1. Structure Refinement
BCHM 5984
September 7, 2009

2. Methods of 3-D Structure Prediction
De novo structure prediction
Derived completely from sequence
Comparative modeling
Fold recognition / Threading (when no homologs exist)
Homology modeling (when clear homologs exist)

3. Steps in Homology Modeling
Identify homologues
Determine sequence identity
Align sequences
Identify conserved / non-conserved regions
Generate model for conserved regions
Generate model for non-conserved regions
Build sidechains
Evaluate and refine

4. Understanding the Problem
Leach, AR (2001) “Molecular Modelling Principles and Applications”. 2nd Ed. Prentice Hall Publishers.

5. Proteins are Complicated
Many more atoms
Must consider:
Ideal bond lengths, bond angles, dihedrals
Electrostatic interaction (hydrogen bonds, ionic interactions) and van der Waals interactions

6. How to Solve the Problem
Given a function f which depends on one or more independent variables x1, x2, …, xi, find the values of those variables where f has a minimum value.
At a minimum point, the first derivative of the function with respect to each of the variables is zero and the second derivatives are all positive:
The function f is the potential energy
The variables xi are the atomic Cartesian coordinates
Change the position of the coordinates (xi) until we find the position with the smallest potential energy

7. Examples of the Functions
E = Ecovalent + Enoncovalent can be further expanded to: Ecovalent = Ebond + Eangle + Edihedral Enoncovalent = Eelectrostatic + Evdw

8. Examples of the Functions
Enoncovalent = Eelectrostatic + Evdw
Coulombic Potential
Lennard-Jones Potential

9. Examples of the Functions
Ecovalent = Ebond + Eangle + Edihedral
Bond Stretching Potential
Harmonic Angle Potential
Dihedral Potential

10. What You Need to Start Cartesian coordinates of your model
(x, y, z) for every atom = 3N variables where N is the number of atoms
Energy minimizer program
Knows potential energy functions for minimization
Knows ideal bond lengths, bond angles, etc. for all atomic interactions, covalent and non-covalent (also called force fields)

11. Energy Minimization Methods
Non-derivative methods
Simplex
First-order derivative methods
Steepest descents
Conjugate gradients
Second-order derivative methods
Newton-Raphson
Quasi-Newton methods
Davidson-Fletcher-Powell (DFP)
Broyden-Fletcher-Goldfarb-Shanno (BFGS)

12. Energy Minimization Methods
Non-derivative methods
Simplex
First-order derivative methods
Steepest descents
Conjugate gradients
Second-order derivative methods
Newton-Raphson
Quasi-Newton methods
Davidson-Fletcher-Powell (DFP)
Broyden-Fletcher-Goldfarb-Shanno (BFGS)

13. Simplex Method
Moves around like an “amoeba”

14. Non-Derivative Methods
Advantages:
Works well when starting configuration is very high in potential energy
Disadvantages:
Surprisingly slow (calculations are fast, but it takes many iterations)
Not good for large biomolecules

15. Energy Minimization Methods
Non-derivative methods
Simplex
First-order derivative methods
Steepest descents
Conjugate gradients
Second-order derivative methods
Newton-Raphson
Quasi-Newton methods
Davidson-Fletcher-Powell (DFP)
Broyden-Fletcher-Goldfarb-Shanno (BFGS)

16. Steepest Descents Method
1.) Evaluate the sum of all forces on the system (first derivative of potential energy functions) 2.) Move in the direction of the force until potential energy stops decreasing 3.) Turn 90° and return to step 2
sx = -gx / |gx|
s = step direction
g = gradient direction
x = coordinates of system
The next step is orthogonal:
gx  gx-1 = 0

17. SD: When to Turn Line Search Arbitrary Step
Find three points along a line where the middle point is less than the other two points
Calculates a function for the three points and determines the minimum
The minimum becomes the middle point, and repeat
Arbitrary Step
Try a small step size to see that potential energy decreases
Iteratively increase step size until potential energy is increased
Multiply the final step size by 0.5

18. SD: When to Stop After a predefined energy minimum has been reached
For example, < 1.0 kJ / mol
After a predefined number of steps
For example, after 1000 orthogonal steps

19. SD: Searching Problem
Does not work well in (relatively) flat energy wells
Takes too many steps / too long to finish

20. Conjugate Gradients Method
1.) Evaluate the sum of all forces on the system (first derivative of potential energy functions) 2.) Move in the direction of the force until potential energy stops decreasing 3.) Return to step 1
Red line = Conjugate gradients
Green line = Steepest descents

21. Steepest Descents vs. Conjugate Gradients
Stable and rigorous
Generally slower and takes more steps than CG in flat wells
Can take bigger steps and finish faster in steep wells
Conjugate gradients:
Slower in the beginning, but can be faster overall (takes fewer steps) in flat wells
Less stable than SD (may need restarting)
Both methods (ideally) converge to the same local energy minimum

22. Energy Minimization Methods
Non-derivative methods
Simplex
First-order derivative methods
Steepest descents
Conjugate gradients
Second-order derivative methods
Newton-Raphson
Quasi-Newton methods
Davidson-Fletcher-Powell (DFP)
Broyden-Fletcher-Goldfarb-Shanno (BFGS)

23. Second-Order Derivative Methods
Use first derivatives to see which way the gradient flows
Use second derivatives to see changes in the way the gradient flows
Tries to predict the best spot to “jump” to
Newton-Raphson method:
xn = current position
xn+1 = next position
f’(xn) = first derivative of energy function
f'’(xn) = second derivative of energy function

24. Newton-Raphson can be Slow
A Hessian matrix is a matrix of second-order derivatives of a function
Must be calculated in each step for Newton-Raphson method

25. The BFGS Assumption
Calculating second-order derivatives is hard and time consuming
Never actually calculates a Hessian matrix, just estimates it as it goes along
Estimated by looking at successive gradients
Not technically a “true” second-order derivative method

26. Second-Order Derivative Methods
Advantages
Takes the fewest steps
Fastest (for small molecules)
Disadvantages
For big systems, can require too much memory
Best suited for small molecules
Red line = BFGS method
Green line = Conjugate gradients

27. Choosing an EM Method Depends on: Storage / computational capabilities
Number of atoms in the system
When working with proteins, always steepest descents or conjugate gradients
Initial Refinement
(Av. gradient < 1 kcal Å-2)
Stringent Minimization
(Av. gradient < 0.1 kcal Å-2)
Method
CPU time (s)
# of iterations
Steepest descents 67 98
1405
1893
Conjugate gradients
149
213
257
367

28. Inherent Problem of EM Only finds local minima
No method available can find the global minimum from any starting point

29. Performing Energy Minimization
Links
Dundee PRODRG2 Server (
Swiss-PDBViewer (
GROMACS (
NAMD (
AMBER (
CHARMM (
Methods of EM in GROMACS
Steepest descents
Conjugate gradients
L-BFGS (limited-memory Broyden-Fletcher-Goldfarb-Shanno quasi-Newtonian minimizer)

30. Discuss Homework 2