F and V

Comparison with the maximum-likelihood estimator:
The performance of the homozygosity-based estimator _F is further compared to that of the maximum-likelihood (ML) estimator _L proposed by NIELSEN 1997 . Assuming the single-step stepwise mutation model, 10,000 samples are simulated for a number of combinations of and sample size. The two estimators, _F and _L, are computed for each simulated sample. The mean value and mean square error (MSE) for the corresponding estimates are then computed and the results are summarized in Table 4. Two conclusions are obvious from Table 4. First, the ML estimator _L is, in general, upwardly biased. Although the bias decreases with sample size, it is still appreciable even when the sample size is 300. In comparison, the mean value of the homozygosity-based estimator _F exhibits little bias, similar to the case of comparing _F and _V. Second, in general the ML estimator _L has a larger MSE than that of _F, except in the cases where is small and sample size is large. It is somehow surprising that as increases, the relative performance of _L, measured by MSE, gets worse compared to _F. Two possible causes might be that the ML estimator implemented by Nielsen may not be a true ML estimator and it is not efficient. Indeed, in Nielsen's algorithm, a k-allele model was used to approximate the stepwise mutation model (NIELSEN 1997 ) in which the accuracy is not well known. Because of a high mutation rate for microsatellites, the value can be quite large even for a modest population size. For example, many samples from human populations have yielded estimates of > 10. This makes _F more preferable in general than _L.

View this table: In this window In a new window   Table 4. Comparison of F and L under various combinations of and sample size (n)

To address the issue of efficiency, we performed a large-scale simulation to see the extent to which performance of the ML estimator is affected by the number of runs through the Markov chain. In the comparison with the ML estimator _L shown in Table 4, the _L was computed using the default Markov chain steps, 100,000 runs. Table 5 shows the results with three different numbers of runs through the Markov chain, 10,000, 100,000 and 1,000,000, where is set to 10.0. It is clear from Table 5 that there is a big improvement in the performance of _L in terms of MSE when the number of runs through the Markov chain changes from 10,000 to 100,000, but only a small improvement when the replicate number changes from 100,000 to 1,000,000. More importantly, even when 1,000,000 replicates were used for the _L, it still has larger bias and MSE than the homozygosity-based estimator _F when = 10.0. An extreme case was carried out in which the number of runs through the Markov chain for _L was set to 10,000,000 when = 10.0 and sample size n = 50. In this case, the MSE of _L was 69.53, which is still >50.62, the MSE of _F.

View this table: In this window In a new window   Table 5. Comparison of F and L when = 100 for different numbers of runs through the Markov chain

Robustness of the estimator:
So far, the analysis is based on the single-step stepwise mutation model. While this may be true for some microsatellite loci, statistical analysis suggests that not all of them adhere to this simple version of the stepwise mutation model (SHRIVER et al. 1993 ; DI RIENZO et al. 1994 ). Furthermore, direct mutation assays at several loci showed that occasionally mutation may lead to jumps of allele sizes beyond one repeat unit (WEBER and WONG 1993 ). On the basis of these lines of evidence, a generalized version of the stepwise mutation model (KIMMEL and CHAKRABORTY 1996 ; FU and CHAKRABORTY 1998 ) was proposed in which each mutation is supposed to change the allele size from X to X + U. The mutation is symmetric and the absolute value of the offset U is sampled from a geometric distribution with parameter ; that is,

(9)

The performance of both estimators under this generalized stepwise mutation model was investigated through computer simulation. A total of 50,000 samples were simulated assuming the generalized model with = 0.67. With this value,

That is, on average each mutation causes a jump of allele sizes of 1.5 repeat units. For each simulated sample, the sample procedure as before was taken to obtain the two estimators, _F and _V. The bias and MSE were also taken for each estimator. The corresponding theoretical values for the bias and MSE of _V were also computed. The details are in the Appendix. The simulation value agrees well with the theoretical value. The results are shown in Table 6.

View this table: In this window In a new window   Table 6. Comparison of F and V under the generalized model

Table 6 shows that under the generalized stepwise mutation model, both estimators are upwardly biased. That is, both estimators on average overestimate the real value. The bias is an increasing function of . When the bias of _F is compared to that of _V, the former always has a smaller bias than the latter, which means that _F is less biased than _V especially when is high. Comparison between the corresponding MSEs also shows that _F has a smaller MSE than _V. These two points make _F still more preferable than _V even when the actual mutation model is the generalized stepwise mutation model.

	APPLICATION

TOP ABSTRACT METHODS AND RESULTS APPLICATION DISCUSSION APPENDIX LITERATURE CITED

To test the performance of the homozygosity-based estimator _F with real data, we use the allele frequency data from the ALFRED database at Yale University (CHEUNG et al. 2000 ). There are altogether 115 dinucleotide repeats with data from 10 worldwide populations. The 10 populations are Biaka, Mbuti, Druze, Danes, Han, Japanese, Melanesian-Nasioi, Yakut, Maya-Yucatan, and Surui. More information about the loci and populations can be found at http://alfred.med.yale.edu/alfred/index.asp.

For each population-locus combination, _F and _V are computed. To compare the consistency of the estimators, one locus is randomly chosen as the base locus and the ratio of the estimate for other loci in the same population is taken over the estimate for the base locus. Since the effective population size is generally supposed to be the same in the same population for all loci from the same sample, we are estimating the ratio of mutation rates using information from different populations. Assuming the mutation rate for a particular locus is constant across the populations, the estimates of the ratio of mutation rates from different populations are the estimates of the same quantity. Consequently, the dispersion of the results is an indicator of the consistency of the estimator. The coefficient of variance (ratio of standard deviation to mean) is taken as a measure of dispersion. In almost all the cases, the coefficient of variance is smaller with _F than with _V, which indicates that the homozygosity-based estimator _F is more stable and more consistent than the variance-based estimator _V. Examples of the results from four loci are tabulated in Table 7, where the base locus (locus 1) is D11S935, locus 2 is D7S640, locus 3 is D6S441, and locus 4 is D5S408, with the corresponding mutation rates denoted as µ₁–µ₄, respectively.

View this table: In this window In a new window   Table 7. Comparison of estimates of ratio of mutation rates with F and V

	DISCUSSION

TOP ABSTRACT METHODS AND RESULTS APPLICATION DISCUSSION APPENDIX LITERATURE CITED

KIMMEL and CHAKRABORTY 1996 showed that sample homozygosity at a microsatellite locus depends not only on , but also on the pattern of allele size change caused by mutation. Therefore, any attempt to estimate on the basis of homozygosity has to be mutation model dependent. Interestingly, the regression formula we found on the basis of the single-step stepwise mutation model is reasonably robust against deviations from the single-step model. This is a useful property since it is very difficult to specify the model with confidence. On the other hand, if one has sufficient confidence in a particular model, a similar approach can be used to derive the regression formula under the model. This can be seen from our simulation study when the mutation model deviates from the single-step stepwise mutation model to the generalized stepwise mutation model.

Although the maximum-likelihood estimator, _L, proposed by NIELSEN 1997 is computationally demanding, its performance was compared to that of the homozygosity-based estimator _F through a large-scale simulation. The ML estimator _L is found to be slightly upwardly biased. This is not too surprising because many maximum-likelihood estimators are known to be biased for small sample sizes. Indeed we found that the _L approaches the true value as sample size (n) increases. However, even when n = 300, there is still an appreciable amount of bias. The MSE of _L decreases with the increase of the sample size. However, in the most likely range of for microsatellites, _L has in general larger MSE than _F unless the sample size is extremely large. _L has a slight advantage when is small. However, in such a situation, the bias of _L may be more of a concern. For example, from Table 4 when = 2.0 and n = 30, the bias can be nearly 18%. All these factors make _F an attractive alternative to _L.

We have relied on regression to find a way to remove bias as an estimator of from _F. It should be pointed out that jackknife is a widely used approach to reduce bias in estimation (e.g., MANLY 1997 ). The underlying theory is that a jackknife estimator removes the bias of order 1/n; that is, if the original biased estimate has the form

(10)

where A is a constant, then the jackknife estimator can remove the bias. However, the relationship between E(_F) and is rather complex. Although the exact relationship is unknown, Equation 7 and Equation 8 indicate that the relationship is certainly not in the form of Equation 10. So the jackknife estimator is unlikely to be able to remove much of the bias in _F. Indeed, when the jackknife method was applied in our simulated sample, we found that it was able to remove only 10% of the bias in many combinations of parameters. Therefore, jackknife is not an appropriate approach to use in this situation.

From Equation 5 of KIMMEL and CHAKRABORTY 1996 , the estimator based on allele size variance under any arbitrary stepwise mutation model is given by

(11)

where V = 2E(V_s) and U₀ is the symmetrized allele size change in a single generation and is mutation model dependent. Consequently, the variance-based estimator _V is mutation model dependent and is applicable to the particular model itself. In the case of the single-step stepwise mutation model, Equation 11 is reduced to Equation 1 since . Therefore, _V is a special case of _V and is mutation model dependent and applicable to the single-step stepwise mutation model. Hence it is no surprise that _V becomes biased under the generalized stepwise mutation model.

RUBINSZTEIN et al. 1995 argued that the mutational transitions may be asymmetric. During the analysis in this article we did not differentiate the asymmetric model from the symmetric model. This is because from KIMMEL and CHAKRABORTY 1996 homozygosity and allele size variance are independent of mutation direction. Indeed, these are confirmed in our simulation (data not shown). Consequently, our homozygosity-based estimator _F is applicable for single-step stepwise mutation, symmetric or not. Computer programs to carry out the analysis and to estimate _F are available upon request.

	ACKNOWLEDGMENTS

We thank R. Nielsen for sharing his ML program. This work was supported partly by National Institutes of Health grants R01 GM50428 and R01 GM60777 to Y.-X. Fu.

Manuscript received May 30, 2003; Accepted for publication September 24, 2003.

	APPENDIX

TOP ABSTRACT METHODS AND RESULTS APPLICATION DISCUSSION APPENDIX LITERATURE CITED

CALCULATING THE BIAS AND MSE OF _V UNDER THE GENERALIZED STEPWISE MUTATION MODEL
Given that

that is, |U| geometric(0.67), since the mutation is symmetric, U₀ = U, we have

(A1)

Since _V = 2V_s and from Equation 4 of KIMMEL and CHAKRABORTY 1996 we have

(A2)

where V is defined in KIMMEL and CHAKRABORTY 1996 ,

(A3)

Substituting = 0.67 into Equation A3, we have

(A4)

Therefore,

(A5)

To calculate the MSE of _V we need to calculate variance of size variance V first. From Equation 16 of KIMMEL and CHAKRABORTY 1996 ,

(A6)

Since U₀ = U and U geometric(0.67), the moment-generating function of U₀ is

(A7)

Taking the fourth derivative of Equation A7 and setting t = 1, = 0.67, we have

(A8)

From Equation 5 of KIMMEL and CHAKRABORTY 1996 , we have

(A9)

Substituting Equation A8, and Equation A9 into Equation A6, we have

(A10)

Since _V = 2V_s, we have

(A11)

Therefore,

(A12)

	LITERATURE CITED

TOP ABSTRACT METHODS AND RESULTS APPLICATION DISCUSSION APPENDIX LITERATURE CITED

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