Optimisation: From theory to better prediction

Transcription

1 Optimisation: From theory to better prediction Claire Adjiman Claire Adjiman 23 May 2013 Christodoulos A. Floudas Memorial Symposium 6 May 2017

2 Why look for the global solution? Deterministic global optimisation (DGO) provides certainty safety and risk: finding the worst-case solution to minimise/control risk thermodynamics: local minima are just wrong model parameterisation/identification: gaining a correct physical understanding DGO can provide enhanced performance (technical or economic) why settle for second (or third, or fourth) best?

3 Deterministic global optimisation: Basic concepts Department of Chemical Engineering min x s. t. f ( x) g( x) 0 h( x) = 0 x X f(x) Nonconvexity f(x) Convex relaxations x 2 x x 2 x x 1 x 1

4 Branch-and-Bound First iteration

5 Branch-and-Bound First iteration Second iteration

6 Branch-and-Bound First iteration Second iteration

7 The ααbb underestimator / algorithm ααbb underestimator for a CC 2 general function ff(xx): nn ff xx = ff xx + αα ii (x ii LL x ii )(x ii UU x ii ) ii=1 Properties of ff xx for αα ii 0 ff xx ff xx, xx [xx L, xx U ] ff xx is convex xx [xx L, xx U ] ff x = sin 5x + x 2, x XX = 2,2 iff αα max 0, 1 2 λλ mmmmmm where λλ mmmmmm is the minimum eigenvalue of HH xx = 2 ff(xx), xx XX Maranas, Floudas, 1992, The Journal of Chemical Physics, 97, Androulakis, Maranas, Floudas, 1995, Journal of Global Optimization, 7, 337. Adjiman, Dallwig, Floudas, Neumaier, 1998, Computers & Chemical Engineering, 22, 1137.

8 αbb algorithm > 1600 citations Snapshot of a PhD with Chris Department of Chemical Engineering Paper co-authors: I.P. Androulakis S. Dallwig C. D. Maranas A. Neumaier C.A. Schweiger

9 The ability to construct convex underestimators enables the solution of more and more problems Nonlinear and mixed-integer nonlinear problems Dynamic problems Papamichail Bounding trajectories Kazazakis New underestimators Bilevel problems Kleniati, Paulavicius Branch-and-Sandwich algorithm

10 Optimisation for better prediction of solid form

11 Polymorphism Molecules can pack in different forms Abbott Laboratories believed that the HIV drug Ritanovir Form I was stable, but Form II appeared unexpectedly in the manufacturing process Form II is more stable and less soluble: bioavailability of the drug changed Ritanovir had to be recalled at a cost of several hundred million dollars

12 Predicting the unpredictable There are many mysteries of nature that we have not solved. Hurricanes, for example, continue to occur and often cause massive devastation. Meteorologists cannot predict months in advance when and with what velocity a hurricane will strike a specific community. Polymorphism is a parallel phenomenon. We know that it will probably happen. But not why or when. Unfortunately, there is nothing that we can do today to prevent a hurricane from striking any community or polymorphism from striking any drug. Dr. Eugene Sun, Abbott Laboratories, 1998, at press conference on ritonavir crisis, when a more stable polymorph of this HIV drug appeared in manufacture

13 Predicting the unpredictable 4/11 1/8 1/11 1/6

14 Ab initio crystal structure prediction: A large and expensive nonlinear optimisation problem Unit cell is determined by: β lattice lengths a, b and c lattice angles α, β and γ positions of all atoms within unit cell rˆ, = 1,..,, = 1,.., ji i N j Z c ˆr ji,z α γ b ˆr ji,y min a, b, c, α, β, γ, r min a, b, c, α, β, γ, r ij ij ( T, P) ( T, P) G U latt = U U intra + P V + U rep/disp inter T S + U All low-energy local minima, including the global minimum ˆr ji,x ele inter a

15 Crystal structure prediction methodology Department of Chemical Engineering isolated-molecule quantum mechanical calculations empirical models for dispersion/repulsion vdw U, ii, ' 1,.., N = ii s CrystalPredictor 1-4 global search for low-energy structures with limited molecular flexibility and simple models molecular connectivity conformational analysis intramolecular energies & intermolecular electrostatic potentials as functions of molecular conformation polymorphs charge density 1 Karamertzanis & Pantelides J. Comp. Chem., 2005, 26, Karamertzanis & Pantelides Mol. Phys., 2007, 105, Habgood, Sugden, Kazantsev, Adjiman, Pantelides, JCTC, 2015, 11, Sugden, Adjiman, Pantelides, Acta Cryst B, 2016, B72, 864

16 Crystal structure prediction methodology Department of Chemical Engineering isolated-molecule quantum mechanical calculations empirical models for dispersion/repulsion vdw U, ii, ' 1,.., N = ii s Lowest-energy structures only CrystalPredictor 1-4 CrystalOptimizer 5 global search for low-energy structures with limited molecular flexibility and simple models local minimization of lattice energy with extensive molecular flexibility and more accurate models molecular connectivity conformational analysis intramolecular energies & intermolecular electrostatic potentials as functions of molecular conformation polymorphs charge density 5 Kazantsev, Karamertzanis, Adjiman, & Pantelides, JCTC, 2011, 7, 1998.

17 Molecular flexibility Intramolecular energy U intra O1-C1-C2-C3 ( o ) O1-C1-C2-C3 ( o ) H1-O1-C1-C2 ( o ) Intermolecular Energy, U inter H1-O1-C1-C2 ( o ) H1 O1 Successful prediction depends on correct balance between inter- and intramolecular forces C1 C3 C2 O1-C1-C2-C3 ( o ) Lattice energy, U inter + U intra H1-O1-C1-C2 ( o ) Accounting for flexibility comes at a huge computational expense

18 Achieving high accuracy at lower cost: Adaptive Local Approximate Models (LAMs) Carry out QM calculations at specific points T2 (º) T1 (º) Use an adaptive grid: o High density of LAMs in chemically interesting areas o A large search space covered with fewer LAMs

19 Accuracy of adaptive LAMs Intramolecular energy for ROY molecule Exact ab initio surface (5 o scan) Uniform coarse grid (40 o spacing) Adaptive grid

20 Target XXV 6 th blind test of crystal structure prediction Overlays of experimental & predicted structures U latt (kj/mol) Experimental Rank=1 rmsd 20 =0.317Å rmsd 20 = Å rmsd 1 = Å including H atoms rmsd 1 =0.035 Å Not including H atoms rmsd 1 = Å including H atoms rmsd 1 = Å Not including H atoms ρ(g/cm 3 )

21 Target XXVI 6 th blind test CrystalPredictor Department of Chemical Engineering U latt (kj/mol) Only 3643 non-uniform LAMS needed A 70% reduction on the 11,858 LAMs that an equivalent uniform grid would require Density (g/cm 3 )

22 -180 Target XXVI 6 th blind test Final results: CrystalOptimiser Department of Chemical Engineering U latt (kj/mol) RMSD 20 : Å RMSD 1 : Å Density (g/cm 3 )

23 Applications of crystal structure prediction Olanzapine 1 Molecule XX 2 (Eli Lilly) (5 th blind test) GSK269984B 3 (GlaxoSmithKline) Tazofelone 4 (Eli Lilly) Axitinib 5 (Pfizer) Target XXV 6 (6 th blind test) BMS (Bristol Myers Squibb) 1 Bhardwaj R.M., et al.,(2013), Cryst. Growth Des.; 2 Kazantsev A.V., et al., (2011), Int. J. Pharm.; Bardwell et al., (2011), Acta Cryst. B; 3 Ismail S.Z., et al., (2013), Cryst. Growth Des.; 4 Price L.S., et al., (2014), J. Mol. Struct. 5 Vasileiadis et al. (2015) Chem Eng Sci; 6 Habgood et al.(2015) JCTC; 7 Sugden et al. (2016) Acta Cryst B

24 Concluding remarks Optimisation is ubiquitous Nonlinear, mixed-integer, large-scale, bilevel, dynamic problems A key tool to predict behaviour/properties and to design better systems Advances are made through better theories, algorithms, and implementations but also via applications: problem formulation, tailored algorithms

25 Thank you Department of Chemical Engineering Students and postdocs Collaborators Ioannis Papamichail Andrei Kazantsev Manolis Vasileiadis Christina-Anna Gatsiou Nikolaos Kazazakis Panos Karamertzanis Polyxeni Kleniati Matthew Habgood Isaac Sugden Remigijus Paulavicius While at Princeton: Ioannis Androulakis Costas Maranas Carl Schweiger Arnold Neumaier (Vienna) Stefan Dallwig (Vienna) Costas Pantelides Sally Price (UCL) EPSRC And above all Thank you, Chris