Download 1 Peptide bond rotation

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Peptide bond rotation
We now consider an application of data mining that has yielded a result that links
the quantum scale with the continnum level electrostatic field.
In other cases, we have considered the modulation of dielectric properties and the
resulting effect on electronic fields.
Wrapping by hydrophobic groups plays a role in our analysis, but we are
interested primarily in a reversal of the interaction.
We consider modulations in the electronic field that cause a significant change in
the electron density of the peptide bond and lead to a structural change in the
principle bond.
We first describe the results and then consider its implications.
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Peptide resonance states
The peptide bond is characterized in part by the planarity of the six atoms shown
in Figure 1.
The angle ω quantifies this planarity.
The bond is what is known as a resonance [2] between two states, shown in
Figure 1.
The “keto” state (A) on the left side of Figure 1 is actually the preferred state in
the absence of external influences [2].
However, an external polarizing field can shift the preference for the “enol” state
(B) on the right side of Figure 1, as we illustrate in Figure 2.
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""
H b
b
C
""
H b
b
C
N+
N
(A)
(B)
"
"
C
C
b
b
b
b
"
"
O
C
C
bb
O−
Figure 1: Two resonance states of the peptide bond. The double bond between the
central carbon and nitrogen keeps the peptide bond planar in the right state (B). In
the left state (A), the single bond can rotate.
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2.1
Effect of external field
The resonant state could be influenced by an external field in different ways, but
the primary cause is hydrogen bonding.
If there is hydrogen bonding to the carbonil and amide groups in a peptide bond,
this induces a significant dipole which forces the peptide bond into the (B) state
shown in Figure 1.
Such hydrogen bonds could either be with with water or with other backbone
donor or acceptor groups, and such a configuration is indicated in Figure 2.
On the right side of this figure, the polar environment is indicated by an arrow
from the negative charge of the oxygen partner of the amide group in the peptide
bond to the positive charge of the hydrogen partner of the carbonyl group.
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O
−
bb
""
H b
b
l
l
C
N+
"
"
C
C
bb
O−
bb
H
C
l H b
"
l b +"
N
l
l
l
l
Cl
"
l
bb
"
l−
O
C
l
l
l
+
Figure 2: The resonance state (B) of the peptide bond shown, on the left, with
hydrogen bonds which induce a polar field to reduce the preference for state (A).
On the right, the (B) state is depicted with an abstraction of the dipole electrical
gradient induced by hydrogen bonding indicated by an arrow.
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2.2
Effect of external field, continued
The strength of this polar environment will be less if only one of the charges is
available, but either one (or both) can cause the polar field.
However, if neither type of hydrogen bond is available, then the resonant state
moves toward the preferred (A) state in Figure 1.
The latter state involves only a single bond and allows ω rotation.
Thus the electronic environment of peptides determines whether they are rigid or
flexible.
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Measuring variations in ω
There is no direct way to measure the flexibility of the peptide bond.
For any given peptide bond, the value of ω could correspond to the rigid (B) state
even if it is fully in the (A) state.
The particular value of ω depends not only on the flexibility but also on the local
forces that are being applied.
These could in priniciple be determined, but it would be complicated to do so.
However, by looking at a set of peptide bonds, we would expect to see a range of
values of ω corresponding to a range of local forces.
It is reasonable to assume that these local forces would be randomly distributed in
some way if the set is large enough.
Thus, only the dispersion ∆ω for a set of peptide bond states can give a clear
signal as to the flexibility or rigidity of the set of peptide bonds.
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3.1
Measuring variations in ω, continued
The assessment of the flexibility of the peptide bond requires a model.
Suppose that we assume that the bond adopts a configuration that can be
approximated as a fraction CB of the rigid (B) state and corresponding the
fraction CA of the (A) state.
Let us assume that the flexibility of the of the peptide bond is proportional to CA .
That is, if we imagine that the peptide bond is subjected to random forces, then
the dispersion ∆ω in ω is proportional to CA :
∆ω = γCA
where γ is a constant of proportionality.
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(3.1)
3.2
Computing CB
We are assuming that the resonant state ψ has the form ψ = CA ψA + CB ψB
where ψX is the eigenfunction for state X (=A or B).
Since these are eigenstates normalized to have L2 norm equal to one, we must
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have CA
+ CB
= 1, assuming that they are orthogonal. Therefore
q
2.
∆ω = γCA = γ 1 − CB
(3.2)
We can thus invert this relationship to provide CB as a function of ∆ω :
p
CB = γ 1 − (∆ω /γ)2 .
(3.3)
The assumption (3.1) can be viewed as follows.
The flexibility of the peptide bond depends on the degree to which the central
covalent bond between carbon and nitrogen is a single bond.
The (A) state is a pure single bond state and the (B) state is a pure double bond
state.
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3.3
Model justification
Since the resonance state is a linear combination, ψ = CA ψA + CB ψB , it follows
that there is a linear relationship between CA and flexibility provided that the
single bond state can be quantified as a linear functional.
That is, we seek a functional LA such that LA ψA = 1 and LA ψB = 0.
If ψA and ψB are orthogonal, this is easy to do.
We simply let LA be defined by taking inner-products with ψA .
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Predicting the electric field
Since the preference for state (A) or (B) is determined by the local electronic
environment, the easiest way to study the flexibility would be to correllate it with
the gradient of the external electric field at the center of the peptide bond.
However, this field is difficult to compute precisely due to need to represent the
dielectric effect of the solvent.
Even doing a dynamic simulation with explicit water representation would be
challenging due to the need to represent the polarizability of water and to model
hydrogen bonds accurately.
But it is clear that the major contributors to a local dipole would be the hydrogen
bonding indicated in Figure 2.
These bonds can arise in two ways, either by backbone hydrogen bonding (or
perhaps backbone-sidechain bonding, which is more rare) or by contact with
water.
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4.1
Approximating the electric field
The presence of backbone hydrogen bonds is indicated by the PDB structure, and
we have a proxy for the probability of contact with water: the wrapping of the
local environment.
So we can approximate the expected local electric field by analyzing the backbone
hydrogen bonding and the wrapping of these bonds.
In [1], sets of peptide bonds were classified in two ways.
First of all, they were separated into two groups, as follows.
Group I consisted of peptides forming no backbone hydrogen bonds, that is, ones
not involved in either α-helices or β-sheets.
Group II consisted of peptides forming at least one backbone hydrogen bond.
In each major group, subsets were defined based on the level ρ of wrapping in the
vicinity of backbone.
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Figure 3: From [1]: Fraction of the double-bond (planar) state in the resonance for
residues in two different classes (a) Neither amide nor carbonyl group is engaged
in a backbone hydrogen bond. As water is removed, so is polarization of peptide
bond. (b) At least one of the amide or carbonyl groups is engaged in backbone
hydrogen bond.
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4.2
Relating the groups to resonant states
Figure 3(a) depicts the resulting observations for group I peptide bonds, using the
model (3.3) to convert observed dispersion ∆ω (ρ) to values of CB , with a
constant γ = 22o .
This value of γ fits well with the estimated value CB = 0.4 [2] for the vacuum
state of the peptide bond [2] which is approached as ρ increases.
Well wrapped peptide bonds that do not form hydrogen bonds should closely
resemble the vacuum state (A).
However, poorly wrapped peptide amide and carbonil groups would be strongly
solvated, and thus strongly polarized, leading to a larger component of (B) as we
expect and as Figure 3(a) shows.
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4.3
Interpreting the figures
Using this value of γ allows an interesting assesment of the group II peptide
bonds, as shown in Figure 3(b).
These are bonds that are participating in backbone hydrogen bonds.
We see that these bonds also have a variable resonance structure depending on the
amount of wrapping.
Poorly wrapped backbone hydrogen bonds will nevertheless be solvated, and as
with group I peptide bonds, we expect state (B) to be dominant for small ρ.
As dehydration by wrapping improves, the polarity of the environment due to
waters decreases, and the proportion of state (B) decreases.
But a limit occurs in this case, unlike with group I, due to the fact that wrapping
now enhances the strength of the backbond hydrogen bonds, and thus increases
the polarity of the environment.
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4.4
Interpreting the figures, continued
The interplay between the decreasing strength of polarization due to one kind of
hydrogen bonding (with water) and the increasing strength of backbone hydrogen
bonding is quite striking.
As water is removed, hydrogen bonds strengthen and increase polarization of
peptide bond.
Figure 3(b) shows that there is a middle ground in which a little wrapping is not
such a good thing.
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Implications for protein folding
After the “hydrophobic collapse” [3] a protein is compact enough to exclude most
water.
At this stage, few hydrogen bonds have fully formed.
But most amide and carbonyl groups are protected from water.
The previous figure (a) therefore implies that many peptide bonds are flexible in
final stage of protein folding.
This effect is not included in current models of protein folding.
This effect buffers the entropic cost of hydrophobic collapse in the process of
protein folding.
New models need to allow flexible bonds whose strengths depend on the local
electronic environment.
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References
[1] Ariel Fernández. Buffering the entropic cost of hydrophobic collapse in
folding proteins. Journal of Chemical Physics, 121:11501–11502, 2004.
[2] Linus Pauling. Nature of the Chemical Bond. Cornell Univ. Press, third
edition, 1960.
[3] Ruhong Zhou, Xuhui Huang, Claudio J. Margulis, and Bruce J. Berne.
Hydrophobic collapse in multidomain protein folding. Science,
305(5690):1605–1609, 2004.
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