- 2011 -
3D Structures of Biological Macromolecules
Part 6: Selected Topics
(Quantum Chemistry, Molecular Dynamics, Statistical Potentials, Lattice Models)
Jürgen Sühnel
jsuehnel@fli-leibniz.de
Leibniz Institute for Age Research, Fritz Lipmann Institute,
Jena Centre for Bioinformatics
Jena / Germany
Supplementary Material: www.fli-leibniz.de/www_bioc/3D/
Quantum Chemistry
Quantum Chemistry
Quantum-chemical Calculations: Telomeric DNA
Quantum-chemical Calculations: Telomeric DNA
Quantum-chemical Calculations: Telomeric DNA
Quantum-chemical Calculations: Telomeric DNA
Molecular Dynamics
Simulation of Protein Folding – Molecular Dynamics
AMBER
GROMOS
CHARMM
TINKER
Molecular Dynamics Simulation
Protein Capsid Of Filamentous Bacteriophage Ph75 From Thermus Thermophilus
1HGV, extended structure
1HGV, actual structure
1HGV, 61% helix, 1.928 ns
1HGV, 75% helix, 3.428 ns
Images created using VMD (Visual Molecular Dynamics) (HUMPHREY, W., DALKE, A. and
SCHULTEN, K., 1996.VMD - Visual Molecular Dynamics. Journal Molecular Graphics,14,
pp33-38).
Molecular Dynamics Simulation
amber.scripps.edu
Molecular Dynamics Simulation
Molecular Dynamics Simulation – GROMOS Package
www.gromos.net
Molecular Dynamics Simulation – GROMOS Package
Molecular Dynamics Packages
www.charmm.org
Molecular Dynamics Packages
dasher.wustl.edu/ffe/
Visualizing and Analyzing Molecular Dynamics Simulations
www.ks.uiuc.edu/Research/vmd/
Folding Surface for Lysozyme
Dobson, Sali, Karplus, Angew. Chem. Int. Ed. 1998, 37, 868.
Protein Folding States
Dobson, Sali, Karplus, Angew. Chem. Int. Ed. 1998, 37
Monitoring Protein Folding by Experimental Methods
Dobson, Sali, Karplus, Angew. Chem. Int. Ed. 1998, 37, 868.
Monitoring Protein Folding by Experimental Methods
Paxco, Dobson, Curr. Opin. Struct. Biol. 1996, 6, 630.
Protein Folding by Molecular Dynamics
Protein Folding by Molecular Dynamics
Protein Folding by Molecular Dynamics
Villin headpiece domain
(PDB code: 1vii)
Actin binding site highlighted
36 amino acids
Protein Folding by Molecular Dynamics
Protein Folding by Molecular Dynamics
Protein Folding by Molecular Dynamics
Radius of Gyration
In a globular protein the radius of gyration Rg can be predicted with reasonable
accuracy from the relationship
Rg(pred) = 2.2 N 0.588
where N is the number of amino acids.
Protein Folding by Molecular Dynamics
Protein Folding by Molecular Dynamics
Statistical Potentials
A statistical potential or knowledge-based potential is an energy function
derived from an analysis of known protein structures.
They are mostly applied to pairwise amino acid interactions. The statistical potential
assigns to each possible pair of amino acids a weight or score or energy.
Statistical potentials are applied to protein structure prediction and to protein
folding.
Their physical interpretation is highly disputed. Nevertheless, they have been
applied with great success, and do have a rigorous probabilistic justification.
Thomas, Dill, J. Mol. Biol. 1996, 257, 457-469
Statistical Potentials
Boltzmann distribution:
The Boltzmann distribution applied to a specific pair of amino acids, is given by:
where r is the distance, k is the Boltzmann constant, T is the temperature and Z is the partition function, with
The quantity F(r) is the free energy assigned to the pairwise system. Simple rearrangement results in the inverse
Boltzmann formula, which expresses the free energy F(r) as a function of P(r):
To construct a so-called Potentail of Mean Force (PMF) , one then introduces a so-called reference state with a
corresponding distribution QR and partition function ZR, and calculates the following free energy difference:
The reference state typically results from a hypothetical system in which the specific interactions between the
amino acids are absent. The second term involving Z and ZR can be ignored, as it is a constant.
Statistical Potentials
In practice, P(r) is estimated from the database of known protein structures, while QR(r) typically results from
calculations or simulations. For example, P(r) could be the conditional probability of finding the Cβ atoms of a valine
and a serine at a given distance r from each other, giving rise to the free energy difference ΔF. The total free energy
difference of a protein, ΔFT, is then claimed to be the sum of all the pairwise free energies:
where the sum runs over all amino acid pairs ai,aj (with i < j) and rij is their corresponding distance. It should be noted
that in many studies QR does not depend on the amino acid sequence
Intuitively, it is clear that a low free energy difference indicates that the set of distances in a structure is more likely in
proteins than in the reference state. However, the physical meaning of these PMFs have been widely disputed since
their introduction. The main issues are the interpretation of this "potential" as a true, physically valid potential of mean
force, the nature of the reference state and its optimal formulation, and the validity of generalizations beyond pairwise
distances.
Statistical Potentials
wij(r)
ij(r)
*
–
-
interaction free energy
pair density
reference pair density at
infinite separation
Statistical potentials can be determined by
simply counting interactions of a specific type
in a dataset of experimental structures.
The distance dependence may or may not be taken
into account. If not, the interaction free energy is usually
called a contact potential. It represents an average over
distances shorter than some cutoff distance rc.
Thomas, Dill, J. Mol. Biol. 1996, 257, 457-469
Lattice Folding
Lattice Algorithm
•
•
•
•
•
•
Red = hydrophobic, Blue = hydrophilic
If Red is near empty space E = E+1
If Blue is near empty space E = E-1
If Red is near another Red E = E-1
If Blue is near another Blue E = E+0
If Blue is near Red E = E+0