Mutual information between DNA bulk profiles and germplasm sources

Bulk DNA fingerprinting in its simplest form is the conversion of allele frequencies into a binary code resulting from band present-band absent scoring, through the processing of DNA samples from a mixture of plants. We expect that only those allele frequencies above a certain threshold will be likely to produce bands. Thus, the presence of a band indicates that a given allele is present in the sampled population, whereas its absence can be interpreted as absence of the given allele or a lower than threshold frequency in the reference population. From the mathematical point of view, we want to maximize the average information gained about the identity of a given germplasm source in the space of populations, with the knowledge of the band profile of the corresponding bulk. In this work, average information will be based on the foundations of information theory [24]. Let be a random variable with possible values and probabilities , respectively. The Shannon entropy of is defined as follows:(1)

The expression equals 0 by definition. This concept leads to the definition of mutual information, i.e. the information conveyed about a random variable by a variable , as the average reduction of the uncertainty or entropy of given knowledge of the value of the variable [25]:(2)where is the average Shannon entropy of , given knowledge of the value of the variable . In the context of this work denotes marker genotypes whereas represents germplasm sources. Mutual information in equation (2) can be interpreted as the expected reduction in the entropy or uncertainty about the identity of a random population caused by the knowledge of its marker genotype . Mutual information is symmetrically defined in terms of entropies; in fact, as we see in in equation (2), it can also be expressed as the expected reduction in the uncertainty about a marker genotype caused by the knowledge of the member identity . Furthermore, if represents either individuals or genetically homogeneous populations then becomes zero, because of null marker diversity within those units. Under this particular situation, mutual information in equation (2) becomes simply , i.e. the Shannon entropy of the distribution of marker genotypes [26].

In DNA bulk fingerprinting for germplasm source identity we want to maximize . However, if genetic heterogeneity is present within the tested populations and thus sampling error, the term is not assumed to be zero by the fact that there is potential variation in marker genotypes provided by different assays. The term will be called diversity, whereas will be called noise. Through optimizing diversity will be increased among the single or multilocus binary profiles of different groups of individuals, while reducing noise in order to carry out a reliable sampling.

To find an analytical solution for an optimal sampling size in bulks, a pattern of distribution of allele frequencies across varieties would have to be assumed. However, a practical approach, which has been used as an example in this work, is to use previous knowledge of allele frequencies in different populations of a species and several loci for a given marker type, and find a consensus sample size that is highly informative. Furthermore, locus selection may be done through estimating the mutual information . If noise, i.e. the term, which represents the variation among DNA bulks from the same source, can be ignored, then marker informativeness can be calculated simply through the entropy of genotypes among germplasm sources, i.e. . The so-calculated value of information is given in bits, and can be interpreted as the number of fully informative binary loci, or briefly, the effective number of binary loci [26].

Average bulked DNA marker information per allele

Consider a random bulk with alleles or individuals in a random mating diploid population, where the absence or presence of a given allele is scored through a marker system. The maximum information that a given allele can provide is 1, and it depends on the variation of its frequency across germplasm sources and the bulk size. Let , , be the allele frequency at the germplasm source. Thus the probability of the allele being absent at the germplasm source is assuming random sampling. If we consider each germplasm source as being equally frequent within a defined set, then the global probabilities of absence and presence of a given allele in a random bulk would be their average probabilities across germplasm sources. Thus the entropy of the binary marker genotypes for a given allele is:(3)

This is the diversity part of equation (2). The entropy of the marker genotypes conditional on germplasm sources , is the average entropy of the binary genotypes across populations:(4)

This is the noise part of equation (2). The information provided by the binary scoring of the given allele is . We expect that the value of equation (4) will descend as we increase the bulk size, becoming more replicable. In fact, the limit of when tends to infinity is zero, and information is .

So far we have considered only the noise due to sampling error. However, there are technical issues that introduce undesirable variation. They are complex and most of them need to be studied on experimental basis [27]. In the case of SSRs, the presence of stutter bands potentially introduces incorrect artificial variation [28]. There are generic problems associated to all electrophoresis-based genetic markers. Two of them are the technical homoplasy, i.e. when two bands are mistakenly considered homologous, and oversplitting, in which bins are too thin and may split variant locations of the same amplicon [27]. Other problems are the occurrence of false positives and false negatives, associated to the so-called bayesian error rates, which represent the probabilities of mis-scoring the presence or absence of alleles. We expect that all the mentioned technical factors reduce the mutual information. For instance, let us consider the case of sensitivity, defined as the probability of an allele being detected given that it is a present in a given bulk. Under the absence of sampling error, if we fix sensitivity to a value of , with being the frequency of bulks with the presence of a certain allele, then mutual information is modified to , where entropies are calculated with the probability vectors in their subscripts. It can be graphically demonstrated that information is monotonically increasing with sensitivity, reaching a maximum of with and a minimum of 0 with . In this work we analyze one aspect of technical noise: the allele dilution effect on its detection.

The problem of allele dilution

We have assumed so far that any allele present in a bulk will be revealed by the given marker system. However, that is not always the case because it has been observed that, depending on the marker system, a low frequency allele in a bulk may not be detected. For RFLP and RAPD in tomato, it was found that an allele was still detectable at a proportion of 0.05 in a DNA pool [1]. Kuboki, Yoshimura and Yano [29] reported a similar threshold of detection in rice. These reports were confirmed by Dubreuil et al. [10], who found that an RFLP allele is barely detectable when present in a proportion of 0.05 within a DNA pooled-sample.

For AFLPs, the lowest ratio at which an allele was still detected in a range of crop species was 1:24, i.e. with a dilution threshold of approximately 0.05 [30]. However, below the 1:4 ratio only a few bands of high original intensity could still be recovered. For SSR markers in maize, Reif et al. [12] found that alleles in a dilution below 0.2 were most often undetected. For SSR markers, Dubreuil et al. [11] tested dilutions of 0.025, 0.05, 0.10, 0.20, 0.30, 0.40 and 0.50 for inferring allele frequencies from DNA bulk data. From the optimal SSR markers they found a range of correlations from 0.79 to 0.99; however they did not report a specific detection threshold.

In view of the above information, we considered a range of threshold dilutions from 0.05 to 0.2 in order to evaluate a strategy for coping with the dilution problem. The selected strategy to overcome the technical problem of rare allele amplification was the use of fractionated bulks, in the same way as the strategy used by Dubreuil et al. [10] for genotyping 10 maize populations from North America, where three independent bulks of 10 plants were randomly selected from each population.

Let be the threshold dilution for a given marker, this being the minimum dilution from which an allele can be detected in a bulk of alleles randomly sampled from a population with allele frequency . The probability of the allele being undetected in a bulk is:(5)where is the number of copies of the allele in the bulk. The size of the bulk is equivalent to plants from a random mating population. If the sampling strategy consists in independent bulks of sample size , the probability of the allele being undetected is . The scoring scheme under this strategy would be to assign the symbol 1 if the allele shows up in at least one of the bulks, and 0 otherwise.

To further check the problem of allele dilution and to validate the tested range, we assayed three SSR loci for corn: phi093, phi072 and phi064. The biological material comprised three maize accessions collected in the state of Puebla in Mexico. The plants were analyzed both individually and in fractionated samples ranging from 22 to 30 plants, divided into bulks of four to 10 plants. The number of different SSR alleles in those samples ranged from 3 to 10. PCR reactions were carried out using standard protocols and run on a 3730 ABI DNA ANALYZER. Scoring was carried out with the GeneMapper Software Version 4.0 (AppliedBiosystems).

Bulk DNA sampling with SSRs across maize populations

As an example application of the model, a set of 56 maize populations was sampled by selecting 10 random plants per germplasm source, which were individually genotyped using microstatellite markers, called SSRs. The source of this genetic material was the active collection of the USA maize germplasm bank, curated by the USDA in the North Central Regional Plant Introduction Station (NCRPIS) at Ames, Iowa. These populations were collected in USA, Mexico, Argentina, Peru, Brazil, Chile, Ecuador, Paraguay and Uruguay. A total of 31 SSR primers were used for microsatellite amplification. Genomic DNA was extracted from leaves of young maize seedlings by using the commercial PUREGENE DNA Isolation Kit [31]. The PCR products were separated and visualized in polyacrylamide gels (4% denaturing 6 M urea, 29:1 acrylamide:bisacrylamide), and a file containing the information for each gel was generated by the ABI GeneScan software. The gel files generated by the ABI GeneScan software were scored using the computer software STRand [32] following the local Southern algorithm for estimating allele sizes. The 31 SSR assays resulted in 198 polymorphic bands. These data have been submitted to USDA; part of them are already in the page http://www.ars-grin.gov/npgs/(Accessed 2013 September 2) and the remaining are in process of being curated.

A total of 191 of the 198 alleles identified in this study were selected for optimization, since complete data sets were available for these alleles. Equations (3) and (4) were written as R functions [33] to estimate diversity, noise and information for bulk sizes of 10 to 100 alleles, with increments of 10. Plots were exhaustively examined to see the trends across increasing bulk sizes, and an average plot was constructed.

To examine the dilution problem, a sample of 60 alleles divided into 1, 2, 3, 5, 6, 10, 15 and 30 bulks, was evaluated for the probability of a given allele being undetected, by equation (5). These numbers of divisions of the sample were restricted to generate bulks with an integer number of plants, i.e. 30, 15, 10, 6, 3 and 1, respectively. The first case is the situation when the 60-allele sample is not divided. The last case represents the genotyping of 30 individual plants. The evaluated threshold dilutions were 0.05, 0.1, 0.15 and 0.02. The allele frequencies were set as 0.1, 0.2 and 0.3.

Information for 30 SSR marker loci was calculated through simulated bulks of 30 plants for each of the 56 maize populations, based on the estimated allele frequencies. A multinomial simulation was performed to generate each set of 60 sampled alleles through the rmultinom command in R [33]; afterwards, the absence or presence of each allele in the sample was tested, and then encoded as either 0 or 1, respectively, thus generating a vector of size equal to the number of alleles of the given SSR in all populations. After bulk simulation for each SSR locus, the entropy of the electrophoretic patterns, coded by binary vectors, was estimated through equation (1). This value of was taken as the amount of information per SSR locus, since we did not consider due to the fact that with 60 sampled alleles this term becomes almost zero. One of the SSR loci from the set of 31 was not considered due to missing data.

Bulk size optimization

Although the sample size of 10 maize plants per population was small, this survey allowed us to evaluate general tendencies of allele frequencies for DNA bulked SSR marker information in this species. Based on the estimated allele frequencies, Fig. 1 depicts plots of information and bulk size for two random alleles and for the average trend, whereas average values for all sample sizes are presented in Table 1. In general, the results showed that from bulks of 60 alleles or more, the noise term is close to zero, with information reaching a near-maximum level, being almost equal to the diversity term, i.e. the entropy of the binary marker genotypes defined in equation (3). The average information per allele for bulks based on 60 alleles, i.e. 30 plants, was 0.568 bits with a diversity value of 0.577, with the noise term oscillating between 0 and 0.048, and a mean of 0.006. The bulks of 100 alleles, i.e. 50 plants, gave the maximum information in the tested range: 0.577 bits with an average noise of 0.002. However, as can be observed in Fig. 1, this does not represent an appreciable advantage over bulks composed of 60 alleles. Thus, at least for maize germplasm sources, our results indicate that bulked DNA from 30 plants can provide a close to optimum information per SSR allele.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 1. Plots of information parameters for several bulk sizes, based on estimated allele frequencies in maize germplasm sources.

https://doi.org/10.1371/journal.pone.0079936.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Average values for information parameters in bulks of 10 to 100 SSR alleles in maize.

https://doi.org/10.1371/journal.pone.0079936.t001

The drawback to this recommended sample size is that for large bulks, e.g. with more than 20 alleles, the dilution problem must be considered by the fact that alleles with a very low frequency within the bulk may not be detected due to insufficient sensitivity of the marker protocol.

Strategy for the problem of allele dilution

We have already found that 60 allele bulks sampled for SSR markers in maize provide a near-optimum amount of information; the question now is what would be an adequate number of independent bulks, such that  =  60, to cope with the dilution problem. Fig. 2 depicts results for different threshold dilutions and population allele frequencies. For an allele frequency of 0.1 the ability of detection with a threshold dilution of 0.05 performs well, and almost perfectly for 60 alleles divided into three bulks or more. For a threshold dilution of 0.1, division into three bulks still works well, giving a probability of non-detection less than 0.1. However, for an allele frequency of 0.1, threshold dilutions of 0.15 or more make the bulk system unreliable, even dividing the 60 allele sample into six bulks of 10 alleles. The oscillation observed for threshold dilutions of 0.1 and more, although apparently illogical, has an explanation based on the minimum number of copies being present in a given bulk, which can be detected. For example, let us compare three and five divisions for a threshold dilution of 0.1. In the first case, the bulks are composed of random samples of 20 alleles; thus, two or more copies of the rare allele are required to be detected, which has a probability of failure of 0.392 for a single bulk and 0.06 for the three bulks. On the other hand, if we use five bulks of 12 alleles, those samples also need at least two copies of the rare allele, having a probability of failure of 0.66 for a single bulk and 0.124 for the five bulks. In other words, the oscillation obeys the discrete nature of the required minimum number of copies of the rare allele to be detected.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. Probability of an allele being undetected for different numbers of fractions of a 60-allele sample, under several dilution thresholds and population allele frequencies.

https://doi.org/10.1371/journal.pone.0079936.g002

For an allele frequency of 0.2, even under a threshold dilution of 0.2, 60-allele samples divided into three or more bulks perform well, with probabilities of non-detection always lower than 0.1. For allele frequencies of 0.3, even undivided samples of 60 alleles perform well. Thus, in general, we can say that, for random mating maize populations genotyped with SSRs, samples of 30 plants fractionated into three bulks of 10 plants are reliable for rare allele detection with allele frequencies of 0.1 or higher, for threshold dilutions of 0.1 or less. Such was the strategy used by [8] in bulked analysis of maize populations. For threshold dilutions above 0.1, failure of detection of rare alleles must be taken into account, according to equation (5) and the tendencies depicted in Fig. 2.

The results of the bulked and individual assays for three SSR loci to evaluate dilution effects are summarized in Table 2. The accessions are coded as PL077, PL092 and PL149. The accession keys are followed by the bulk numbers. The rows with bold face characters show the results of pooling the information of the three bulks for each combination of marker and accession. The column of alleles gives their numbers based on individual plant assays. The dilution range was calculated in basis of allele frequencies from individual assays within bulks. Detection percentages were calculated from the fraction of alleles that were amplified in the bulked analyses. The dilutions in undetected cases are the frequencies of the alleles that the bulk-based analyses were unable to detect in each assay. The minimum allele dilution was 0.05, with a maximum of 0.70. In most cases the bulk-based approach allowed detection of all the alleles present in the individual plants. The only exception was the locus phi093, where alleles with frequencies of 0.10, 0.11 and 0.15 were not amplified. However, the combined information of the three bulks within the sample allowed detection of the three present alleles, thus reaching a 100% rate of detection. The primers for loci phi064 and phi072 allowed detection of all alleles through bulk-based analyses, even those with a dilution of 0.05. The presence of multiple allelism, e.g. in bulks PL077—3 and PL149—1 for marker phi064 with nine alleles, did not affect their amplification in the PCR experiments. In all cases, alleles with dilutions of 0.2 or more were detected, with most bulks having a successful threshold dilution of 0.05. These results show the usefulness of dividing samples into bulks of 10 plants.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Analysis of three SSR loci for their amplification under allele dilution.

https://doi.org/10.1371/journal.pone.0079936.t002

Information of SSR loci

Results of the estimation of SSR information are shown in Table 3. The average information per SSR locus was 3 bits, i.e. it was equivalent to 3 binary, fully informative markers, and ranged from 0.795 (phi213984) to 4.951 (phi064), with the highest frequency being placed in the interval from 3 to 4 bits (Fig. 3A). The number of alleles per locus ranged from 3 (phi121) to 11 (phi96100). Information was positively correlated with the number of alleles (Fig. 3B), with a linear correlation , x for a two-tailed test, and a 95% confidence interval of 0.616 to 0.900, thus indicating that allelic number may work at least as a rough indicator of SSR informativeness.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Informativeness of SSRs.

Distribution of informativeness in 30 SSR loci measured in bits (a). Association between SSR marker informativeness and number of alleles per locus (b).

https://doi.org/10.1371/journal.pone.0079936.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Informativeness of SSR loci for bulk DNA genotyping.

https://doi.org/10.1371/journal.pone.0079936.t003

The goal of calculating individual marker information is to have a criterium for locus selection; such as in the case of the choice of SNP markers in association studies [19], [20]. However, the advent of high-throughput technologies and informatics tools make possible to analyze simultaneously a high number of markers and select the subset related to a given trait [34]. This work however is not aimed at association, but rather the optimization of the necessary number of informative markers to enable rapid large scale population monitoring.

Suggestions for bulk sampling optimization

For bulk sampling with binary scoring with SSRs in corn, we recommend using 30 plants per accession, divided into three pools of 10 plants. Comparing allele presence among the three bulks of 10 plants per accession, will give an idea of the robustness of allele detection. The cases were results are consistent among the three bulks would be instances with low genotyping error. When only one of the bulks allows detection of a given allele whereas the others give a negative value, the detection of this allele may be subject to a high sampling error or technical problems. Furthermore, validation of the method requires individual genotyping in a fraction of the sampled accessions.

When working with a different species or marker system, previous allele frequency data for a certain number of accessions can be utilized to check the effect of sampling size on mutual information and its components of diversity and noise for several loci and alleles. This is done by calculating the result of equation (3) minus the one of equation (4). To facilitate calculation, the following R code [33] can be used:

#Basic functions for entropy

MyLog2p<-function(x){if(x =  = 0) 0 else x*log(x,2)} entropy<-function(x){-sum(sapply(x,MyLog2p))}

#Calculator of information, diversity and noise

InfoBulkBinary<-function(p,n)

{n<-2*n;v1<-(1-p)ˆn;v2<-1-v1;mymat<-cbind(v1,v2); div<-entropy(c(mean(v1),mean(v2)));a<-NULL;

for (i in 1:length(p)){a[i]<-entropy(mymat[i,])};

noise<-mean(a);c(div,noise,div-noise)}

#p is the vector of frequencies of a given allele across populations

#n is the bulk size in number of diploid plants.

As an example, for a vector of allele frequencies in 10 accessions: 0.1,0,0.4,0.3,0,0.25,0.5,0.23,0.15,0, the information for a bulk of 20 plants is calculated as follows:

InfoBulkBinary(c(0.1,0,0.4,0.3,0,0.25,0.5,0.23,0.15,0),20)

The result:

[1] 0.88327703 0.01279625 0.87048078,

is given in the following order: diversity, noise and information.