Optimising selection in tree breeding with constraints on relatedness and operational flexibility

Transcription

1 Optimising selection in tree breeding with constraints on relatedness and operational flexibility T.J. Tim Mullin 1,2, Makoto Yamashita 3, and Pietro Belotti 4 1 Skogforsk (The Forestry Research Institute of Sweden), P.O. Box 3, SE Sävar, SWEDEN rue du Grand-Royal Est, Shefford, QC, J2M 1R5 CANADA 3 Tokyo Institute of Technology, W8-29 Ookayama, Meguro-ku, Tokyo JAPAN 4 Xpress Optimizer Team, FICO, Starley Way, Birmingham, UK

2 Acknowledgements Föreningen Skogsträdsförädling (The Swedish Forest Tree Breeding Association) JSPS KAKENHI (Grant 15K00032) GenTree - Optimising the management and sustainable use of forest genetic resources in Europe (H2020 Grant Agreement SFS )

3 Accurate genetic evaluation now what? Genetic evaluation: the collection and processing of phenotypic and genomic data to infer genetic value. Next step is to select populations for use as breeding parents or for deployment Identifying the single best candidate is easy, but the nextbest candidates will often be closely related Simple truncation selection? Generally unacceptable Restrictions? Limits on contributions of relationship groups

4 What does optimal look like? Adjusting selection contributions to manage diversity/relatedness described as early as 1958 Ideally, mean genetic value of selected population maximised, while satisfying a relatedness criterion Mean genetic value: average of estimated breeding value (or index value), weighted by contribution to the population (number of ramets, number of crosses, etc.) Relatedness: variety of identity-by-descent (IBD) statistics available James and McBride J Genet 56: 55-62

5 Describing relatedness (diversity) Census size (N) doesn t consider relatedness among genotypes (clones) or unequal contributions Effective population size various calculations Status number (N s ) from average coancestry between all individuals in population, or group coancestry (θ) N s = 1 2θ Proportional gene diversity (GD) GD = 1 θ θ GD N s

6 Linear deployment 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 so that their frequency is linearly related to their genetic value Provided that genotypes are NOT related Applied operationally to establish seed orchards in Sweden Lindgren et al TAG 77:

7 Frequency Linear deployment Use linear deployment to optimise unrelated clones N = 2800 N s = 14 θ = best used equally 23 best linearly deployed X X Breeding value

8 Group-merit selection Formulate group merit of a population as function of average genetic value and diversity: B ω = g cθ ω where: B ω is a group merit selection criterion ω a set of genotypes g average breeding value of ω θ ω group coancestry of ω a constant converting θ ω to units of breeding value c Choose a value for c, then conduct an iterative search for a group of genotypes that maximises the function Must repeat with varying weights c, until desired θ ω is obtained Lindgren and Mullin Silvae Genetica 46: Brisbane and Gibson TAG 91:

9 Gain % Group-merit selection Applied operationally to jack pine in Ontario, Canada Group-merit selection Uniform family restrictions Effective population size (Ns) Mullin Unpublished file report.

10 Meuwissen s Optimum contributions Meuwissen introduced a quadratic object function to impose a constraint on change in inbreeding or diversity Maximise: c T g Subject to: c T Ac/2 θ c T 1 = 1 Developed what has become known as the Optimum Contributions OC algorithm to optimise contributions by an iterative algorithm using Lagrangian multipliers Implemented in freeware app GENCONT, currently standard selection procedure for many livestock breeding programs Time-consuming, iterative procedure, with limitations True optimum NOT guaranteed. Constraints may be violated. Meuwissen J. Anim. Sci. 75:

11 Some characteristics of early attempts Iterative search algorithms can bypass the true optimum Some solutions are integer ideal if selections to contribute equally (mainline BP, conservation, simple clone mix, etc.): Restricted selection Group-merit selection Some solutions are actually continuous distributions of contribution proportions (seed orchards, mate planning, etc.): Linear deployment Meuwissen s Optimum Contributions Often (usually) difficult to impose operational constraints on maximum or minimum contributions Enter.. a new era of exact solutions through mathematical programming

12 Mixed-integer quadratically constrained optimization (MIQCO) Selecting a fixed-size breeding population with uniform mating is an integer problem formulate as MIQCO and solve Iterative search algorithms can bypass the true optimum z Branch-and-bound algorithms perform a systematic evaluation of all possible solutions without explicitly evaluating all of them. suppose a feasible solution is at hand with objective value 100. now suppose we solve a relaxation, such as fixing some of the variables and solving the resulting linear program, and we obtain a value of only 90. we can ignore all possible solutions having the fixed values in the relaxation since they must all have an objective value of at most 90. Mullin and Belotti TGG 12: 4

13 Mixed-integer quadratically constrained optimization (MIQCO) Each subproblem in branch-and-bound is solved as a nonlinear problem, which requires considerable computational effort. The culprit: the single nonlinear constraint on relatedness Solution: approximate the nonlinear constraint with a set of linear ones, known as Outer Approximation Developed a heuristic that iteratively swaps elements to find feasible solution Implemented on the generic freeware branch-and-bound framework COIN-OR Cbc ( Provides exact optimum solution to the fixed-size-equalcontribution flavour of population selection Mullin and Belotti TGG 12: 4

14 Semidefinite programming Optimising unequal contributions perhaps of greater interest Pong-Wong and Woolliams proposed that optimisation of the convex quadratic function could be approached through Semi-definite programming SDP 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 Efficient freeware solver known as SDPA (Yamashita et al. 2012) Provides an exact solution to the unequal contributions flavour of population selection (contributions expressed as proportions) Pong-Wong and Woolliams GSE 39: 3-25 Woolliams et al J Anim Breed Genet 132: Yamashita et al Pp In: Anjos and Lasserre(eds) Handbook on Semidefinite Springer Science

15 Semidefinite programming 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: c T g genetic value (g) Subject to: A 1 c c T 0 constraint on group coancestry (θ) 2θ c T c T c u 0 m c 0 constraints on minimum contributions constraints on maximum contributions First application in breeding in Sweden (Lindgren Seed Orchard) Now incorporated in the TREEPLAN system for selection of seed orchards and genetic contributions to breeding plans Ahlinder et al TGG 10: Kerr et al J Anim Breed Genet 132:

16 Second-order cone programming Mathematically, a second-order cone is a special case of a positive semidefinite constraint The numerator relationship matrix A is positive definite, but we can apply the Cholesky factorization to derive a second-order condition corresponding to the constraint on group coancestry Because SOCP is a special case of SDP, a specialised SOCP solver might find a solution more quickly. Freeware solver known as ECOS (Domahidi et al. 2013) Compared with SDP, our first tests using a simple SOCP formulation were up to 250 times faster for small problems (~2000 candidates), but only 7 times faster when number of candidate (~ candidates) Domahidi, Chu and Boyd European Control Conference, Zurich Yamashita, Mullin and Safarina Cornell Univ arxiv: v2

17 Second-order cone programming Observed that x A is much more dense than its inverse A A -1

18 Second-order cone programming We introduced a new variable y = Ac and derived an equivalent sparse SOCP using A -1 Maximise : (A 1 g) T y Subject to : (A 1 e) T y = 1 y T A 1 y 2 θ l A 1 y u Made the problem an even more compact SOCP using the Quaas-Henderson method to construct A -1 directly, without having to construct A Provides extremely fast, exact solution to the unequal contributions flavour of population selection Yamashita, Mullin and Safarina Cornell Univ arxiv: v2

19 Second-order cone programming Formulation Pedigree size = 2045 Non-zero elements Total time (seconds) SDP simple SOCP compact SOCP Pedigree size = SDP simple SOCP compact SOCP Pedigree size = SDP Out of memory simple SOCP Out of memory compact SOCP Yamashita, Mullin and Safarina Cornell Univ arxiv: v2

20 Selection optimisation by SOCP and MIQCO freely accessible to breeders OPSEL : freeware available to carry out optimal selection (publically available from Skogforsk at ) Formulates problem, calls appropriate solver, interprets solution, calculates contribution by genotype, and provides summary statistics For a list of candidates: 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 selected population Constraint on group coancestry (status number) MIQCO or SOCP solution? Mullin Arbetsrapport från Skogforsk Nr

21 Candidate file Genotype Female Male EBV Max Min

22 Parameters given through screen dialogue or by a parameter file Candidate_file.csv 1 : equal contributions = 0, unequal contributions = 1 Next line should specify drive and folder containing solver application S:\WSFchange\Treeplan\Ass_programs\OPSEL 1 : complete pedigree check = 1, else = : census size of selected group : maximum group coancestry constraint 0 : no minimum = 0, minimum contribution specified = 1 Next line should specify option switches for solver 1 : output as txt = 0, output as csv = 1 1 : clean up work files = 1, else 0

23 A case study Client wants to establish a Scots pine seed orchard with 2800 planting positions, and with Status number N s = 14 (θ = ) Search our data records and retrieve 2000 F 1 candidates, as well as F 0 ancestors (founders), for total pedigree of 2045 genotypes Scion material for F 0 s is virtually unlimited, if they still exist, but for the younger F 1 s, scions effectively limited to 50 per genotype. Compare alternative selection solutions: 1. Use 14 best unrelated genotypes in equal proportions (restricted selection) 2. Optimise number and contribution of selected genotypes with OPSEL, with no limits on availability of scion material for F 0 or F 1 genotypes 3. Optimise with OPSEL, but limit F1 contributions to no more than 50

24 Frequency 14 best unrelated F 1 s Select 14 best unrelated genotypes (no full or half sibs) and use in equal proportions (200 ramets per) N = 2800 # genotypes = 14 N s = 14.0 θ = Mean BV = best unrelated F1s Breeding value X

25 Frequency OPSEL F 0 s, F 1 s, no limits optimise with OPSEL allowing unlimited F 0 and F 1 s N = 2801 # genotypes = 56 N s = 14 θ = Mean BV = best unrelated F1s OPSEL unlimited F0 and F1s Breeding value X X

26 Frequency OPSEL limit F 1 s 50 optimise with OPSEL allowing unlimited F 0 s, but limiting F 1 s 50 (insufficient scion material!) N = 2796 # genotypes = 71 N s = 14 θ = Mean BV = best unrelated F1s OPSEL unlimited F0 and F1s OPSEL F1s limited X XX Breeding value

27 How much better? Top 14 unrelated F 1 s used equally OPSEL solution with unlimited F 0 s and F 1 s OPSEL solution with unlimited F 0 s and F 1 s 50 N # genotypes N s θ Average BV Difference +17.3% +13.0%

28 Operational flexibility Reoptimise to update solution 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 as the maximum constraint Specify minimum contributions One or more genotypes selected as standards and are required to appear in some minimum number of times You wish to add material to replace mortality in a recently established orchard. The counts of surviving grafts per clone are specified as the minimum to appear in the solution. New selections for annual crossing programme, accounting for parents already represented in earlier crosses

29 Rather than selecting a BP select parent contributions for mating OPSEL can optimise a fixed-size breeding population, breeders more likely interested in optimising mating SOCP can be used to quickly optimise parental contributions to next round of breeding Breeder identifies desired operational crossing effort, say 100 crosses, equivalent to 200 parent contributions (2 per cross, one as female, the other as male) Perform optimised selection of 200 genotypes to serve as parents, and round the contributions to whole numbers Allocate to crosses using a mating algorithm (e.g., positive assortative, minimum coancestry, random, etc.)

30 Thanks for listening!