======================
Solution to exercise 4
======================

----------------------
Water and Biomolecules
----------------------

.. contents::
.. section-numbering::
.. include:: ../charents.txt





The N-methylacetamide (NMA) dimer
=================================

As one can see from the structures and from the following schemes, the 
N-methylacetamide (NMA) molecule can be understood as one peptide residue terminated with
two CH\ :sub:`3` ligands. Generally, a peptide chain consisting of amino
acid residues looks like this

.. figure:: peptidechain.jpg
   :width: 450
   :align: center

The side chain marked *R* distinguishes the 20 different amino acids. In general, there
are hydrophobic, polar and charged amino acids, depending on the properties of the side
chain. The amino acids used in this exercise are glycine (Gly) with R=H and alanine (Ala)
with R=CH\ :sub:`3`.

A dimer consisting of two NMA molecules looks like this

.. figure:: nma_model.jpg
   :width: 250
   :align: center

Therefore the NMA molecule corresponds to a Gly residue terminated by two CH\ :sub:`3` groups.

The NMA model is a good model for a peptide chain or a |beta|\ -sheet in the sense that it
models the peptide unit well. In the NMA dimer, the most important hydrogen bond, the
N-H...O=C bond, is also well modeled. This hydrogen bond is also very important for the
stability of |beta|\ -sheets. On the other hand, also a C-H...O=C hydrogen bond plays an
important role in some |beta|\ -sheets, and this hydrogen bond is not present in an NMA
dimer. Another difference is that in the NMA dimer, the two molecules are rather flexible
and can orient themselves ideally for forming an N-H...O=C hydrogen bond. In a real peptide
chain, however, the residues experience numerous other constraints and can therefore not form
ideal hydrogen bonds.

The interaction energy between the two NMA molecules is calculated as


  |Delta|\ E=E\ :sub:`NMA-dimer`-2*E\ :sub:`NMA-single`


where E\ :sub:`NMA-dimer` is the total energy of the NMA dimer and E\ :sub:`NMA-single`
is the total energy of a single NMA molecule. We can calculate the interaction energy
for the different XC functionals, where only PW91 is calculated self-consistently and
therefore marked with (sc).

=============  ===============
XC functional  |Delta|\ E [eV]
=============  ===============
PW91 (sc)      -0.32          
PBE            -0.30     
RPBE           -0.23 
=============  ===============

The negative sign of |Delta|\ E means that the two NMA molecules bind to each other.
The hydrogen bond mainly responsible for the attractive interaction is the
N-H...O=C hydrogen bond. One would expect the interactions of a real |beta|-sheet to be
weaker, as the peptide chains are more constrained and therefore cannot orient themselves
ideally for the formation of hydrogen bonds. In this sense the NMA binding energy is an
upper bound for the binding energies in a |beta|-sheet. Concerning the different XC-functionals
one observed that PW91 and PBE give similar results. This is to be expected, as the PBE functional
was constructed to be similar to PW91. The RPBE functional gives a significantly smaller binding
energy. While the RPBE functional performs very well for surfaces, for hydrogen bonds the
PW91 functional is the better one.

Now we plot the difference in electron density of two interacting NMA molecules and of
two single NMA molecules with their positions frozen to the ones from the NMA dimer.
We use the script `<diff_density_plot.py>`_ and the plot looks like

.. figure:: NMA_densplot.jpg
   :width: 300
   :align: center

Blue marks the contour for -15 (see note for units) and denotes electron density depletion. 
Yellow marks the contour at +15 (see not for units) and denotes electron density addition.
Thus we can see a clear polarization along the N-H...O=C bond, which confirms that this
hydrogen bond is the main reason for the interaction.


Parallel and antiparallel two-stranded beta-sheets
====================================================

The structure of |beta|-sheets is depited below. The *antiparallel* |beta|-sheet
looks like

.. figure:: scheme_anti.jpg
   :width: 450
   :align: center

and the *parallel* |beta|-sheet looks like

.. figure:: scheme_par.jpg
   :width: 450
   :align: center

One can see that the peptide chains have been terminated with structure *a* for the
N-end and structure *b* for the C-end.

.. figure:: termination.jpg
   :width: 450
   :align: center


One can see that in the *antiparallel* |beta|-sheet, the peptide chains are
ideally positioned for forming a N-H...O=C hydrogen bond. In the parallel chains
the N-H...O=C bond geometry is less optimal. As the NMA molecules can assume ideal
positions for a N-H...O=C hydrogen bond, the NMA dimer resembles most an antiparallel
|beta|-sheet.

The C-N, C-C and C=O bonds in the peptide unit are almost equal for the different
|beta|-sheet structures. Therefore, we measure the bond legths for the alanine chains
only.

=====================  ================================  ============================  ===========================
Bond                   antiparallel Ala sheet [|angst|]  parallel Ala sheet [|angst|]  organic compounds [|angst|]
=====================  ================================  ============================  ===========================
C\ |alpha|-N           1.46                              1.46                          1.47
N-C                    1.34                              1.34                          1.47
C-C\ |alpha|           1.54                              1.54                          1.54
C=O                    1.23                              1.23                          1.20
=====================  ================================  ============================  ===========================

The C\ |alpha|-N and C-C\ |alpha| bonds have clearly single bond character. Therefore the peptide
chain can rotate around these angles and assume different conformations. The angles describing rotation around
these bonds are called *dihedral angles*. The N-C bond in a peptide chain is significantly shorter than a corresponding
single bond in organic compounds. This shows that the N-C bond in an amino acid has partly double bond character and
therefore is rigid. In turn, the C=O bond in an amino acid has double bond character, but is slightly longer than in
other organic compounds. Therefore the N-C and C=O bands form two resonating structure with delocalized |pi|-orbitals.
Thus, no rotations around the N-C bond is possible and one peptide unit itself is rather stiff.

For the bonds and angles of the N-H...O=C hydrogen bonds, we concentrate on the Ala peptide chains. We get the
following results:

=====================  ======================  ==================  =========
Parameter              antiparallel Ala sheet  parallel Ala sheet  NMA dimer
=====================  ======================  ==================  =========
NH...OC [|angst|]      1.96                    2.19                2.07
N...O [|angst|]        2.94                    3.00                3.09
<NH...O [|deg|]        163                     147                 176
<CO...H [|deg|]        158                     136                 144
=====================  ======================  ==================  =========

One can see that the bond geometries for the NMA dimer and for the |beta|-sheet
structures are nonetheless quite different. This shows that the NMA dimer is useful
as a minimum model but limited for bond geometry predictions.

Usually, |beta|-sheet structures are found in hydrophobic regions of the protein.
Therefore, hydrophobic amino acids have a bigger probability of being located in
a |beta|-sheet than polar amino acids. Therefore, in some situations, water can
stabilize |beta|-sheets even more by keeping them together from the outside.
It is observed that *parallel* |beta|-sheets always are buried in hydrophobic regions,
whereas antiparallel |beta|-sheets can be exposed to solvent. This suggests that
antiparallel |beta|-sheets are intrinsically more stable than parallel |beta|-sheets.



The water dimer
===============

The bond length of the water hydrogen bond (O...H) is 1.97 |angst|. This hydrogen bond
is certainly the most important one in nature, as it determines the properties of water.

The two frozen water molecules are calculated with the scripts `<H2O_freeze1.py>`_
and `<H2O_freeze2.py>`_. The density difference plot is done with the script
`<H2O_density_plot.py>`_ and looks like

.. figure:: H2O_densplot.jpg
   :width: 450
   :align: center

Again, blue denotes electron density depletion and yellow denotes electron density addition.

To calculate the binding energy of the water dimer one has to relax a single water molecule.
This can be done using the script `<2H2O.py>`_ and deleting one water molecule in it. The
binding energy is then calculated analogous to the NMA dimer and the result is |Delta|\ E=0.23eV
for the PW91 functional. This shows that the water molecule binds more weakly than the NMA dimer,
but that the binding energies are of the same order of magnitude.

