麻省理工大学：《Foundations of Biology》课程教学资源（英文版）Lecture 3 For a molecular simulation or model

7.91 Amy Keating How do we use computational methods to analyze, predict, or design protein sequences and structures? Theme Methods based on physics vs, methods based on our accumulated empirical knowledge of protein properties

How do we use computational methods to analyze, predict, or design protein sequences and structures? Theme: Methods based on physics vs. methods based on our accumulated empirical knowledge of protein properties 7.91 Amy Keating

For a molecular simulation or model you need 1. a representation of the protein 2. An energy function 3. a search algorithm or optimizer

For a molecular simulation or model you need: 1. A representation of the protein 2. An energy function 3. A search algorithm or optimizer

Covalent Potential Energy Terms Covalent U,+U1 bond angle +u tU orion bond ∑ k,(b-b)2 bonds bond angle ∑k(0-20 bond angles z mproper dihedral ∑k improper dihedrals 2(- torsion ∑k1+ cos(no-6) torsions Brooks et al, J. Comput. Chem. 4: 187-217 (1983)

Covalent Potential Energy Terms UCovalent = U bond + U angle bond + Uimproper dihedral + U torsion 2 U bond = ∑ 1 kb (b − b 0 ) bonds 2 2 U angle bond = ∑ 1 kθ (θ − θ ) bond angles 2 0 Uimproper dihedral = ∑ 1 improper dihedrals 2 k Φ ( Φ − Φ 0 ) U torsion = ∑ 1 kφ[1 + cos( n φ − δ )] torsions 2 Brooks et al., J. Comput. Chem. 4: 187-217 (1983) 2

Non-Covalent Potential Energy Terms Lennard-Jones potential Non-covalent U vdw +U elec B vdw accurate approximate q91 Coulombs law i>J y

Non-Covalent Potential Energy Terms UNon -covalent = UvdW +U elec Lennard-Jones potential 12 ij ij r B 6 ij ij r C − ⎛ Bij U vdW = ∑⎜ 12 − Cij ⎞⎟ i; j ⎝ rij rij 6 ⎠ “accurate” approximate qi q j U elec = ∑ Coulomb’s law i ; j εrij

The potential energy surface is a 3N-6 dimensional space For a protein, we assume a single native-structure minimum There are many local minima, and some may be close in energy to the global minimum

The potential energy surface is a 3N-6 dimensional space. For a protein, we assume a single native-structure minimum. There are many local minima, and some may be close in energy to the global minimum. Energy X

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