Download Bond angles

Una maniera nuova di guardare le proteine
globulari
Cercare le ”strutture a strato"
•
Le proteine globulari possono essere
rappresentate come formate da “strati” di
scheletro peptidico :
– Gli strati possono essere rappresentati dal
piano dei legami H nei foglietti b o da una serie
di a-eliche parallele impaccate contro i
foglietti
Simplified layer structure of proteins.
Layers of secondary structure (b green; a •
red) are combined to make globular
protein domains. The b-sheets are
represented as bars and circles, as they •
would appear when viewed looking along
their component strands. Each sheet has
a left-handed twist between the strands
•
(not depicted) onto which can be added
curl and stagger
Redrawn with permission from Nature, 2002,
416, 657
• Possono essere piatti, curvati o cilindrici
I residui idrofobici vengono nascosti tra gli
“strati”
Nelle proteine solubili, l’esterno e’ costituito
principalmente da residui polari che
interagiscono favorevolmente con il solvente
E’ impressionante come la maggioranza delle
strutture proteiche sia classificabile in questo
modo: solo poche strutture non possono
esserlo
Esempi di domini proteici con diverso
numero di strati dello scheletro proteico
Residui idrofobici in
giallo
due strati di a-elica
Una inusuale
struttura a
5 strati: uno
strato di
filamenti b
chiuso a
sandwich
tra 4 strati di aeliche
uno strato di foglietti b fra due
strati di elica, in tutto 3 strati
strati concentrici
di filamenti b
all’interno e di a
eliche all’esterno
(struttura nota
anche come
barile a-b)
TIPI DI FOLD PIU’
FREQUENTI
Fold delle proteine e funzione
In natura forse esistono 10000 diversi fold ovvero forme in cui possiamo
trovare le proteine che hanno una organizzazione tipica della propria
componente di struttura secondaria.
In realtà possiamo dire che solo alcune centinaia sono realmente diffuse
mentre le altre sono rare o specie specifiche.
La natura ottimizza le proprie energie e non crea nuove forme se piccole
modificazioni di strutture preesistenti consentono di ottenere funzioni diverse
(Coulson, Proteins 2002)
Il concetto dei
SUPERFOLDS poche centinaia (forse 400) presenti in molte proteine tante funzioni
MESOFOLDS una via di mezzo. Fold mediamente frequenti
UNIFOLDS presenti in un’unica specie di proteine. Forse la maggior parte e specie specifiche.
Possono essere oltre 10000
Fold delle proteine e funzione: Superfolds, mesofolds, unifolds
I Superfolds
TIM barrel fold
The structure consists of an eight fold repeat of beta/alpha
units.
1) Eight parallel beta strands on the inside are covered by
2) Eight alpha helices on the outside.
The fold was first seen in triose phosphate isomerase. All
known TIM barrel structures are enzymes, except for the
narbonin family. Many of these enzymes are glycosyl
hydrolases (EC 3.2.x.x). The fold is highly versatile, being found
in single-domain monomeric enzymes and as the catalytic
domain of larger enzymes. The active site is found at the Cterminal end of the barrel in a series of loops, hence it is very
easy to alter the function and/or specificity without altering the
core structure.
Alpha/beta hydrolase fold
The structure is an eight stranded, mostly parallel
alpha/beta structure. The fold is tolerant to large
insertions and is very plastic. All proteins known so far
containing this fold are enzymes. The enzymatic
properties of this fold are formed by a catalytic triad of a
nucleophile, acid and a histidine residue. The
nucleophile is found in a "nucleophilic elbow" turn
located just after the fifth beta strand.
NAD binding domain (Rossman fold)
This is a double beta-alpha-beta-alpha-beta motif, and is a
common structural motif of enzymes binding NAD, NADP
and other related cofactors for example, NAD is found in
dehydrogenases as the hydrogen acceptor. The domain is
found as a common core unit in many structures, with other
structural units at the periphery.
P-loop NTP hydrolase fold
This fold consists of alpha/beta/alpha, parallel or mixed
beta sheets of variable size. The fold binds the phosphate
of ATP or GTP and is found in ATP and GTP binding
proteins such as adenylate kinase. The P-loop is
phosphate binding loop that binds the phosphate groups
of ATP and GTP,and is a glycine-rich sequence with the
consensus sequence (A,G)xxxxGK(T,S). The P-loop
residues are shown in detail (left) in guanylate kinase.
Ferredoxinlike fold
This fold consists of an alpha/beta sandwich with
an antiparallel beta sheet The ferredoxinlike fold is
associated predominantly with nonenzymatic
ferredoxins. Ferredoxins are iron-sulphur clusters
invovled in electron transport, and often form part
of multisubunit assemblies.
DISTRIBUZIONE DI FUNZIONE
NEI FOLD
Classi strutturali e funzione enzimatica
(Hegyi & Gerstein, JMB 1999)
1)
alpha/beta (α/β): proteine con struttura alpha e beta ben distinte. sono prevalentemente enzimi specialmente
trasferasi e idrolasi
2)
Tutte alpha (α) e piccoli fold: associate a proteine non-enzimatiche
Da queste considerazioni, però, non si è certi di associare univocamente ad una classe strutturale una funzione
basata sul contenuto di struttura secondaria.
Proteine da swissprot vs. i fold di SCOP usando BLAST e divise in 92 categorie funzionali (divise in 6
supergruppi) per riga contro 229 fold per colonna. La prima riga con i non-enzimi. Ci sono 21068
(=92x229) combinazioni possibili ma solo 331 sono state osservate (quadratini pieni).
Calcolo delle forze di
interazione
What is a Force Field ?
• A force field is a set of equations and parameters which when
evaluated for a molecular system yields an energy
• A force field is a specific type of vector field where the value of a
given force is defined at each point in space. Examples include
gravitational fields and electrostatic fields
• The space around a radiating body within which its electromagnetic
oscillations can exert force on another similar body not in contact
with it
It is all about time versus accuracy
•
•
•
•
•
•
Quantum chemistry
Approximations
Force Fields
Hybrid methods
Molecular dynamics and energy calculations
Minimizers
Quantum chemistry is accurate, but slow
Quantum chemistry is accurate, but slow
The largest ‘thing’ that can realistically be worked-out using the
Schödinger equation is hydrogen. Other applications are the particle
in a box that is mainly of theoretical importance
Actually, pure quantum chemistry cannot be applied in our
(protein) world.
Approximations can make quantum chemistry software faster, but
at the cost of accuracy.
The basic functional form of a force field encapsulates both bonded terms relating to atoms that are
linked by:
1. covalent bonds
2. nonbonded (also called "noncovalent") terms describing the long-range electrostatic and van der
Waals forces.
The specific decomposition of the terms depends on the force field, but a general form for the total
energy in an additive force field can be written as where the components of the covalent and
noncovalent contributions are given by the following summations:
The bond and angle terms are usually modeled as harmonic oscillators in force fields that do not
allow bond breaking. The functional form for the rest of the bonded terms is highly variable. Proper
dihedral potentials are usually included. Additionally, "improper torsional" terms may be added to
enforce the planarity of aromatic rings and other conjugated systems, and "cross-terms" that describe
coupling of different internal variables, such as angles and bond lengths. Some force fields also include
explicit terms for hydrogen bonds.
The nonbonded terms are most computationally intensive because they include many more
interactions per atom. A popular choice is to limit interactions to pairwise energies. The van der Waals
term is usually computed with a Lennard-Jones potential and the electrostatic term with Coulomb's law,
although both can be buffered or scaled by a constant factor to account for electronic polarizability and
produce better agreement with experimental observations.
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
The shape of the potential
energy well will be parabolic
and the motion will therefore
tend to be harmonic
Note that when the bond is at its
equilibrium length i.e., r = r_eq
the potential energy is assigned
to be zero and when r
approaches large values (i.e. the
bond breaks) the energy goes
infinite. This kind of approach
does not attempt to reflect the
energy of formation of the bond
- it only seeks to reflect the
energy difference on a small
motion about the equilibrium
value
Atom pair
r_eq in Å
K_r in kcal/(molÅ2)
C=O
C-C2
C-N
C2-N
N-H
Typical values for bond constants and
equilibrium bond lengths taken from
the AMBER potential energy function
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)
united atom means that instead of representing
the carbon atom and the two hydrogen atoms
which are bonded to it separately only one
centre is considered
Note that oxygen atoms are shown in red, nitrogens are in blue, hydrogens in white
and carbon atoms are in light green. Only the hydrogen atoms bonded to an oxygen
or nitrogen are shown. This is fairly routine approximation called the "united-atom"
or "extended-atom" representation, which is an option in many potential energy
functions. The
hydrogen atoms
which should be
attached to
carbons are not
explicitly
represented but
instead a small
adjustment is
made to the LINK
NEEDED van der
Waals parameters
of the carbon
atom.
Bond angles
A bond angle theta  between atoms A-B-C is defined as the
angle between the bonds A-B and B-C:
As bond angles are found
(experimentally and
theoretically) to vary around a
single value it is sufficient in most
applications to use a harmonic
representation (in a similar
manner to the bond potential):
Angle eqin 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
These are the bond angle parameters necessary in representing a glycine residue and its
connections to neighbouring residues. A bond angle around 109 degrees means that the
central atom is tetrahedral (with four other atoms bonded to it):
The angle around the C beta atom of an alanine residue - showing the three hydrogen
atoms bonded to the carbon. In contrast an angle around 120 degrees indicates a flat
(sp2) central atom with three other atoms bounded to it:
This shows the angles made around a main chain nitrogen atom are all approximately
equal to 120 degrees: consequently the group is planar.
The source of bond angle parameters is the same as for bonds: high resolution small
molecule X-ray structures for eqilibrium values and either spectroscopic data or ab
initio calculations for force constants.
Potential energy curve for
the N-C-O Bond Angle
Dihedral angles
The standard functional form for representing the potential energy
for a torsional rotation was introduced by Pitzer (1951)
Vn gives the energy barrier to rotation, n the number of maxima
(or minima) in one full rotation and  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
n
Vn
(in kcal/mol)
1
2
3
1.3
5.0
0.0
γ
(in degrees)
0.
180.
-
NON-BONDED INTERACTIONS
As the name implies non-bonded interactions act between atoms
which are not linked by covalent bonds. Like most things this is
simple to state but can be confusing to apply in practice! In most
approaches then atoms which are involved in a bond angle are
also not regarded as having a non-bonded interaction. 1-4
interactions (those between the end atoms involved in a dihedral
angle) are sometimes given an additional scaled down nonbonded
interaction
Interazioni elettrostatiche
Non-bonded Interactions
Electrostatic interactions
electromagnetic interactions dominate on the molecular scale and
provide the fundamental basis for all the different bonded and
non-bonded interactions discussed here. This is clearest in the
case of electrostatic interactions where charges on nuclei and
electrons interact according to Coulomb's law
where qi and qj are the magnitude of the charges, rij is their
separation, 0 the eletric constant in vacuum and r the relative
dielectric constant of the medium in which the charges are
placed
The source of dielectric effects is that the
electric field polarizes the material involved.
Suppose that we have two charges interacting
in a vacuum. We can draw the electric field
lines (the direction in which a positive charge
would be forced to move)
These charges are then placed in a dielectric
medium - which can be thought of as being
composed of a large number of microscopic
dipoles (a little rod with positive charge at one
end and negative at the other). 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.
that the electric field between charges
permeates the whole of space - it does not only
depend on what is immediately in between the
charges
The dielectric constant of selected materials
Material Dielectric constant
Water (20 C)
Water (0 C)
Ice (-10 C)
Methanol
Liquid H2S (-85.5 C)
Beeswax
Paraffin
Liquid Argon(-191 C)
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 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:
SALT BRIDGE
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
Hydrogen bonds 2.8 Å 6kcal/mol
The electrostatic interactions between groups which
carry no formal overall electrical charge
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.
1. The most common approach is to place a partial charge at each
atomic centre (nucleus).
2. These charges then interact by Coulomb's Law.
3. The charge can take a fractions of an electron and can be positive or
negative.
4. 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.
1. 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).
2. 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.
INDUCTION
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
(London forces)
The London dispersion force is the weakest intermolecular
force. The London dispersion force is a temporary attractive
force that results when the electrons in two adjacent atoms
occupy positions that make the atoms form temporary
dipoles. It exploits the INDUCTION.
Imagine that we have an atom of argon. It can
be considered to be like a large spherical jelly
with a golf ball embedded at the centre. The
golf ball is the nucleus carrying a large positive
charge and the jelly represents the clouds of
electrons whizzing about this. At a point
external to the atom the net average field will
be zero because the positively-charged nucleus'
field will be exactly balanced by the electron
clouds:
However, atoms vibrate (even at 0K) and
so that at any instant the cloud is likely to
be slightly off centre. This disparity
creates an "instantaneous dipole":
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 dispersion and repulsion terms discussed above are
commonly grouped together into the Lennard-Jones or 6-12 potential
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 known 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.