PHYSICAL REVIEW B 84, 139902(E) (2011)

Erratum: Mixed quantum-classical dynamics of an amide-I vibrational excitation in a proteinα-helix [Phys. Rev. B 82, 174308 (2010)]

Holly Freedman, Paulo Martel, and Leonor Cruzeiro(Received 28 July 2011; revised manuscript received 7 September 2011; published 7 October 2011)

DOI: 10.1103/PhysRevB.84.139902 PACS number(s): 82.20.Ln, 82.20.Wt, 87.15.bd, 87.10.Hk, 99.10.Cd

In the GROMACS code modifications, instead of the nanome-ter unit for the distance that is standard in GROMACS, a unitof 1 A was previously assumed. This led to dipole-dipoleinteractions between amide I vibrations at different sites andthe interaction energies of the amide I vibration with theprotein hydrogen bonds being overestimated, respectively, bythree orders and by one order of magnitude. In addition, thequantum mechanical force terms were overestimated because,through the same error, the sites defined as hydrogen-bondedin the protein were not properly identified. Because theinfluence of the quantum vibration on the conformation of thepolypeptide is less pronounced when these errors are corrected,the simulation times are increased to 10 ps. Moreover, eachof the 20 simulation runs, obtained by varying the seed ofthe random number generator for the initial velocities, wasrepeated ten times in order to sample different quantum MonteCarlo paths and to yield better statistical estimates. Thus, theresults presented here, for each value of χ , represent averagesover 200 simulation trajectories.

Two other changes are made in the current modifiedGROMACS code. The first of these makes sure that theinteraction energy term varies continuously with the lengthof the hydrogen bond, even when the hydrogen bond breaks.This term is calculated as

Hint = χ

N∑

n=1

[(dn − deq)a+n an], (2)

where χ = dε/d(|ROn − RN

m |) is the parameter of nonlinearityand deq denotes the equilibrium length of the hydrogen bondbetween the carbonyl oxygen atom at site n at positionRO

n andthe amine nitrogen atom at site m at position RN

m . deq is set

FIG. 1. Cumulative average of the number of helical residues as afunction of simulation time (ps) determined by the g helix GROMACS

analysis tool.

FIG. 2. Cumulative average of the number of helical residues as afunction of simulation time (ps) determined by the do dssp GROMACS

analysis tool.

equal to 0.3 nm according to the force field. dn is the smallesthydrogen-bond distance between sites n and m, as the sites mare varied, assuming that a hydrogen bond only exists when|RO

n − RNm | < 0.35nm and n > m + 3; if no such hydrogen

bond exists, dn is set equal to 0.35 nm.Second, a slight modification is made in the Monte Carlo

selection of the quantum vibrational eigenstate. We stipulatethat the state into which the amide-I vibration is given achance to propagate, according to a Metropolis-Monte Carlostep, should be most localized, i.e., should have the greatestprobability amplitude, at a site different from the previous one.This is performed to allow for a greater chance of propagationof the quantum vibration to another site, at each time step.

D

T

FIG. 3. Cumulative average of the minimum distance from theamino-terminal Ace group to the carboxy-terminal NH2 group asa function of simulation time (ps) determined by the g mindistGROMACS analysis tool.

139902-11098-0121/2011/84(13)/139902(2) ©2011 American Physical Society

ERRATA PHYSICAL REVIEW B 84, 139902(E) (2011)

TABLE I. Average localization (second column). The thirdcolumn gives the standard deviation taken over the ten sets of repeatedsimulation runs.

χ (pN) Localization Standard deviation

−800 0.334 0.007−400 0.348 0.006−200 0.372 0.007−100 0.422 0.008−50 0.436 0.01450 0.573 0.020100 0.663 0.036200 0.873 0.025400 0.975 0.010800 0.994 0.003

Rejection of this Monte Carlo step leads, as before, to theselection of the eigenstate that has the greatest probability inthe same site as the previous state.

After the corrections and changes mentioned above, we stillfind that the quantum amide-I excitation tends to increase thedegree of helicity of the peptide but to a lesser extent thanour data had previously suggested. The number of residuesbelonging to the longest helical segment at the end of the10-ps simulation, as determined by the g helix program, arefound, from smallest to highest, according to the values of χ

in piconewtons, in the following order −800 < −400 < −200< 0 < −100 < 50 < 100 < −50 < 200 ≈ 400 < 800, whereχ = 0 corresponds to the classical simulation (see Fig. 1).This ordering is similar, with a few small exceptions, whenthe number of helical residues is determined by the do dsspprogram (see Fig. 2).

Recalculation of the distances from the acetyl group, atthe amino-terminal end of the peptide, to the NH2 cap atthe carboxy-terminal end, indicates a contractile effect on thepeptide for all the values of χ tested here (see Fig. 3). Orderingfrom smallest to highest, the end-to-end distances found at theend of 10-ps simulations shows that the values follow the trendχ = 800 ≈ 400 < 200 < −200 ≈ −100 ≈ −800 < −400 ≈100 ≈ 50 < −50 < 0, where χ is in piconewtons and χ = 0corresponds to classical simulation.

Computed values of localization, averaged over time andover each of the 200 data sets (Table I), show the oppositetrend to what was reported in our paper. As the magnitudeof χ is increased, states become less localized for negativeχ , but they become more localized for positive χ so that thevalues of localization increase monotonically from 0.33 to0.99 as the value of χ is raised from −800 to 800 pN.

FIG. 4. Residue number at which the amide-I excitation isprimarily located, i.e., for which the coefficient in the expansionin terms of basis functions localized on individual residues islargest, plotted as a function of simulation time (ps) for trajectoriescorresponding to (a) χ= 800 pN and (b) χ= 100 pN.

Finally, the trajectories of the quantum amide-I vibration,represented in the new Fig. 4 by the site where probability ishighest, show that, for χ = 800 pN, the amide-I excitationtends to be trapped at hydrogen-bonded sites where localiza-tion is close to one (〈L〉 = 0.996). This is because the on-siteenergy at these sites is large and negative, compared to theenergies of non-hydrogen-bonded sites. On the other hand, forχ = 100 pN, the on-site energies of hydrogen-bonded sites aresmaller, the amide-I excitation is less localized (〈L〉 = 0.676),and it is transmitted throughout the entire peptide during the10-ps simulation.

To summarize, our conclusion is qualitatively the same asbefore, namely, that the amide-I vibration tends to increasethe helicity and to lead to a contraction of the peptide,but the quantitative effect is weaker than was reported. Thevibrational excitation acts mainly to trap any transient helicalcharacter that arises from thermal fluctuations. Moreover,because trapping and/or self-trapping limits the mobility of thequantum excitation and trapping increases as χ increases, theeffect of the quantum vibration on helicity is not as dependenton the magnitude of χ as was suggested in our paper.

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