The basic idea of our efficient strategy is to record the starting positions and founder origins (founder chromosome identifiers) of each chromosome segment rather than the allele state at each locus for the whole genome.

An example to illustrate the simulation strategy is shown in figure 1. The pairs of starting base pair position and origin vectors for founder 1 are {[0], [a]} and {[0], [b]}. During meiosis, assuming a crossover occurs at base pair position 12 (e.g. 12.0 Mb), the pair of position and origin vectors for one of the resulting recombinant chromosomes is {[0, 12], [a, b]}. Similarly, pairs of position and origin vectors for founder 2 are {[0], [c]} and {[0], [d]}. Assuming a crossover occurs at base pair position 47 (e.g. 47.0 Mb), the pair of position and origin vectors for one of the resulting recombinant chromosomes is {[0, 47], [c, d]} . Thus, the chromosomes of the offspring of founder 1 and founder 2 are {[0, 12], [a, b]} and {[0, 47], [c, d]}. Suppose a crossover occurs at base pair position 32 during meiosis in this offspring. Then, the pair of position and origin vectors for one of the resulting recombinant chromosomes are {[0, 12, 32, 47], [a, b, c, d]}. Then, given the positions of the crossover sites and corresponding origins of chromosomes, the entire genome of any non-founder can be constructed to the density of the founder genomes. In the classical gene drop method, all allele states are dropped down sequentially from founders all the way to last generation. However, in the drop-down strategy proposed here, what are dropped down sequentially from founders to the last generation are sparse vectors containing only founder origins and crossover positions. Thus, in the absence of mutation, computing time and memory requirement to drop down this genomic information over generations is free of the number of loci. Once the origin and position vectors are available in the latter generations of interest, allele state information from founders can be dropped down directly to this generation.

figure 1