Modeling Biological Systems Opportunities for Computer Scientists
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1 Modeling Biological Systems Opportunities for Computer Scientists Filip Jagodzinski RBO Tutorial Series 25 June 2007 Computer Science Robotics & Biology Laboratory
2 Protein: πρώτα, "prota, of Primary Importance Mmmm steak Immune Response Structural & Mechanical Functions Cellular Division Computer Science Robotics & Biology Laboratory 2
3 Tutorial Overview Proteins: Their structure, function, and dynamics Proteins as building blocks Protein dynamics Proteins and their functions The physics of protein motion Modeling & analysis of proteins Snapshots of a protein: Crystallography Numerical methods of protein motion: Molecular Dynamics Description of protein motion: Normal Mode Analysis and the Elastic Network Model Opportunities for Improvement How can we improve upon MD and NMA Are we asking the right questions? What features of protein motion should we be looking at? Utilize ideas from operational space control; kinematics from mechanics? Computer Science Robotics & Biology Laboratory 3
4 Part I: Protein Structure, function, and dynamics Proteins as building blocks Protein dynamics Proteins and their functions The physics of protein motion Computer Science Robotics & Biology Laboratory 4
5 Proteins: Interaction on an Atomic Level Computer Science Robotics & Biology Laboratory 5
6 Proteins: Elements Combine R O R C C N C H H H Computer Science Robotics & Biology Laboratory 6
7 Proteins: Backbone Constraints R R Computer Science Robotics & Biology Laboratory 7
8 Proteins: Elements Combine Computer Science Robotics & Biology Laboratory 8
9 Proteins: From Amino Acids Pehnyalaninie Tyrosine Tryptophan Computer Science Robotics & Biology Laboratory 9
10 Proteins: to Secondary Structure Computer Science Robotics & Biology Laboratory 10
11 Proteins: MANY! Atom-Atom Interactions Computer Science Robotics & Biology Laboratory 11
12 Proteins: Physical Constraints Backbone angles Sidechain attraction Sidechain repulsion Bending Twisting Rigidity Non-Twisting C H H C N φº H C ωº C N N O H C H ψº C N Computer Science Robotics & Biology Laboratory 12
13 Protein motion Is there movement? Where does movement occur? What is the rate of change? Are there rigid regions? Can we make atom substitutions? Computer Science Robotics & Biology Laboratory 13
14 Protein Ligand Compatibility C C C N C C H O C C C O Tylenol Which ligands are effective? Does a ligand fit into a cleft? Is a ligand compatible? Computer Science Robotics & Biology Laboratory 14
15 Protein Ligand Interaction Is there a feasible approach path? Can we modify a known effective ligand and will that new ligand still be effective? Computer Science Robotics & Biology Laboratory 15
16 Protein Structure, function and dynamics Computer Science Robotics & Biology Laboratory 16
17 Part II: Modeling and Analysis of Proteins Snapshots of a protein: Crystallography Numerical methods of protein motion: Molecular Dynamics Description of protein motion: Normal Mode Analysis and the Elastic Network Model Computer Science Robotics & Biology Laboratory 17
18 Observing the behavior of atoms Computer Science Robotics & Biology Laboratory 18
19 Bragg Scattering 2d sin θ = nλ Computer Science Robotics & Biology Laboratory 19
20 Crystallography making the crystal Computer Science Robotics & Biology Laboratory 20
21 Crystallography Experimental Setup Computer Science Robotics & Biology Laboratory 21
22 Crystallography Experimental Setup Computer Science Robotics & Biology Laboratory 22
23 Crystallography Solve for Electron Densities Fitting X-Rays Electron Density Map Fitting Diffraction Pattern Refinement Fitting Computer Science Robotics & Biology Laboratory 23
24 Molecular Dynamics Protein Rigidity, Protein Flexibility Binding Site / Ligand Compatibility Binding Site / Ligand Interaction Computer Science Robotics & Biology Laboratory 24
25 Molecular Dynamics H N O C H C N φº H C ωº C N H C H C N Statistical Mechanics Pressure Energy Temperature Heat capacities ψº Atomic Positions & Velocities Molecular Dynamics provides the means to solve the equations of particle motion and evaluate these mathematical formulas. Computer Science Robotics & Biology Laboratory 25
26 Potential Energy; Classical Mechanics M is the number of configurations in the molecular dynamics trajectory and V i is the potential energy of each configuration. Classical Mechanics Relate the potential energy of a system to the changes in position as a function of time. Why using energy function? This makes sure that we maintain the conservation of energy. Computer Science Robotics & Biology Laboratory 26
27 Molecular Dynamics 1 2 F 1 (X) = m 1 a(t) 1 F 2 (X) = m 2 a(t) 2 F 3 (X) = m 3 a(t) 3 F(X) = Ma(t) F(X) = MV(t) F(X) = - V(X) 3 V(X) = of the system is a function of the particle coordinates X = "potential" in Physics = "force field" in Chemistry Given initial positions (X-ray Crystallography) and velocities (randomized Gaussian) of the atoms, we can calculate all future (or past) positions and velocities for all atoms. Computer Science Robotics & Biology Laboratory 27
28 Molecular Dynamics Force Fields V(X) = description of the terms by which the particles in the system interact r(å) 2 1 Repulsion Attraction Computer Science Robotics & Biology Laboratory 28
29 Molecular Dynamics - Motion Computer Science Robotics & Biology Laboratory 29
30 Molecular Dynamics Sample Run 1. Atom Coordinates 2. Add hydrogen bonds 3. Solvate Protein 4. Energy Minimization 5. Dynamics of water 6. Dynamics of protein Configuration calculate forces Update configuration Computer Science Robotics & Biology Laboratory 30
31 Molecular Dynamics Sample Run Output Computer Science Robotics & Biology Laboratory 31
32 Molecular Dynamics Sample Run 100,000 integration steps 2.0 femtoseconds 200 picoseconds 22,009,885 x 10 6 fp ops 3 hours 37 minutes, 1CPU Flap Curling 10 nanoseconds 10,000 picoseconds ~170 hours Flap Closing/Opening 10 microseconds 10,000,000 picoseconds ~7,000 days Computer Science Robotics & Biology Laboratory 32
33 Molecular Dynamics Is it good? Bad? Feasible? Local Motions (0.01 to 5 Å, to 10-1 s) Atomic Fluctuations Not practical Practical Femtosecond: Picosecond: Nanosecond: Microsecond: Millisecond: seconds seconds 10-9 seconds 10-6 seconds 10-3 seconds Sidechain Motions Loop Motions Rigid Body Motions (1 to 10Å, 10-9 to 1s) Helix Motions Domain Motions Subunit Motions Large-Scale Motions (> 5Å, 10-7 to 10 4 s) Helix coil transitions Dissociation/Associations Folding and unfolding Computer Science Robotics & Biology Laboratory 33
34 Normal Modes What are they? A normal mode in an oscillating system is the frequency at which a deformable structure will oscillate when disturbed. A set of these frequencies are unique to each structure. Why is this relevant to proteins? Computer Science Robotics & Biology Laboratory 34
35 Normal Mode Analysis Harmonic Oscillations t = 0 t = 1 M 1 M 2 M 1 M 2 x 1 x 2 F = -kx F = -dv/dx = -kx V = 0.5k x 2 k = d 2 V/dx 2 F = ma md 2 x/dt 2 = -kx x(t) = A sin(2πνt) -4π 2 ν 2 mx = -kx ν = Frequency of vibration Computer Science Robotics & Biology Laboratory 35
36 Normal Modes for Proteins? Low Frequency High Frequency C H Global motions Local motions 1. Each normal mode acts like a simple harmonic oscillator. 2. A normal mode is a concerted motion of many atoms. 3. Normal modes are independent; they don t interact. 4. Normal mode analysis is less expensive than MD simulation, but requires much more memory. Approximate the potential energy and describe the motion of the molecules as collection of independent harmonic oscillators. Computer Science Robotics & Biology Laboratory 36
37 Normal Mode Analysis Molecular orbital calculations are used to find the potential energy of the molecules as a function of the position of each atom. The second derivative of the potential energy can then be used to calculate the force constants. The change of the force on atom 1 in the x-direction when you move atom 1 in the x-direction. The change of the force on atom 1 in the x-direction when you move atom 2 in the y-direction. Computer Science Robotics & Biology Laboratory 37
38 Normal Mode Analysis derivation 3 symmetric matrix The complete list of force constants eigenvalue-eigenvector equation Eigenvalues = normal mode frequencies Eigenvectors = coordinate displacements Computer Science Robotics & Biology Laboratory 38
39 Normal Mode Analysis RNAP Example Structure Elastic Network Model Normal Mode Analysis Computer Science Robotics & Biology Laboratory 39
40 Modeling and Analysis of Proteins Crystallography Molecular Dynamics Normal Mode Analysis C H Computer Science Robotics & Biology Laboratory 40
41 Part III: Opportunities for Improvement How can we improve upon MD and NMA Are we asking the right questions? What features of protein motion should we be looking at? Utilize ideas from operational space control; kinematics from mechanics. Choose the simplest representation that will illustrate the property of interest satisfactorily. -Tsjerk Wassenaar, Alex de Vries, Dr. Alan E. Mark Computer Science Robotics & Biology Laboratory 41
42 What should we study? What model should we use? 100% Precision & Accuracy 0% Modeling Tools Image interpolation Backbone Kinematics + Rigidity + Hydrophobic Interactions + Side Chain Steric Interactions? Molecular Dynamics Months Time/Comp Requirements Seconds Which model best maximizes accuracy while minimizing computational requirements? Computer Science Robotics & Biology Laboratory 42
43 Operational Space Control, Dynamics Observation: Result: Observation: Result: Length of covalent bonds of the protein backbone are relatively fixed; only in the most extreme circumstances do the lengths of these bonds change. Fix the length of the covalent bonds and have them assume the links in a kinematic chain. Hydrogen bonds have a higher degree of flexibility than do covalent bonds. Model hydrogen bonds as springs; any deviation of the springs from the stable state results in an appropriate restoring force Computer Science Robotics & Biology Laboratory 43
44 Motion on the Motion manifold Mechanics: transform information about the interactions of individual atoms into valid conformational changes of the kinematic structure. F Use the Jacobian Matrix, J, to transform a force applied in Cartesian space to an equivalent torque in joing space τ = J T F We do not need to calculate the force between all possible atom-partners Consider only how a displacement of an atom affects the connected atoms Computer Science Robotics & Biology Laboratory 44
45 Protein Motion; Using Kinematic Representation Computer Science Robotics & Biology Laboratory 45
46 Future Directions Etc. Hydrophobic Interactions Sidechain Steric Clashes Hydrogen Bonds = Springs Kinematic Serial Chain Robot F Domain A Domain B F Side Chain X Side Chain Y F (x,y,z) Molecular Dynamics Engine (GROMACS) Computer Science Robotics & Biology Laboratory 46
47 Modeling Biological Systems Opportunities for Computer Scientists Filip Jagodzinski RBO Tutorial Series 25 June 2007 Computer Science Robotics & Biology Laboratory
48 Statistical Mechanics Statistical mechanics: the study of the macroscopic system from a molecular point of view. Mechanical (microscopic) state defined by the atomic positions, q, and momenta, p - Configuration Thermodynamic (macroscopic) state a small set of parameters, including T, P, and number of particles, N - Energy A single point in mechanical space, denoted by Γ, describes the state of the system. An ensemble is a collection of points in mechanical space satisfying the conditions of a particular thermodynamic state. The dilemma appears to be that one can calculate time averages by molecular dynamics simulation, but the experimental observables are assumed to be ensemble averages. <A> ensemble = <A> time Ensemble average = Time average Ergodic Hypothesis The trajectory of an isolated mechanical system runs through all states compatible with the total energy of the system. Computer Science Robotics & Biology Laboratory 48
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