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Msi 0112 t2
Msi 0112 t2
Msi 0112 t2
Msi 0112 t2
Msi 0112 t2
Msi 0112 t2
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Msi 0112 t2

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molecular simulations: molecular interactions, docking, binding

molecular simulations: molecular interactions, docking, binding

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  • 1. MSI, MScBIOINFO 2012 Solvation methods in molecular simulations Not just a speed issue Jordi Villà i Freixa jordi.villa@upf.edu // cbbl.imim.es Programa de Recerca en Informàtica Biomèdica Universitat Pompeu Fabra Barcelona Barcelona, January 2012
  • 2. MSI, MScBIOINFO 2012 1 Table of contentsOutline 2 Coulomb’s law of electrostatic forcesCoulombFEPIon 3 Investigating processes: free energy perturbationschannels Microscopic or discrete methodsnon-equilibrium Semimacroscopic methodsGramicidin 4 Ion channels 5 Exploring the use of non-equilibrium MD 6 Example: Gramicidin A
  • 3. Binding of cetuximabMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Shoichet, 2007]
  • 4. Electrostatics principlesMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin
  • 5. Electrostatics principlesMSI, MScBIOINFO 2012OutlineCoulombFEP Charge interactions obey Coulomb’s lawIonchannelsnon-equilibriumGramicidin
  • 6. Electrostatics principlesMSI, MScBIOINFO 2012OutlineCoulombFEP Charge interactions obey Coulomb’s lawIonchannels When more than two charges interact, the energies arenon- sums of Coulombic interactionsequilibriumGramicidin
  • 7. Electrostatics principlesMSI, MScBIOINFO 2012OutlineCoulombFEP Charge interactions obey Coulomb’s lawIonchannels When more than two charges interact, the energies arenon- sums of Coulombic interactionsequilibrium Electric field, Gauss’s law and electrostatic potentialsGramicidin allow us to do such calculations
  • 8. Coulomb’s law IMSI, MScBIOINFO 2012Outline 1 q1 q2 U(r ) = (1)Coulomb 4πε0 εr rFEPIon Clearly long ranged (r −1 ) with respect to dispersionchannels forces (r −6 ) and strongnon-equilibrium Polarizable media (charges redistribute in response to anGramicidin electric field) shield charges strongly (large εr ) As εr depends on temperature, Eq. 1 resembles, actually, a free energy expression. Polarizability arises from: permament dipoles, atomic polarizabilities, hydrogen bonds
  • 9. Coulomb’s law IIMSI, MScBIOINFO 2012 The Bjerrum length lB describes the charge separation atOutline which the Coulomb energy U(r ) equals the thermalCoulomb energy RT [?] For example, for q1 = q2 = e: .FEPIonchannels 1 e2 N lB =non- 4πε0 εr RTequilibriumGramicidin At vacuum this occurs around 560 (in water this needs to be divided by εr = 80). At bigger distances, the interactions are weaker than thermal energy RT , and particles are governed by Brownian motion.
  • 10. Charges interact weaker in waterMSI, MScBIOINFO 2012 U(r )OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin RT vacuum r water lB , vacuum
  • 11. Electrostatic force and fieldMSI, MScBIOINFO 2012 1 q1 q2 r F=− U= 4πε0 εr r 2 rOutlineCoulomb which, for a unit charge, becomes the electrostatic field ifFEP dependent on just one particle with charge q:Ionchannels F(r) 1 q r E(r) = =non-equilibrium qtest 4πε0 εr r 2 rGramicidin For more complex settings of charges, one may use Gauss’ law, which equals the flux of the electrostatic field through any bounding surface to the sum of all charges enclosed: n 1 φ= εE · ds = qi (2) surface ε0 i=1
  • 12. Electrostatic potentialsMSI, MScBIOINFO Electrostatic field describes forces (vectorial). Electrostatic 2012 potential describes energies (scalar). We start by:Outline δw = −Fdl = −qEdlCoulomb which allows us to define the work done against the fieldFEP when moving a charge q between two points as:Ionchannels Bnon- wA→B = −q Edlequilibrium AGramicidin and the corresponding difference in electrostatic potentials as: B wAB ψB − ψA = =− Edl qtest A (equivalent to E = − ψ). Thus, the electrostatic potential qfixed around a point charge, ψtest = 4πε0 εr r , and that produced by a charge density, ψtest = 4πε0 εr V ρrfixed dV (r12 , distances 1 12 between all the charges and the test charge).
  • 13. Electrostatic potential surfacesMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin The work along equipotential curves is zero. In addition, electrostatic interactions are conservative forces.
  • 14. Poisson equationMSI, MScBIOINFO Gauss theorem states that the flux of a vector field through a 2012 closed surface is equal to the divergence of the same field throughout its volume:OutlineCoulomb v · ds = · vdVFEP S V ∂ ∂ ∂Ionchannels where = ( ∂x , ∂y , ∂z ). For electric fields:non-equilibrium εr E · ds = εr · EdVGramicidin S V which, upon use of Gauss law in the left hand side for a given charge density ρ becomes the differential form of Gauss law: ρ εr · E = ε0 Finally, considering E = − ψ one obtains Poisson’s equation: 2 ρ ψ= (3)
  • 15. Poisson-Boltzmann equationMSI, MScBIOINFO 2012OutlineCoulomb What does it happen if we have a given ion concentration?FEP Poisson equation transforms into the Poisson-BoltzmannIon expression:channels 2 ρr ψ(r) =non-equilibrium ε0 εrGramicidin −zi qψ(r) · [ε(r ψ(r)] = −4πρr − 4π ci∞ zi qλr exp (4) kB T i
  • 16. Mixing thermodynamics with electrostaticsMSI, MScBIOINFO 2012 We shall investigate two situations:OutlineCoulomb Moving charges from one point to another within a fixedFEP electrostatic field (Session V on ion channels)Ionchannels Computing free energies for creating the electrostaticnon- fields: "charging up" an assembly of originally unchargedequilibrium particles:Gramicidin 1 1 qi qj ∆Gel = wel = qi ψi = 2 8πε0 εr rij i i j=i
  • 17. Ion solvation: Born Energy IMSI, MScBIOINFO 2012Outline Charging up a continuous distribution resembles the aboveCoulomb equation, but using a continuous description of the charge:FEPIon 1channels ∆Gel = ρψV dVnon- 2 VequilibriumGramicidin or, for a charged sphere or radius a: 1 1 q 1 q 1 q2 ∆Gel = σψS dS = 4πa2 = 2 S 2 4πa2 4πε0 εr a 4πε0 εr 2a
  • 18. Ion solvation: Born Energy IIMSI, MScBIOINFO 2012 Discharge ∆G1 q = +1 q=0OutlineCoulombFEP TransferIon ∆Gchannels ∆G2non-equilibriumGramicidin q = +1 q=0 Charge ∆G3 q2 1 1 ∆Gel = ∆G1 + ∆G2 +∆G3 = − 8πε0 a ε0 εw ≈0
  • 19. Investigating processes: free energy perturbations (FEP)MSI, MScBIOINFO 2012 Absolute value of A (Helmholtz free energy, the “natural”Outline quantity in the canonical, NVT , ensemble) is difficult to get,Coulomb but its relative value is easier:FEPmicro- QYsemi- ∆A = AY − AX = −kB T lnIon QXchannelsnon- Zwanzig developed a better way to evaluate this quantity:equilibriumGramicidin ∆A = −kB T ln < exp[(HY − HX )/kB T ] >X Most times X and Y do not overlap in phase space and thus the evaluation of the above average is dificult.
  • 20. Free energy perturbations (FEP)MSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin A is a state function, which implies we can use any path to go from one state to the other with identical result.
  • 21. Free energy perturbations (FEP)MSI, MScBIOINFO 2012 Thus, we can envisage an imaginary path that slowly leads from X to Y:OutlineCoulomb ∆A = AY − AXFEP = (AY − A1 ) − (A1 − AX )micro-semi- QY Q1Ion = −kB T lnchannels Q1 QXnon- = −kB T ln < exp[(H1 − HX )/kB T ] >XequilibriumGramicidin −kB T ln < exp[(HY − H1 )/kB T ] >1 which can be extended to as many states as we need: QY Q3 Q2 Q1 ∆A = AY − AX = −kB T ln ··· QY−1 Q2 Q1 QX
  • 22. Free energy perturbations (FEP)MSI, MScBIOINFO To efficiently implement a Free Energy Perturbation method, 2012 we need to move the systems from X to Y in small steps (even if they have no real physico-chemical meaning!)Outline through a mapping potential:CoulombFEP Em = λm EY + (1 − λm )EXmicro-semi-Ion where λm goes from 0 to 1.channelsnon-equilibriumGramicidin
  • 23. Alchemical transformationsMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ion We can use FEP tochannels do alchemicalnon-equilibrium transformationsGramicidin
  • 24. Alchemical transformationsMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin
  • 25. MSI, MScBIOINFO 2012OutlineCoulomb ExampleFEPmicro- Create a code that can evaluate the free energy differencessemi-Ion between two states U1 and U2 characterized by the potentialschannels U1 = 10x 2 and U2 = (x − 20)2 + 40, with U resulting in kcalnon-equilibrium mol−1 , using free energy perturbations. ConsiderGramicidin temperatures T = 10, T = 50 and T = 300.
  • 26. Molecular representation needs to include molecule size... and moreMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin [Kitchen et al., 2004]
  • 27. Summary of solvation methodsMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin To learn more: [Warshel, 1991, Grochowski and Trylska, 2008, Ytreberg et al., 2006, Dill et al., 2005]
  • 28. Linear Response approximationMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin
  • 29. Linear Response approximationMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibrium λa =< Ub − Ua >b +∆Ga→bGramicidin
  • 30. Linear Response approximationMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibrium λa =< Ub − Ua >b +∆Ga→bGramicidin λb =< Ub − Ua >a −∆Ga→b
  • 31. Linear Response approximationMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibrium λa =< Ub − Ua >b +∆Ga→bGramicidin λb =< Ub − Ua >a −∆Ga→b 1 ∆Ga→b = [< Ub − Ua >a + < Ub − Ua >b ] 2
  • 32. ExerciseMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi- ExampleIon Add the possibility to obtaining LRA results in your FEP codechannels and compare with the FEP result.non-equilibriumGramicidin
  • 33. Linear Interaction EnergyMSI, MScBIOINFO 2012OutlineCoulombFEP Why not expanding the LRA idea to the non-polar part?micro-semi-Ionchannels ∆G = α E vdW b − E vdW f +β E el b − E el f +γnon-equilibrium [Hansson et al., 1998, de Amorim et al., 2008]Gramicidin
  • 34. Protein dipoles Langevin dipolesMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin [Warshel, 1991]
  • 35. pKa calculationsMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin
  • 36. PDLD/S-LRAMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin
  • 37. PDLD/S-LRAMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin [Sham et al., 2000]
  • 38. Semiexplicit methodMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin [Fennell et al., 2010]
  • 39. Semiexplicit methodMSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin [Fennell et al., 2010]
  • 40. MSI, MScBIOINFO 2012OutlineCoulombFEPmicro-semi-Ionchannelsnon-equilibriumGramicidin
  • 41. Special structures: ion channelsMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin
  • 42. Special structures: ion channelsMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin
  • 43. Special structures: ion channelsMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin
  • 44. From PMF to CrooksMSI, MScBIOINFO 2012 ξ = ξ(R N )OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin
  • 45. From PMF to CrooksMSI, MScBIOINFO 2012 ξ = ξ(R N )Outline d Rd Pδ(ξ(R N ) − ξ) exp(−βH)Coulomb ρ(ξ) =FEP d Rd P exp(−βH)Ionchannelsnon-equilibriumGramicidin
  • 46. From PMF to CrooksMSI, MScBIOINFO 2012 ξ = ξ(R N )Outline d Rd Pδ(ξ(R N ) − ξ) exp(−βH)Coulomb ρ(ξ) =FEP d Rd P exp(−βH)Ionchannels ρ(ξb ) ∆Aξb −ξa = −kT lognon- ρ(ξa )equilibriumGramicidin
  • 47. From PMF to CrooksMSI, MScBIOINFO 2012 ξ = ξ(R N )Outline d Rd Pδ(ξ(R N ) − ξ) exp(−βH)Coulomb ρ(ξ) =FEP d Rd P exp(−βH)Ionchannels ρ(ξb ) ∆Aξb −ξa = −kT lognon- ρ(ξa )equilibrium z(t) z(t)Gramicidin ∂Vz (ξ) W (z(t)) = dz =k dz(z − ξ) z0 ∂z z0
  • 48. From PMF to CrooksMSI, MScBIOINFO 2012 ξ = ξ(R N )Outline d Rd Pδ(ξ(R N ) − ξ) exp(−βH)Coulomb ρ(ξ) =FEP d Rd P exp(−βH)Ionchannels ρ(ξb ) ∆Aξb −ξa = −kT lognon- ρ(ξa )equilibrium z(t) z(t)Gramicidin ∂Vz (ξ) W (z(t)) = dz =k dz(z − ξ) z0 ∂z z0 ∆A(z) = (< WF >z − < WR >z ) /2 Wd (z) = (< WF >z + < WR >z ) /2
  • 49. From PMF to CrooksMSI, MScBIOINFO 2012 ξ = ξ(R N )Outline d Rd Pδ(ξ(R N ) − ξ) exp(−βH)Coulomb ρ(ξ) =FEP d Rd P exp(−βH)Ionchannels ρ(ξb ) ∆Aξb −ξa = −kT lognon- ρ(ξa )equilibrium z(t) z(t)Gramicidin ∂Vz (ξ) W (z(t)) = dz =k dz(z − ξ) z0 ∂z z0 ∆A(z) = (< WF >z − < WR >z ) /2 Wd (z) = (< WF >z + < WR >z ) /2 kB T ∆z D(z) = =v
  • 50. Crooks provides the most accurate free energyMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 51. Gramicidin AMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin
  • 52. Gramicidin AMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 53. Channel occupationMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 54. Forward and reverse workMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 55. Forward/Reverse work probabilitiesMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 56. Final free energy profileMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 57. Final dissipative workMSI, MScBIOINFO 2012OutlineCoulombFEPIonchannelsnon-equilibriumGramicidin [Fabritiis et al., 2008]
  • 58. MSI, MScBIOINFO de Amorim, H. L. N., Caceres, R. A., and Netz, P. A. (2008). 2012 Linear interaction energy (lie) method in lead discovery and optimization. Curr Drug Targets, 9(12):1100–1105. Dill, K. A., Truskett, T. M., Vlachy, V., and Hribar-Lee, B. (2005).Outline Modeling water, the hydrophobic effect, and ion solvation.Coulomb Annu Rev Biophys Biomol Struct, 34:173–199.FEP Fabritiis, G. D., Coveney, P. V., and Villà-Freixa, J. (2008). Energetics of k+ permeability through gramicidin a by forward-reverse steered molecular dynamics.Ion Proteins, 73(1):185–194.channels Fennell, C. J., Kehoe, C., and Dill, K. A. (2010).non-equilibrium Oil/water transfer is partly driven by molecular shape, not just size. J Am Chem Soc, 132(1):234–240.Gramicidin Grochowski, P. and Trylska, J. (2008). Continuum molecular electrostatics, salt effects, and counterion binding–a review of the poisson-boltzmann theory and its modifications. Biopolymers, 89(2):93–113. Hansson, T., Marelius, J., and Åqvist, J. (1998). Ligand-binding affinity prediction by linear interaction energy methods. J. of Comput-Aided Mol. Design, 12(1):27–35. Kitchen, D., Decornez, H., Furr, J., and Bajorath, J. (2004). Docking and scoring in virtual screening for drug discovery: Methods and applications. 3:935–949.
  • 59. MSI, MScBIOINFO 2012Outline Sham, Y., Chu, Z., Tao, H., and Warshel, Ash, E. L. (2000). Examining methods for calcultions of binding free energies: Lra, lie, pdld-lra and pdld/s-lra calculations ofCoulomb ligand binding to an hiv protease.FEP Proteins, 39:393–407.Ion Shoichet, B. K. (2007).channels No free energy lunch. Nat Biotechnol, 25(10):1109–10.non-equilibrium Warshel, A. (1991). Computer Modeling of Chemical Reactions in Enzymes and Solutions.Gramicidin John Wiley & Sons. Ytreberg, F. M., Swendsen, R. H., and Zuckerman, D. M. (2006). Comparison of free energy methods for molecular systems. J Chem Phys, 125(18):184114.

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