Optimization and Frustration:
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1 Optimization and Frustration: The Dynamical Lattice Model of Proteins Sigismund Kobe and Frank Dressel Institut für Theoretische Physik, Technische Universität Dresden, D Dresden, Germany (
2 Outline» Protein and models of proteins» Dynamical Lattice Model» Global optimization» Results» Protein design» Energy landscapes» Summary, perspectives, conclusion
3 Optimization and Frustration Molecular modeling of biological structures in:
4 Protein» 3d complex structure formed by (different) amino acids (aa).» biological function: machines in cells. Chemical structure of aa R: one of 20 possible sidechains (SC)
5
6 Protein models high complexity Known models:» Lattice» Off lattice computability lattice model off lattice model
7 Dynamical Lattice Model Assumptions:» Atoms: hard spheres.» Atomic bonds are fixed in length and orientation» Sidechains: spheres (diameter according to their Van der Waals volume), touching the Cα.
8 Dynamical Lattice Model Cα ' lies in the plane Cα C N': C-N' / Å 1.32 CN'Cα ' / º 123 Cα C / Å 1.53 Cα CN' / º 114 N-Cα / Å 1.47 NCα C / º 110 ω 180» two degrees of freedom (ϕ, Ψ)
9 Dynamical Lattice Model Task: calculate the next Cα position reduction of Ramachandran map (ϕ vs. ψ ) rrm Cluster analysis Ramachandran map of Arginin (G.J. KLEYWEGT, T.A. JONES, 1996) rrm of Arginin
10 Dynamical Lattice Model Twist according to one of the rrm possibilities twist rrm of Arginin» Spatial (twisted) position of the Cα '» Continue with Cα ' as the new starting point
11 Dynamical Lattice Model Crucial points: reduced Ramachandran map» few but relevant Cα ' positions» few but relevant next aa positions» Pairwise interaction: Etotal = E<ij>» different structures per aa mixed q state Potts model qav = 3.65
12 Global optimization Interaction matrix a c d e f g h i k l m n p q r s t v w y O a c d e f g h i k l m n p q r s t v y G. SETTANNI et al. (1999)» Pairwise interaction energy between Cβ : Eij= (eij+ei0+ej0) {tanh[(8.0 r1 r2 )/2]/2+0.5} (cutoff: 8 Å) w
13 distance of Cβ -atoms/å
14 Global optimization Problem:» Find the structure of a protein with N aa, which belongs to the exact minimum of energy Egs Solution:» Use branch and bound algorithm developed for spin glass models S. KOBE, A. HARTWIG (1978)
15 Results Input: Sequence of the protein only. Used model parameters:» Interaction matrix of the aa (cutoff = 8 Å)» rrm: single linkage cluster analysis» Cα distance constraints: dmin, C C ' = 3 Å» Side chain overlap exclusion α α Output: 3d structure, Egs
16 Results Name: Polyalanine Sequence: A40 Sequence length (N): 40
17
18 Results Name: Trp cage Sequence: NLYIQWLKDGGPSSGRPPPS Sequence length (N): 20 PDB id: 1L2Y RMSD: 5.90 Å
19
20 Results Name: Insulin Sequence (Chain A): GIVEQCCTSICSLYQLENYCN N = 21 PDB id: 1B19:A RMSD: 5.54 Å Note: Cystein interactions 6 20, 7 11, 7 20, deleted!!!
21
22 Results Name: Alzheimer disease Amyloid A4 Sequence: DAEFRHDSGYEVHHQKLVFF AEDVGSNKGAIIGLMVGGVV N = 40 PDB id: 1AML RMSD: 5.95 Å
23
24 PDB structure:
25 Results RMSD: Comparison with real structure name PDB N RMSD/Å synth. α -helix 1AL compstatin 1A1P conotoxin 1AKG trp cage 1L2Y cecropin magainin hybrid 1D9J insulin A 1B19:A insulin A (red.) 1B19:A Alzheimer A4 1AML
26 Protein Design Name: hypothetical Telluride 2005 protein Sequence: ENERGYLANDSCAPESANDDYNAMICS N= 27 PDB id: to be announced Biological function:???
27
28 ENERGYLANDSCAPESDYNAMICS (N = 24)
29 Energy Landscapes ordinate: abscissa: lines: example: energy HAMMING distance [number of different (POTTS) states related to the ground state] connect structures, which differ from each other in only 1 state 1D9J (N = 20)
30 1D9J ground state
31
32 Dynamical Lattice Model summary Lattice» no real structure» good computational performance Off lattice» real structure conformations» high computational effort Dynamical Lattice Model» similarity with real structure» good computational performance allows global optimization» secondary structure of proteins (in the native state) is obtained
33 Dynamical Lattice Model outlook» real structure dynamics» heuristic algorithms (ground state with high probability)
34 Myoglobin (N = 154)
35 Conclusion: Optimization and Frustration are basic concepts in nature DLM spatial structure of proteins energy landscapes and dynamics Acknowledgment: A. HARTWIG
36
37 ERNST ISING 1925
38 Ernst Ising ( ) Johanna Ising (*1902) (photo: Peoria/IL, U.S.A., March 1996)
39 Method: branch-and-bound example: N = 8 Jij =( ) )
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