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Covalent bonds
In simple terms a covalent bond exists between two atoms if they share
electrons between them. In contrast, an ionic bond is formed if an
electrons are transferred between atoms (e.g., in sodium chloride an
electron is given up by the sodium atom to form a Na+ ion and accepted by
the chlorine atom to form a chloride, Cl-ion.
A single bond is formed when one pair of electrons is involved and a double
bond when two pairs are involved.
In quantum chemical terms such a picture is overly simplistic as the
although a bonding orbital results in an increase in electron density
between the atoms it also spreads over the rest of the molecule. This is
particularly in the case of delocalized bonding. The "classical" example of
delocalized bonding is the benzene molecule - which can be described as
resonance hybrid between a number of alternate structures:
The standard way to approximate the potential energy for a bond in a protein
and most other molecules is to use a Hooke's law term
where r is the length of the bond (i.e., the distance
between the two nuclei of the atoms between which
the bond acts), r_eq is the equilibrium bond length
and K_r is a spring constant. This basically represents
the bond as a spring linking the two atoms.
Graph of the potential energy dependence for a C=O Bond
Atom pair
r_eq in Å
K_r in kcal/(molÅ2)
C=O
C-C2
C-N
C2-N
N-H
1.229
1.522
1.335
1.499
1.010
570
317
490
337
434
O
a carbonyl oxygen
C
a sp2 carbon (such as that attached to an O)
N
a main chain nitrogen atom
H
a hydrogen atom attached to the N
C2
a "united atom"
group(CH2)
Bond angles
A bond angle between atoms A-B-C is defined as the angle
between the bonds A-B and B-C:
Angle Θeqin degrees
Kein kcal/(mol.degrees2)
C-N-H
119.8
35.0
C2-N-C
121.9
50.0
C2-N-H
118.4
38.0
C-C2-N
110.3
80.0
C2-C-O
120.4
80.0
C2-C-N
116.6
70.0
O-C-N
122.9
80.0
Potential energy curve for
the N-C-O Bond Angle
Dihedral angles
Vn gives the energy barrier to rotation, n the number of maxima
(or minima) in one full rotation and γ determines the angular
offset. The use of the sum allows for complex angular variation
of the potential energy. Barriers for dihedral angle rotation can
be attributed to the exchange interaction of electrons in adjacent
bonds. Steric effects can also be important.
Potential energy curve for the omega dihedral angle
Interazioni elettrostatiche
Non-bonded Interactions
Electrostatic interactions
where qi and qj are the magnitude of the charges, rij is their
separation, ε0 the permittivity of free space and εr the relative
dielectric constant of the medium in which the charges are
placed
The dielectric constant of selected materials
Material
Dielectric constant
Water (20oC)
Water (0oC)
Ice (-10oC)
Methanol
Liquid H2S(-85.5 oC)
Beeswax
Paraffin
Liquid Argon(-191oC)
Vacuum
80.3
87.7
~98
33.6
9.3
2.9
2.0-2.5
1.5
1.0 (by definition)
The strictly correct way to use the law would be to
consider every nucleus and electron separately, plug it
into the Schrödinger equation and apply quantum
chemical methods to solve the equation for the spatial
configuration of nuclei we are interested.
As already mentioned this is completely impractical
for biomolecular systems. So instead we wish to
develop a useful model for the interactions between
nuclear centres (commonly called "atoms") without
having to explicitly deal with the electrons in a system.
The simplest approach is to just to consider the formal charges
of the protein. Formal charges show whether chemical groups
are ionized i.e., whether an atom or set of atoms has lost or
gained an electron. Isolated amino acids (in neutral solution)
are zwitter ionic - this means that although the molecule has no
overall charge it carries both a negatively charged group and a
positively charged group:
In practice salt bridges are relatively rare in proteins and in
practice they normal occur on the surface as opposed to
internally. An exception is when an internal salt bridge is
involved in the catalytic mechanism of an enzyme such as in
the asp-his-ser triad of serine proteases (a classic example of
the structural basis of enzyme activity):
The reason for this is that although an internal salt bridge is a
strong interaction in comparison to having the isolated
residues widely separated in a vacuum it is normally
destabilizing for a protein. This apparent paradox is due to
that fact that when considering the effect of an interaction
one must consider the difference in the (free) energy between
the folded and unfolded but solvated states.
In the unfolded state the residues involved in a salt bridge
would be widely separated but each making very favourable
interactions with water molecules (there is an entropic
contribution to this).
These interactions are lost when the same residues are buried
in the largely hydrophobic core of the protein.
Similar arguments apply to practically all considerations of
elucidation the energetic contributions to protein folding or
ligand binding - normally a small overall free energy
advantage arises from the balance between large but
cancelling contributions.
Hydrogen bonds
2.8 Å 6kcal/mol
Partial Charges
We have seen that electrostatic interactions are of fundamental
importance to proteins. We shall now briefly examine the
manner in which they are normally treated in computational
studies.
The most common approach is to place a partial charge at each
atomic centre (nucleus). These charges then interact by
Coulomb's Law. The charge can take a fractions of an electron
and can be positive or negative. Charges on adjacent atoms
(joined by one or two covalent) bonds are normally made
invisible to one another - the interactions between these atoms
being dealt with by covalent interactions.
Note that the concept of a partial charge is only a convenient
abstraction of reality. In practice many electrons and nuclei
come together to form a molecule - partial charges give a crude
representation of what a neighbouring atom will on average
"see" due to this collection.
The standard modern way to calculate partial charges is to
perform a (reasonably high level) quantum chemical
calculation for a small molecule which is representative of
the group of interest (e.g., phenol is considered for tyrosine).
The electrostatic potential is then calculated from the
orbitals obtained for many points on the molecular surface. A
least squares fitting procedure is then used to produce a set of
partial charges which produce potential values most
consistent with the quantum calculations.
The normal treatment for partial charges is to assume they are
fixed. In practice the electric field caused by other atoms and
molecules will polarize an atom effecting its electron
distribution and thus its partial charge. In turn the partial charge
produces an electric field which affects neighbouring charges
and thus fields.
The process of polarization has an energetic effect. In practice
it is difficult to find adequate parameters to treat systems as
complex as proteins.
Induction effects can be shown to decay by a r-6 relations so
they can normally be regarded as implicitly corrected for when
the dispersion term is fitted.
Dispersion
The Dispersion interaction can be shown to
vary according to the inverse sixth power of
the distance between the two atoms.
The factor Bij depends on the nature of the
pair of atoms interacting (in particular their
polarizability)
The factor Bij depends on the nature of the pair of atoms
interacting (in particular their polarizability). It is normal to
parameterize the dispersion empirically using structural and
energetic data from crystals of small molecules. It is not
possible to use simple quantum chemical calculations to find
parameters.
In this each electron is solved independently keeping the
other orbitals frozen (in a self consistency). This effectively
means that electrons only experience a time averaged picture
of other electrons - so that dispersion cannot come into
effect. More advanced methods in quantum chemistry
introduce methods to tackle "electron correlation" to avoid
this.
Repulsion
When two atoms are brought increasing close together
there is a large energetic cost as the orbitals start to overlap.
In the limit that the atomic nuclei where coincident the
electrons of the two atoms would have to share the same
orbital system. The Pauli exclusion principle states that no
two electrons can share the same state so that in effect half
the electrons of the system would have to go into orbitals
with an energy higher than the valence state. For this reason
the repulsive core is sometimes termed a "Pauli exclusion
interaction".
The Lennard-Jones potential
and van der Waals Radii
The equation can be rewritten in an equivalent more
instructive form (choosing the case for an interaction be two
atoms of the same type):
The minimum of the function is at r = 2R* and has an energy
of minus E*. The distance R* is know as the van der Waals
radius for an atom and E* is its van der Waals well depth.
atom type
van der Waals radius
in Å
C (aliphatic)
O
H
N
P
S
1.85
1.60
1.00
1.75
2.10
2.00
van der Waals well depth
in kcal/mol
0.12
0.20
0.02
0.16
0.20
0.20
It is important to note that the Lennard-Jones interaction between uncharged
atoms (such as CH3 groups) is less attractive than that between charged
groups such as oxygens. The difference is that the contribution from
electrostatics will dominant the L-J interactions.
In cases where uncharged groups form compact structures van der Waals
energies are often cited as stabilizing the conformation. Although partly true
very often the major contribution comes rather from hydrophobic exclusion.
Solvent effects and the hydrophobic interactions
The fact that proteins normally function in an aqueous
environment considerably complicates the understanding of the
interactions between different groupings. We have already seen
that when thinking about the effect that various interactions
(such as hydrogen bonds) have on the overall stability of a
protein one must compare the free energy contribution to the
folded state with that of the random coil.
In this section we will examine two important contributions that
solvent makes to protein interactions.
• every dipole lines up so that its
positive end points toward the
negative charge and vice-versa.
This means that the electric field
caused by the dipoles will oppose
the original electric field at all
places. This reduction in field
causes a reduction in electric
potential and thus a reduction in the
interaction energy.
• the electric field between charges
permeates the whole of space - it
does not only depend on what is
immediately in between the
charges.
So we have seen that the dielectric constant of materials is
caused by microscopic dipoles in the material. These have two
sources:
1. Electronic polarizability. As we have seen in both the
sections on induction and dispersion when an electric field is
applied to an atom a dipole is induced. This is because the
electron cloud surrounding the nucleus tends to be displaced by
the field.
This induced dipole contributes to the dielectric constant of any
material.
2. Orientational polarizability. If the molecules composing
the material have an intrinsic dipole moment and they are free to
rotate these will have a tendency to rotate so as to oppose the
external field. The larger the dipole moment the larger the
induced field and so the larger the dielectric constant.
In the early days of protein simulation (1980's) it was often assumed that
the effective dielectric "constant" between two charges at a separation of
R angstroms varied as R. This had the fairly dodgy justification that one
could expect electronic polarizability to dominate when two atoms are in
proximity, when R is around 3 (not necessarily true as the medium
surrounding the charges as well as between them has a dielectric effect).
At large distances the dielectric would get larger reflecting the fact that
increasing amounts of water would tend to surround the charges. The
real justification for the model was that it save a lot of computer time
when this was a really scare commodity. It changes Coulomb's law into
a R-2 dependence so that it avoids the necessity of finding using the
square root to work out the distance between two atoms. Recent work
has found that using a more sophisticated function to represent
"dielectric constant" (usually with a sigmodial form) has yielded good
results.
A more normal approach today is to actually include a large number of
water molecules in a molecular dynamics simulation. A dielectric
constant of 1 or 2 (to represent electronic polarizability) is used. An
alternative approach is to use a "reaction field".
The hydrophobic effect
This is in essence a simple observation that a apolar molecules
tend to aggregate in the presence of water. It is known to be
entropically rather than enthalpically driven. For this reason it is
not usual to incorporate a term for hydrophobic energy into
potential energy functions. In fact simulations can be performed
using molecular dynamics methods showing hydrophobic
aggregation for small molecules using a conventional potential
energy function
Potential energy functions
We have briefly reviewed the variety of interactions which are important in
protein interactions and seen suitable simple mathematical forms for their
representation. These are drawn together to form a potential energy function:
This function can be used to calculate a value for the potential
energy (PEF(R)) for any conformation of a given protein defined by the (normally Cartesian) coordinate vector R.
A number of important points can be made:
*
We called the function a "potential" energy function as it
does not contain contributions made to the total energy made by
the motions of the atoms involved. It is possible to calculate these
using molecular dynamics methods.
•The function aims to give reasonable values for the difference in
"microstate" energies between two different conformations. The
absolute value for the energy given does not mean anything
(certainly NOT the free energy of formation). Only differences
have meaning. The above equation also does not allow the
examination of any process which involves the change in
chemical bonding, e.g., one cannot simulate chemical reactions in
an enzyme active site with it.
•This kind of function is normally of little use in estimating
whether a protein adopts a particular fold
To be able to calculate the potential energy of a
protein using the above equation a large number of
parameters (equilibrium bond lengths beq, bond
stretching constants Kb ...) is required. The process of
finding these is arduous and in there are only around
four potential energy functions in common usage for
proteins (CHARMm, AMBER, GROMOS and
ECEPP).
•Although results obtained with current potential energy
functions are only approximate they have one great advantage
- they are computationally cheap. This allows the introduction
of realstic representation of environment - such as having
large numbers of explicitly modelled water molecules
surronding a protein.
• It also allows the calculation of the potential energy for
many different conformations of the same molecule. This
facilitates the use of techniques such as molecular dynamics
which allows the thermal motions of a system to be explored.
• This can be contrasted with quantum chemical methods
which even for small systems are so expensive that only a
limited number of calculations can be made but produce very
accurate energies.
Energy minimization
This is in many ways the simplest simulation procedure. The
basic idea is that starting from some structure (R) we find its
potential energy using the potential energy function given as
equation (1) above. The coordinate vector R is then varied
using an optimization procedure so as to minimize the potential
energy PEF(R).
Very often these methods are used if a distorted structure is
produced - e.g. a homology based model. Energy minimization
can then relieve short interatomic distances while maintaining
important structural features.
Energy minimization can be used to help to solve
experimental structures:
*
In X-ray crystallography measure a set of intensities
I(h,k,l) for a large number of reflections (h,k & l are Miller
indices listing the reciprocal lattice points of the crystal).
These are proportional to Fobs(h,k,l)2 the observed structure
functions. Once there is an approximate idea of the structure
(an initial model) the model's electron density can be used to
calculate expected structure factors Fcalc(h,k,l). The
structure can be refined by minimizing:
The program XPLOR is commonly used to do this
In NMR (nuclear magnetic resonance) experiments give
approximate distances between hydrogen atoms (NOE's) and
some dihedral angles. To obtain a 3D structure a similar
process is performed in which the objective function is the sum
of PEF(R) and restraint terms for the distances and dihedral
angles. Molecular dynamics simulated annealing is used to
optimize the function to obtain a set of 3D structures consistent
with the experimental data.
*
Denaturazione delle proteine
Temperatura
Pressione
Acido
Base
Agenti caotropici
Salting out
Detergenti