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Physics-based scale- and geometry-consistent coarse-grained potentials

This workshop paper discusses the use of physics-based, scale- and geometry-consistent coarse-grained potentials in molecular simulations, including smoothing energy surfaces and reproducing structure and properties of training molecules.

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Physics-based scale- and geometry-consistent coarse-grained potentials

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  1. Physics-based scale- and geometry-consistent coarse-grained potentials Adam Liwo,1 Adam K. Sieradzan,1 Emilia Lubecka,2 Agnieszka Lipska,1 Łukasz Golon,1Cezary Czaplewski1 1Faculty of Chemistry, University of Gdańsk, Gdańsk, Poland 2Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, WitaStwosza 57, 80-308 Gdańsk, Poland adam.liwo@ug.edu.pl Workshop Multi-scale Modelling, Leiden, The Netherlands, June 24-June 28, 2019

  2. Coarse-grainingis an approach to reducethecomplexity of a system (larger system, longer time scales)

  3. Twobasic types of (not structure-based) coarse-grained potentials • Physics-based potentials • smoothing the energy surface by treating rigid objects as single interaction sites (e.g., the Kihara potentials), • averaging out non-essential degrees of freedom, • reproducingstructure and propertiesof trainingmolecules. • Statistical potentialsbased on the „Boltzmann principle” • database information implicit in the potentials, • database information explicit in the potentials.

  4. Examples of coarse-grainedforcefields Recentreview: Kmiecik et al., Chem. Rev., 2016, 116, 7898–7936

  5. Physicalorigin of theall atom and coarse-grainedforcefields All-atom forcefields Coarse-grainedforcefields X: coarse-grained; y fine-grainedd.o.f’s U(X): Potential of MeanForceSurface (PMFS) R: nucleard.o.f’s; relectronic d.o.f’s E(R): Potential Energy Surface (PES) ExactPESs and PMFSs: expensive to determine/use and system specific Classical energy expression E = Estretch + Ebend + Etor + Enb + Eel [+ Esolv] Neo-classical energy expression U = Ustretch + Ubend + Utor + Unb + Uel [+ Usolv]

  6. Prototype of effective energy functions(U) in physics-basedcoarse-grained modeling F: RestrictedFree Energy (RFE); Potential of MeanForce (PMF)

  7. The probability density of a coarse-grained conformation to occur can be computed directly as the Boltzmann factor of the coarse-grained energy. The free energy is consistent with that computed from all-atom energy. A(b)

  8. Scheme of CG force field construction Define CG interactionsites and d.o.f’s Partitionthe energy site-wise Split the PMF: defineeffective energy terms Derivescale- and geometry-consistentanalyticalexpressions Parameterize energy terms Calibrate CG force field

  9. Scheme of CG force field construction Define CG interactionsites and d.o.f’s Partitionthe energy site-wise Split the PMF: defineeffective energy terms Derivescale- and geometry-consistentanalyticalexpressions Parameterize energy terms Calibrate CG force field

  10. Scheme of CG force field construction Define CG interactionsites and d.o.f’s Partitionthe energy site-wise Split the PMF: defineeffective energy terms Derivescale- and geometry-consistentanalyticalexpressions Parameterize energy terms Calibrate CG force field

  11. Define CG interactionsites and d.o.f.’s

  12. Scheme of CG force field construction Define CG interactionsites and d.o.f’s Partitionthe energy site-wise Split the PMF: defineeffective energy terms Derivescale- and geometry-consistentanalyticalexpressions Parameterize energy terms Calibrate CG force field

  13. Define CG interactionsites and

  14. Breaking down the energy intocomponents . . . … …

  15. Scheme of CG force field construction Define CG interactionsites and d.o.f’s Partitionthe energy site-wise Split the PMF: defineeffective energy terms Derivescale- and geometry-consistentanalyticalexpressions Parameterize energy terms Calibrate CG force field

  16. Factorization of the PMF into Kubo clustercumulantfunctions ‘classical’ correlation, order 2 correlation, order 3 correlation, order N-1 … A. Liwo, C. Czaplewski, H.A. Scheraga, J. Chem. Phys., 115, 2323-2347 (2001).

  17. Example: factors of order 1 (f (1) = F) 1 2 1

  18. Example: factor of order 2 (torsionalpotential) 2 2 2 1 1 3 3 - [ + ] g

  19. Multibodytermsincoarse-grainedpotentials (no sharedsecondaryvariables) (sharedsecondaryvariables) (sharedsecondaryvariables)

  20. Scheme of CG force field construction Define CG interactionsites and d.o.f’s Scale- and geometry-consistency: embedtheatomicdetails of thecoarse-grainedsitesintheeffectivepotentials Partitionthe energy site-wise Split the PMF: defineeffective energy terms Derivescale- and geometry-consistentanalyticalexpressions Parameterize energy terms Calibrate CG force field

  21. Factorization of the PMF into Kubo clustercumulantfunctions ‘classical’ correlation, order 2 correlation, order 3 correlation, order N-1 … A. Liwo, C. Czaplewski, H.A. Scheraga, J. Chem. Phys., 115, 2323-2347 (2001).

  22. Rationalapproach to derivingfunctionalexpressions of effective energy terms Generalizedcumulantapproximations (Kubo, 1963) of PMF factors

  23. Degrees of freedom to averageover • Major:angles for collectiverotation of theunitsaboutthevirtual-bondaxes {l}, CG geometry unchanged. • Minor:internalbond, bond-angle and dihedralangles {x} (ideally, onlyfluctuateaboutequilibriumvalues). lJ-1 lJ lI+1 lI

  24. General expression for theall-atom energy lJ lI RIJ = … … … = E({RIJ;lI,lJ;xIk;Jl}) E({rIk;Jl}) = (No external field Þ energy dependsonly on interatomicdistances)

  25. Interatomicdistancesinterms of group rotationangles A.K. Sieradzan, M. Makowski, A . Augustynowicz, A. Liwo, J. Chem. Phys., 2017, 146, 124106.

  26. Site-site energy expansion eIJ(RIJ) …

  27. Energy moments … … … … … Energy moments and, thus, theexpressions for coarse-grained energy termscanthus be obtained by integratingthe products of powers of f’s and g’s .

  28. Graphicalrepresentation of theintegrals J.K. J.K.

  29. General algorithm to computetheintegralsoverthe products of thepowers of f’s and g’s • Usethe Euler formula to expressthecosines. • Do allmultiplications. • Collectthetermsinwhichalll’sintheimaginaryexponentialscancel out.

  30. TheUnifiedCoarse-Grained Model of BiologicalMacromolecules Proteins (UNRES) Nucleicacids (NARES-2P) Polysaccharides (SUGRES-1P) Liwo et al., J. Mol. Modeling, 20, 2306 (2014); UNRES packageavailablefromwww.unres.pl

  31. Backbone-electrostaticinteractionsinproteins: averaged dipole-dipole interactions lJ CaJ+1 mI pJ CaJ RIJ mJ lI pI CaI CaI+1 Liwo et al., Prot. Sci., 2, 1697 (1993); J. Phys. Chem. B 108, 9421 (2004)

  32. capturestheorientationdependence of backbone-electrostaticinteractions lJ PMF CaJ+1 mI pJ CaJ mJ lI pI CaI CaI+1 BothCa…Cavirtualbondsinplane (FIJ=0) Liwo et al., Prot. Sci., 2, 1697 (1993); J. Phys. Chem. B 108, 9421 (2004)

  33. Examplepredictionwithusinglocal+averageH-bondingpotentials: CASP3 target T0061 RMSD=4.2 Å for 61 residues (80%, residues 25-85) RMSD=2.9 Å for 27 residues (36%, residues 16-42)

  34. NARES-2P results: simpleDNAs, ab initio simulations 9BNA (2 x 12) <rmsd> = 4.5 Å 2JKY (2 x 21) <rmsd> = 8.1 Å Y. He, M. Maciejczyk, S. Ołdziej, H.A. Scheraga, A. Liwo, Phys. Rev. Lett., 110, 098101 (2013)

  35. MREMD simulations: simpleRNAs 2KX8 (44) <rmsd> = 9.8 Å 2KPC (17) <rmsd> = 5.1 Å Y. He, M. Maciejczyk, S. Ołdziej, H.A. Scheraga, A. Liwo, Phys. Rev. Lett., 110, 098101 (2013)

  36. Mixedcumulants. Example: thelowest-ordercontribution to mixedcumulant, siteJshared J=I+1, K=I+2 KJ KJ KJ … A.K. Sieradzan, M. Makowski, A . Augustynowicz, A. Liwo, J. Chem. Phys., 2017, 146, 124106.

  37. + + + = + + + + Subtractsfromthemixedcumulant

  38. Torsionalpotentials (twopairs of adjacentunits) I I+1 A.K. Sieradzan, M. Makowski, A . Augustynowicz, A. Liwo, J. Chem. Phys., 2017, 146, 124106.

  39. Torsional PMF profiles for proteins Ala-Ala Ca Ca Sieradzan, et al., J. Chem. Phys., 2017, 146, 124106. Ca Ca

  40. Third order backbone-local and backbone-electrostatic (hydrogenbonding) correlationterms eJ eJ+1 eI eI eJ eJ+1 eI+1 eI+1

  41. Third order backbone-local and backbone-electrostatic (hydrogenbonding) correlationterms

  42. Istheorientation of peptidegroupsgoverned by optimum dipole alignment? b-sheets a-helices Green dots: anglesfromthe PDB; surface: gcorresponding to thealignment of thefictitiousdipoles of theinteractingpeptide group. A.K. Sieradzan, M. Makowski, A . Augustynowicz, A. Liwo, J. Chem. Phys., 2017, 146, 124106.

  43. Dal Pararo’sforce field, explicitpeptide-bonddipoles dependent on virtual-bondangles Alemani et al., J. Chem. TheoryComput., 6, 315-324 (2010)

  44. Effect of third-order backbone-local – bacbkbone-electrostatic correlations on force field ability to reproduce b-structure With correlation terms Without correlation terms Liwo et al., J. Chem. Phys., 115, 2323 (2001)

  45. Improvement of UNRES performance in CASP followingfullimplementation of scale-consistentterms A total of 98 evaluationunits Lubecka et al., J. Mol. Graphics Model., 2019, submitted

  46. Sample CASP13 UNRES predictions Rainbow: experimental; gray: model UNRES Othergroups Rainbow – experimental; Gray: model T0960-D1 (FM) : Model 1 RMSD = 8.2 A, GDT_TS = 48 T0955-D1 (TBM/FM) : Model 2 RMSD = 2.0 A, GDT_TS = 78

  47. Take-homemessage

  48. Want to try UNRES/NARES? • Description, download, instructions and more: www.unres.pl • A. Liwo et al., J. Molec. Modeling, 2014, 20, 1-15. • Server available • http://unres-server.chem.ug.edu.pl (Author: Czarek Czaplewski) • C. Czaplewski, A. Karczyńska, A.K. Sieradzan, A. Liwo, Nucleic Acids Research, 2018, 46, W304-W309(webserverissue)

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