Measuring the representativeness of a germplasm collection

Abstract

Many germplasm collections aim to preserve most of the genetic diversity present in a population so that the population could be regenerated, which provides genetic resources to ensure food security. This paper proposes a way to measure how well a germplasm collection achieve this goal. In the most common scenario, one has little information regarding the number and statistical distribution of alleles at every locus, and it is thus very difficult to measure the representativeness of the accession. Here, we show how to use samples of allelic diversity at a sample of loci to estimate the representativeness of an accession based on the coverage of a sample with point and interval estimates. Our approach avoids making unrealistic assumptions regarding the number of loci, the bounds for the number of alleles or their frequency distributions. Depending on the sampling scheme of a collection, we differentiate between absolute or relative coverage. Here, we demonstrate this methodology using data from the germplasm collection at the Leibniz Institute of Plant Genetics and Crop Plant Research.

Appendix: Proof of properties of the coverage of several populations

• 1. If we select an individual at random from the population and then select one of its attributes X or Y, this attribute will be included in the sample with respective probabilities $C_X$ and $C_Y$. Because the selected attribute is equally likely to be X or Y, the probability that the attribute selected is in the sample is $(C_X + C_Y)/2$.

• 2. If two populations of sizes $N_1$ and $N_2$ are mixed and a sample of size n is taken from the mix, the probability that an individual selected at random from the mixed population is represented in the sample is defined as the coverage of the sample, this follows from the fact that a randomly selected individual from the mix of populations belongs to each initial population with respective probabilities $f= N_1/(N_1 + N_2)$ and $1-f=N_2/(N_1 + N_2)$. It follows that there is no need to mix both populations as long as each population is sampled with sample sizes $n_1=N_1/(N_1+N_2)$ and $n_2$, respectively.

• 3. Suppose we have two populations 1 and 2 of sizes $N_1$ and $N_2$, respectively, where $N_1 = N_2$. Suppose we take a sample of size n from each population and let C represent the coverage of the mixed sample of size 2n. By property 2, C can be interpreted as the probability that a random individual selected from the mix of both populations is represented in the sample, i.e., the absolute coverage. Now suppose that the size of population 2 is increased by a factor of k, where $k >1$, keeping the relative frequency of alleles fixed. Clearly, the previous interpretation of the coverage (absolute coverage) no longer holds because an individual selected randomly from the mixture of populations 1 and 2 is k times more likely to come from population 2. But if we can guarantee that the individual selected is equally likely to come from either population, then the probability that this individual is already represented in the sample is still C. The restriction imposed by requiring that it must be equally likely that the individual comes from either population defines the relative coverage. It follows that if we have two populations of general sizes $N_1$ and $N_2$, $N_1 \ne N_2$, and take a sample of the same size n from each population, the coverage of the sample mix follows the definition of relative coverage.

• 4. This property follows from properties of random sampling.