What would happen? + -

Transcription

1 What would happen? + -

2 Lecture Plan Molecules spend more time in low energy conformations Predicting a structure is approximated as the search for the lowest energy conformation on a protein model Modeling requires data, a molecular model and a strategy to explore protein conformations All practical methods are heuristics; which means that they aren't guaranteed to find THE lowest energy conformation Bigger, faster computers will not get ourselves out of this situation

3 What is a model? Model: a hypothetical description of a complex entity or process. wordnet.princeton.edu/perl/webwn Sir John Kendrew and his model of myoglobin, 1958

4 To model a protein we need? A description of the system. (atom coordinates) An abstraction of what is going on. A way to know whether we are going anywhere. (Force field) A process to find the best protein structure. (optimization)

5 Force fields A set of mathematical functions and parameters to model each relevant forces (VdW, etc.). The energy of 1 atom is the sum of all these terms. The energy of a protein is the sum of energies of all atoms. Force field are mathematical expressions that include all known important factors for molecular interactions. The Natural choice is to use the chemical energy as a score and try to find the structure with the lowest free energy.

6 It doesn t work with free energies G = H TS Free energy is temperature-dependent. Entropic contribution cannot be calculated from a snapshot. Something else must be used.

7 Modeling all relevant energy terms. Energy function A sum of terms that approximate the contributions of known theoretical microscopic forces. EFF = Estr + Ebend + Etors + EVdW + Eel + Ecross

8 Modeling bond stretch/compression In the case of bond stretching/compression, we need to measure the distance r between two atoms, and get from the force field what should be the optimal distance ro for a given pair of atoms.

9 Modeling bond stretch/compression Realistic models of bond stretch-compression are computationally expensive. This figure show how simpler model fare at modeling bond stretching. ab ab ab ab 2 P2 E str R =k 2 R R 0 ab ab 2 ab ab ab 3 ab ab ab 4 P4 E str R ab =k ab R R k R R k R R ab morse E str R =D 1 e R ab

10 Three factors to consider Parameterization nightmare Can someone come up with all these numbers? Generalization How robust is the simulation in a range of conditions. Efficiency The longer it takes to perform a single task, the fewer iterations will be computed in the same amount of time.

11 Modeling VdW Lennard-Jones R0 12 R 0 6 E str R = [ ] R R Rij = ( xi x j ) + ( yi y j ) + ( zi z j ) Is actually a computational stunt so there is no need to compute R but rather use Rn where n is an even factor. EXP6 ( R ) = Ae BR C R6

12 Modeling Electrostatic interactions Modeling electrostatic interaction is critical in many situations. Why? Electrostatic fields decay with 1/ distance. Which makes them the longestranged interactions. QAQB Eel ( RAB ) = ε RAB Coulomb s Law

13 Non-bond interactions create a computational bottleneck Computational cost of non-bonded energy (VdW, El) ~99.88% of computation in protein-sized models. Most of this is very small and does not contribute to the total energy significantly. The number of non-bonded increase to the square of the number of atoms while bonded interaction are increasing roughly linearly.

14 Hydrophobic forces This is not an explicit term. Hydrophobic interaction are due to the difference in free energy between water molecules and polar/non-polar side chains. The effect is thus intrinsic to the computations of electrostatic forces.

15 A process to find the best possible conformation Force fields provide all the functions and parameters to compute the energy of 1 structure. Finding the best structure comes down to finding the structure with the lowest energy. Need a process to change the conformation toward a better structure: optimization. The optimization is iterated until it is reasonable to believe that no better structure can be found.

16 Principle of optimization You start with a protein for which you know all coordinates. Evaluate the energy Find a better structure, usually with small changes Repeat until no better structure can be found. This task is usually NEVER straightforward, unless the system would be made of a small number of atoms.

17 Minimizing The gradient method For each atoms: 1. Computer the force vectors for each term 2. Sum all vectors 3. Move the atoms over a small distance along their resultant vectors. Repeat until all resultant force vectors are of length 0.

18 Optimizing simple functions Energies R Minimum (Ro) R

19 Energies Optimizing more complex functions Conformations Minimum

20 Molecular Simulations Time dependent methods (Molecular Dynamics) Make use of classical mechanics equations such as: F = ma Each atom gets a random kinetic energy vector which is added to the resultant force vector. This simulates thermal motion.

21 Molecular Simulations Verlet Algorithm Numerical solution to Newton s equations r i 1=r i v i t 1/ 2 ai t 2 r i 1=r i v i t 1/2 ai t 2 ri+1 = ( 2ri ri 1 ) + ai t 2 a can be computed from F = ma

22 Molecular Simulations t Reasonable: Femtoseconds Scope of simulation (ideal): (practical): Millisecond 10-3 Microsecond 10-6

23 Molecular Simulations Timesteps To simulate a microsecond, it takes a very, very long time To simulate slow processes, the time scale isn t realistic.

24 Other Optimization strategies Simulated Annealing Scaling down the energy landscape makes the crossing of barrier more probable. Time to do so it in short supply, however!

25 Why modeling proteins Anticodon binding site on erf3 2 possibilities. From phylogenetic information, a few residues were identified as players. Use molecular mechanics to see whether the surface of the protein can accommodate an anti-codon. Inagaki, Y., Blouin, C., Doolittle, W.F., and Roger, A.J Convergence and constraint in eukaryotic release factor 1 (erf1) domain 1: the evolution of stop codon specificity. Nucleic Acids Res 30:

26 Why modeling proteins Modeling a weird substrate into an active site. Mandelate racemase can bind a substrate with two rings! Is there room for this in the wild type active site? The answer is yes, although a bit counter-intuitive. Siddiqi, F, Bourque, J., Jiang, H., Gardner, M., St. Maurice, M., Blouin, C., and Bearne S.L., Perturbing the Hydrophobic Pocket of Mandelate Racemase to Probe Phenyl Motion During Catalysis. Biochemistry 44(25):

27 Folding polypeptide isn t expected to work out as well because Empirical models are parameterized with pre-folded proteins. Role of water in partially folder proteins is significant. Time scale for folding a protein is still a bit out of range for simulation. Assistance in folding, either from Chaperones, other monomers isn t there. Folding process is seeded by the chain extension during translation. Folding of peptide occurs an a time scale COMPLETELY beyond what we can get away with today.

28 Protein folding from Scratch Must be restrained to a limited scope Two genes: TC5b and TC3b Both have references structure for validation. Sequences NLYIQWLKDGGPSSGRPPPS (TC5b; 304 atoms) NLFIEWLKNGGPSSGAPPPS (TC3b; 289 atoms) Software: AMBER 6.0 Model: AMBER Solvation: Generalize Born/solvent-accessible surface area This means that the water molecules are not explicitly defined in the simulation and the effect of the solvent is treated as a macro property.

29 Protein folding from Scratch Must be restrained to a limited scope Understanding folding and design: Replica-exchange simulation of Trp-Cage miniproteins. Pitera, JW., Swope, W Proc. Natl. Acad. Sci. USA, 100: The GIST Run 23 simulation (4ns) in parallel, each at a different temperature from hot to cold. At every 5ps, redistribute the best conformation to the coldest simulation. This is much more effective at exploring solutions than a 23 X 4ns simulation

30 Protein folding from Scratch Impact RED is the reference, GREEN is the computed model. Large Energy barriers are not as high in small, isolated structures. It is reasonable to limit the scope of these simulations to protein domains.

31 Protein folding from Scratch Validation The root mean square deviation RMSD ( n RMSD = atom =i v v wi i iref n n wi ) RMSD from any of the snapshots i =1 2.0 RMSD from the average snapshot.

32 Why multiple temperature sampling work Total energy remains constant: ET = EFF + Ek So, at higher temperatures, higher energy structures are more often observed. This means more alternative conformation sampled in the same amount of time. Each of these conformational groups gets in turn refined at lower temperature. Statistically, most of the simulation time of the coldest chain will be spent in the energy well with the lowest energy.

33 Building a large machine for protein folding IBM Blue Gene project (65K processors, never enough, however)

34 High Performance achievement in MD NAMD Open source University of Illinois, Dept. of theoretical physics Benchmark system (their big one)

35 Parallel computing and Molecular dynamics Folding proteins from an extended conformation is a difficult problem because of the crossing of energy barriers. The following slides describe the limitations of simulating the crossing of energy barrier using massively parallel techniques.

36 Limitations of Parallel computing It takes 1500 days to complete a thesis for one student If the student is helped by someone, the work may go 2X as fast: 750 days. What if 1500 students are working on the same thesis? Overhead Communication Load balancing

37 Parallel computing Factors that complicate parallelization: Some work have to be executed in a sequence Communicating the task and the results becomes an increasingly important time step as the task become small. Each individual process have to wait for the slowest one to finish, leading to a loss of efficiency. It doesn't make sense to have much more CPU than atoms in the system!

38 Time scale in protein folding In the order of micro to milliseconds This is not achievable by modern computers. ~ days for 1 experiment (~28 years) folding@home Using unspent cycles from idle hardware (PS3, Xboxes, PC (Screensaver) )

39 Crossing energy barrier Most of the time is spent waiting for the thermal motion to topple a structure over a barrier. Principle of Ensemble dynamics M CPU should take M X less time to go over a barrier. For breaking 3-10 H-bonds (~22.3 kj/m) If an even occurs on average every 10,000 ns, the chance to witness this event during a 30 ns simulation is 0.3%. If the same simulation runs on 10,000 machines, one expect to observe the event ~30 times over 30 ns.

40 Ensemble Dynamics Start M dynamic calculations with the same initial structure. Once 1 thread finds a barrier and go over it, copy the state of this thread into all other M1 replicate processes. The communication overhead is negligible if the crossings are rare events, which is the case.

41 Villin's headpiece Note how most of the interactions in the partially folded protein are nonnative. This means that in order to resume folding, these must be broken. The Villin headpiece is one of the fastest (known) folding peptide!! What about simulating anything else?

42 Energy Landscape Observe that in this figure that there are: one folding pathway One intermediate Two energy barriers

43 Progress in last 2 years Rates of protein folding appear to be correctly predicted using ensemble dynamics.

44 Progress on large systems Multiplicate simulation of SNARE-mediated vesicle fusion have been published. Ensemble molecular dynamics yields submillisecond kinetics and intermediates of membrane fusion Peter M. Kasson, Nicholas W. Kelley, Nina Singhal, Marija Vrljic, Axel T. Brunger,,, and Vijay S. Pande PNAS August 8, 2006 vol. 103 no

45 Summary Biological models are assumed to have the lowest energy. Optimization is used to find the lowest energy structures, and thus the biologically relevant conformation. Simulation time is the bottleneck. The more you sample, the more likely that the solution will be good. There are some progress in solving protein folding using heuristics and parallel computing, but the solution depend on theoretical breakthroughs, not the addition of hardware.