1. Conformational Sampling
Problem:
How to find all of the possible conformations for a flexible molecule (protein, nucleic acid, polysaccharide, ligand, drug)
The selected approach will depend on several things including:
The size of the molecule, and particularly the number of expected conformational states
The ability to define the states in obvious internal coordinates, such as torsion angles

2. Conformational States
How to find the stable states (conformations) of a molecule?
What defines the state (or conformation) as “stable”?

3. Using Grid Searching to Find Conformational States
If the states are related by simple internal coordinates, such dihedral angles, the states can be found by searching all of the dihedral angle space.
I.e. vary the dihedral angle and look for low energy structures – this is known as Grid Searching
The number of conformations

4. Grid Searching and Combinatorial Explosions
Number of rotatable bonds
Step size (angle increment)
Number of conformations to generate
Total number of conformations 1 10
360/10 = 36 36 30
360/30 = 12 12 2
= 12*12
144 3
= 12*12*12
1,728 4
= 12*12*12*12
20,736 5
= 12*12*12*12*12
248,832 6
= 125
2,985,984
The principle problem with Grid Search methods is that the number of structures to be evaluated increases rapidly – this is the “Combinatorial Explosion” problem
If it takes 1 second to compute the energy of each conformation, how many days will it take to perform a Grid Search of 6 bonds?
The number of conformations

5. Stochastic Conformational Sampling
An alternative to Grid Searching, is to generate structures by randomly changing the atomic positions either in Cartesian space, or in torsion space. Random methods are also known as Stochastic Sampling methods.
The initial structures are usually energy minimized and then sorted with some sort of energy cut-off. I.e. Only low-energy conformations are kept – but the choice of what is “low-energy” is arbitrary. Often 10 – 20 kcal/mol above the minimum. All others are rejected.
The user decides how many random structures to generate.
For this reason, Stochastic Sampling can be much more efficient than grid searching, since it can avoid the Combinatorial Explosion problem.

6. Stochastic Conformational Sampling

7. Stochastic Conformational Sampling

8. Stochastic Conformational Sampling
Both Stochastic and Systematic Searching work “OK” for small molecules

9. Levinthal’s Paradox – Why Nature can’t use Grid Searching to Fold a Protein
In 1969, Cyrus Levinthal noted that, because of the very large number of degrees of freedom in an unfolded polypeptide chain, the molecule has an astronomical number of possible conformations [1].
For example, a polypeptide of 100 residues will have 99 peptide bonds, and therefore 198 different phi and psi bond angles. If each of these bond angles can be in one of three stable staggered conformations, the protein may fold into a maximum of 3198 different conformations.
If a protein were to attain its correctly folded configuration by sequentially sampling all the possible conformations (i.e. by Grid Searching), it would require a time longer than the age of the universe to arrive at its correct native conformation.
This is true even if conformations are sampled at rapid (nanosecond or picosecond) rates. The "paradox" is that most small proteins fold spontaneously on a millisecond or even microsecond time scale.
The fact that many naturally-occurring proteins fold reliably and quickly to their native state despite the astronomical number of possible configurations has come to be known as Levinthal's Paradox.
Levinthal, Cyrus (1969). "How to Fold Graciously". Mossbauer Spectroscopy in Biological Systems: Proceedings of a meeting held at Allerton House, Monticello, Illinois: 22–24.

10. Conformational States
Energy
(kcal/mol)
Torsion Angle (degrees)
What defines the state (or conformation) as “stable”?
At a given temperature which states are likely to be populated?

11. Conformational States Depend on Temperature
Energy
(kcal/mol)
Torsion Angle (degrees)
Average Kinetic Energy = 3/2kBT
kB = Boltzmann’s constant = 0.001 987 kcal/mol/K
At 300K how much kinetic (thermal) energy is available to a molecule?

12. Which Conformational States Are Relevant?
360
330
300
270
240
210
180
150
120 90 60 30
Simulation Time
Energy
(kcal/mol)
Torsion Angle (degrees)
Not all possible states will be populated (observed) at room temperature
For this reason room-temperature MD is inefficient at finding conformational states

13. Which Conformational States Are Relevant?
360
330
300
270
240
210
180
150
120 90 60 30
Simulation Time
Energy
(kcal/mol)
Torsion Angle (degrees)
By raising the temperature it is possible to find other states
This approach can be employed in either MD simulations or MC sampling

14. Increasing Temperature Increases Sampling
State
Simulation Time
Increasing the temperature will enable more states to be detected during the simulation – this is known as Simulated Annealing
But for how long should the simulation be run?
To what temperature should the system be heated?

15. Lowering Internal Barriers Increases Sampling
Energy
(kcal/mol)
Torsion Angle (degrees)
An alternative to raising the energy is to lower the barriers
But how do you know what barriers to lower?
Must be able to identify simple internal coordinates that are related to the states, such as torsion angles

16. Conformational Sampling with Reduced Barriers

17. Conformational Sampling with Reduced Barriers

18. Conformational Sampling with Reduced Barriers

19. Conformational Sampling with Reduced Barriers

20. Choice of Conformational Sampling Method
Thus the problem of conformational sampling is different for a small molecule (with few rotatable bonds) than for a macromolecule, such as a protein
Small molecule – can use Grid or Stochastic Searching to generate an ensemble of structures
Macromolecule – use Simulated Annealing, or Monte Carlo (MC) Sampling, or long MD simulations
In the limit – that is, once all of the stable states have been identified and their populations weighted by their relative energies – each method should give the same answer – this is related to the “Egrodic Hypothesis”
Ergodic Hypothesis: the time average property (from MD) is the same as the ensemble average property (from MC)