Sampling the Potential Energy Surface Energy minimization downhill search, generally to nearest local minimum can be used to relax structures might be useful to define local changes due to mutation Normal mode analysis defines "characteristic motions which are distortions about a local minimum structure orders motions "easy(low frequency to hard"(high) Molecular dynamics movie of motion at given temperature (300 k) equivalent to statistical mechanical ensemble Monte Carlo/ simulated Annealing Describe properties of the landscape and thermodyanmic parameters without simulating how the molecular actually moves
Sampling the Potential Energy Surface • Energy minimization – “downhill” search, generally to nearest local minimum – can be used to relax structures – might be useful to define local changes due to mutation • Normal mode analysis – defines “characteristic motions”, which are distortions about a local minimum structure – orders motions “easy” (low frequency) to “hard” (high) • Molecular dynamics – movie of motion at given temperature (300 K) – equivalent to statistical mechanical ensemble • Monte Carlo/Simulated Annealing – Describe properties of the landscape and thermodyanmic parameters without simulating how the molecular actually moves
Energy minimization Potential Energy, U(R Conformational Space, R X-ray structure Iterative procedures terminate when reach F=-VU(r) tolerance, such as small gradient M +I=r+sF Poor initial structure leads to poor local minimum Multiple minimum problem ONLY FINDS LOCAL MINIMA!
Energy Minimization Conformational Space, R Potential Energy, U(R) X-ray structure X • Iterative procedures; terminate when reach tolerance, such as small F ( i = −∇ r U )i gradient • Poor initial structure leads ri+1 = ri + δ Fi to poor local minimum • Multiple minimum problem ONLY FINDS LOCAL MINIMA!
Uses of simple minimization 1. The"minimum perturbation approach"to modeling a mutation Assume structure of single-site mutant is close to known Wild-type structure Find stable conformations for mutant side chain in context of wild-type protein Use energy minimization to relax candidate structures (all degrees of freedom) Shih, brady and Karplus, Proc. Nat. Acad. Sci. USA82: 1697-1700 (1985); hemagglutinin Gly to Asp mutation modeled accurately 2. Relieving strain before analyZing the energy of an experimental or predicted structure 3. Structure building and refinement when solving structures using X-ray crystallography or NMR
Uses of simple minimization 1. The “minimum perturbation approach” to modeling a mutation – Assume structure of single-site mutant is close to known wild-type structure • Find stable conformations for mutant side chain in context of wild-type protein • Use energy minimization to relax candidate structures (all degrees of freedom) Shih, Brady, and Karplus, Proc. Natl. Acad. Sci. USA 82: 1697–1700 (1985); hemagglutinin Gly to Asp mutation modeled accurately 2. Relieving strain before analyzing the energy of an experimental or predicted structure 3. Structure building and refinement when solving structures using X-ray crystallography or NMR
Normal Mode analysis Characteristic Motions and their relative ease Thermodynamic Properties Mathematical Approximation: Series of Independent Harmonic oscillators U ()=U(R)+VVRR-R)+,∑∑ r-r0r7-r0) ar ar Corresponds to local Expansion of potential Surface as parabolic U(R) c harmonic approximation R
Normal Mode Analysis • Characteristic Motions and their Relative Ease • Thermodynamic Properties Mathematical Approximation: Series of Independent Harmonic Oscillators 0 ∂ 2 U ( ( R U ) = R U 0 ) + ∇ ( R U 0 )( − R R 0 ) + − 1 2 ∑∑ ∂ ∂ ( r r i 0 , )( rj − rj 0 , ) +" i j >i i r ri j Corresponds to Local Expansion of Potential Surface as Parabolic U(R ) actual harmonic approximation R
Normal Modes locate"easy"Deformations U(R R Low-frequency, energetically easy motions will dominate the dynamical behavior of macromolecules
Normal Modes Locate “Easy” Deformations U(R) R • Low-frequency, energetically easy motions will dominate the dynamical behavior of macromolecules