1 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. 1
2 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. 2
(A) H N C (B) H N + C C C O C C 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. 3
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. 4
O H C N + C C O H C H N + C C O 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. + 5
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. 6
3 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. 7
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 C B of the rigid (B) state and corresponding the fraction C A of the (A) state. Let us assume that the flexibility of the of the peptide bond is proportional to C A. That is, if we imagine that the peptide bond is subjected to random forces, then the dispersion ω in ω is proportional to C A : where γ is a constant of proportionality. ω = γc A (3.1) 8
3.2 Computing C B We are assuming that the resonant state ψ has the form ψ = C A ψ A + C B ψ B where ψ X is the eigenfunction for state X (=A or B). Since these are eigenstates normalized to have L 2 norm equal to one, we must have CA 2 + C2 B = 1, assuming that they are orthogonal. Therefore ω = γc A = γ 1 CB 2. (3.2) We can thus invert this relationship to provide C B as a function of ω : The assumption (3.1) can be viewed as follows. C B = γ 1 ( ω /γ) 2. (3.3) 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. 9
3.3 Model justification Since the resonance state is a linear combination, ψ = C A ψ A + C B ψ B, it follows that there is a linear relationship between C A and flexibility provided that the single bond state can be quantified as a linear functional. That is, we seek a functional L A such that L A ψ A = 1 and L A ψ B = 0. If ψ A and ψ B are orthogonal, this is easy to do. We simply let L A be defined by taking inner-products with ψ A. 10
4 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. 11
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. 12
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. 13
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 C B, with a constant γ = 22 o. This value of γ fits well with the estimated value C B = 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. 14
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. 15
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. 16
5 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. 17
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. 18