T.J. Mullin 1 , J. Hallander 2 , M. Yamashita 3 , and O. Rosvall 1

1. Maximising the genetic value of seed orchards and other production populations subject to constraint on gene diversity T.J. Mullin1, J. Hallander2, M. Yamashita3, and O. Rosvall1 1Skogforsk (The Forestry Research Institute of Sweden), P.O. Box 3, SE-918 21 Sävar, Sweden 2 FOI, Cementvägen 20, SE - 901 82 Umeå, Sweden 3Tokyo Institute of Technology, 2-12-1-W8-29 Ookayama, Meguro-ku, Tokyo 152-8552, Japan

2. What does an “optimal” seed orchard look like? • Mean genetic value maximised, while satisfying a diversitycriterion • Mean genetic value: average estimated breeding value (or index value), weighted by number of ramets in the orchard • Diversity: variety of “identity-by-descent” statistics available

3. Describing diversity • Census size (N) – doesn’t consider relatedness among genotypes (clones) or unequal contributions • Effective population size – various calculations “Status number”(Ns) – from average coancestry between all individuals in population, or “group coancestry” (θ) “Proportional gene diversity” (GD) θ ├ GD├ Ns

4. Maximising genetic value with a constraint on diversity • “linear deployment” ideas of Lindgren et al. (1989) • When diversity is constrained, better genotypes should occur more frequently than poorer ones. • Genotypes are best deployed in an orchard so that their frequency is “linearly related to their genetic value” • Provided that genotypes are NOT related Lindgren et al. 1989. TAG 77: 825-831.

5. Linear deployment example • Use linear deployment to optimise unrelated clones • N = 2800 • Ns = 14 • = 0.03571

6. How to optimise when candidate clones are related? • Requires more complex formulation • Easiest when number of candidates is limited: • Can use trial and error approaches • Can define the optimum as a Lagrange function (Bondesson and Lindgren 1993) Bondesson and Lindgren. 1993. Silvae Genet. 42: 157-163

7. Meuwissen’s “Optimum contributions” • Lagrangian multipliers also used by Meuwissen (1997) who introduced a quadratic object function to apply a fixed diversity constraint • Developed what has become known as the “Optimum Contributions” algorithm to optimise selection and contributions of candidates Maximise: Subject to: • Recommended for orchard selection by Kerr et al. (1998) and then applied to selection of a Scots pine seed orchard (Hallander and Waldmann 2009) • Time-consuming, iterative procedure Kerr et al. 1998. Silvae Genet. 47: 165-173 Hallander and Waldmann. 2009. TAG 118: 1133-1142 Meuwissen. 1997. J. Anim. Sci. 75: 934-940.

8. Other mathematical programming • Pong-Wong and Woolliams (2007): recognised serious drawbacks to the Lagrangian multipliers • Iterative removal of candidates might also bypass the true optimum • No restriction on maximum contribution – can’t accommodate pragmatic operational constraints • Proposed that optimisation of “convex quadratic function” could be approached through “Semi-definite programming” (Vandenberghe and Boyd 1996) • Additional constraints can be described, declaring a maximum and even a minimum contribution for each candidate • Optimum solution guaranteed with use of interior-point algorithms • There exist efficient solvers for such problems Pong-Wong and Woolliams. 2007. GSE 39: 3-25 Vandenberghe and Boyd. 1996. SIAM Review 38:49-95

9. Orchard selection as an SDP • To formulate the SDP, we define the problem as a minimisation, where the quadratic constraint on group coancestry is expressed in linear form by its “Shur complement” as an inequality: Minimise:  genetic value (g) Subject to:  constraint on group coancestry(θ)  constraints on minimum contributions  constraints on maximum contributions • Matrix notation – these matrices must be provided to the solver, which is not a trivial matter when matrices are large  contributions must sum to 1

10. OPSEL describes the problem and interprets the solution • OPSEL: “open-source” software available to carry out selection for orchards and other production populations with unequal contributions from candidate genotypes (available from Skogforsk) • For a list of candidates, orchard manager provides: • Complete pedigree (Me+Mum+Dad) for candidates and ancestors • Genetic values (usually EBVs) • Maximum (and optionally minimum) contribution for each candidate • OPSEL also needs to know: • Census size of orchard population • Constraint on group coancestry (status number) • Formulates SDP problem to be solved by SDPA (Yamashita et al. 2010), interprets solution, calculates integer contribution by genotype, and summarises orchard statistics

11. Candidate file

12. A case study • Our client wants to establish a Scots pine seed orchard with 2800 planting positions, and with Status number Ns = 14 ( = 0.03571) • We search our data records and retrieve 2000 F1 candidates, as well as their ancestors, for a total pedigree of 2045 genotypes

13. 14 best unrelated F1s • Select 14 best unrelated genotypes (no full or half sibs) and use in equal proportions (200 ramets per) • N = 2800 • # genotypes = 14 • Ns = 14.0 • = 0.03571 • Mean BV = 381

14. OPSEL – F0s, F1s, no limits • Optimise with OPSEL allowing unlimited F0 and F1s • N = 2801 • # genotypes = 56 • Ns = 14 • = 0.03571 • Mean BV = 447

15. OPSEL – limit F1s ≤ 50 • Optimise with OPSEL allowing unlimited F0, but limiting F1s ≤ 50 (insufficient scion material!) • N = 2796 • # genotypes = 71 • Ns = 14 • = 0.03573 • Mean BV = 439

16. Let’s run this last example! • Orchard with 2800 planting positions • Unlimited ramets available per F0 genotype, but only 50 ramets available from the smaller F1s • Client has specified minimum Ns of 14 • Candidate pedigree, EBVs, etc. found in file as comma-separated values

17. How much better?

18. When to reoptimise? • After scion collection but before grafting, using actual scion inventory as “maximum” constraint • Shipment of surviving grafts to orchard, using current nursery stock inventory as “maximum” constraint • Thinning of orchard, using updated breeding value estimates and current ramet count

19. When to specify a “minimum”? • One or more genotypes selected as “standards” and are required to appear in some minimum number of times

20. Other applications? • Optimise genetic value of seedlots with constraint on gene diversity, when mixing single-family seed collections