The nonlinear response function formalism [30
] was utilized to interpret the experimental data, with the specific aim of determining the extent to which 3PEPS can distinguish between the different unfolding models. Simulations of photon echo experiments by this method, also employing the high temperature approximation, have been described in many publications [11
]. Here, we use this approach with modifications to account for the non-Gaussian inhomogeneous broadening due to the distribution of conformers. We simulated the APS as a function of GuHCl concentration and pump laser wavelength for both two- and three-state models.
In our simulation, the line broadening function is expressed as:
) is the time-correlation function of the electronic transition frequency. In our previous work, we gave the M
) for the native Zn-cyt c [25
], as a sum of Kubo oscillators and damped vibration oscillators:
Here, we used these parameters, which are collected in . This M(t) does not include a Gaussian component, because its exponential terms reflect the high frequency vibrational modes, which dominate the protein inertial motions, usually reflected by a fast Gaussian component.
For simplicity, we use the same M
) for the native, unfolded and intermediate conformers. We are therefore assuming the same homogeneous dynamics (solvent dynamics and chromophore vibrations) for the three conformers. This assumption is reasonable because although the conformational state has a measurable effect on the porphyrin vibrations and the solvent contributions, their impacts on the lineshape are small relative to the change in the magnitude of inhomogeneous broadening. However, since the APS of the native and unfolded protein are different, we add different inhomogeneous terms when calculating g
). To reproduce the APS, we incorporate Δin
= 150, 260 and 200 cm−1
, respectively, for native, intermediate and unfolded conformers. The intermediate is attributed the largest inhomogeneous broadening because its absorption spectrum (shown in ) has the widest FWHM. The peak extinction coefficient of the three conformers is slightly different (). Although it would be more accurate to weight the contributions to the optical response accordingly, we neglect this small effect. There is some uncertainty in the exact values of ωeg
, the 0–0 transition energy, however, as they are not experimentally available. Nevertheless, as ωeg
should always be at the red side of the absorption maximum, we chose the following values for the simulation: 420, 426 and 427 nm for the unfolded, intermediate and native conformers, respectively. For each GuHCl concentration and folding model (two- or three-state), the population is calculated based on the global fitting parameters (Cm(1,2)
) shown earlier. Then, the corresponding optical response function is a sum of these conformers, with the response of each conformer weighted according to its population. For example, since the midpoint sample consists of a nearly 1:1 mixture of native and unfolded forms in the two-state model, its total optical response function is a sum of native and unfolded samples, with equal contributions from the two conformers. Finally, the calculation includes the effects of finite pulse duration (45 fs), bandwidth, and center wavelength, as implemented in previous work [19
shows the simulated 3PEPS for native, unfolded states and midpoint in the two-state model. At T = 0, the three states have similar initial peak shifts: ~25 fs. As mentioned earlier, the initial peak shift is inversely proportional to the homogenous line broadening, which is characterized by M(t). Actually the simulated initial peak shifts are slightly greater than the experimental values ~20 fs. The spectrum of vibrational mode-specific displacements for Zn-cyt are not available, thus we do not attempt to quantitatively reproduce the homogeneous lineshape. With our model M(t), the homogeneous broadening is smaller than the actual value, so the simulated initial peak shifts are expected to be larger than measured values. The simulation reproduces the prominent oscillation in the experimental data, and the three samples have identical peak and trough positions, although the fluctuations are not as well-resolved in the experiment due to limitations in the signal to noise. The APS from the simulations are 4.2, 5.7 and 7.8 fs for native, unfolded and midpoint states, respectively. These values match the experiment, within the measurement error (), which indicates that this folding process can be explained by a two-state folding model.
Simulated 3PEPS traces for native, midpoint and unfolded Zn-cyt c.
We calculated the asymptotic peak shift as a function of GuHCl concentrations for both two- and three-state models (). In three-state model, the APS increases from 0 to 2.5 M GuHCl, then decreases and levels off at GuHCl concentrations above 3.5 M where the population is constant. Interestingly, for GuHCl concentrations below 2.5 M, the difference between two and three-states is insignificant. From our experimental data, we cannot rule out the possibility of a three-state model, because this model predicts an APS at 2.5 M GuHCl concentration that differs from the experimental value by only 0.3 fs. This difference is within the error of our measurement. However, the predictions of the two models differ significantly in the 2.5–3.5 M GuHCl range. Therefore, a titration measurement in this concentration window could be useful for distinguishing between two and three-state models, since the APS values differ significantly, and exceed the error of the experiment (typically less than ±0.5 fs) [19
Comparison of two-and three-state models. (a) Simulated asymptotic peak shift of Zn-cyt c with varying GuHCl concentration. (b) Simulated asymptotic peak shift for the midpoint state of Zn-cyt c (2.5 M GuHCl) with varying laser wavelength.
Next, we investigated the effect of laser wavelength on the APS of the midpoint sample by varying the laser wavelength both experimentally and numerically at a GuHCl concentration of 2.5 M, as shown in . At this GuHCl concentration, the sample consists of a 1:1 mixture of native and unfolded conformers for two-state model, or an approximately 1:1 mixture of intermediate and unfolded conformers for the three-state model, as shown in . For the three-state model, the APS increases when the laser wavelength is tuned to 428 nm. This result is physically reasonable because the ωeg of intermediate is assumed to be 426 nm. Therefore, pump laser bandwidth is tuned to preferentially probe this conformer, which has the largest inhomogeneous broadening conformer. For the same reason, the APS reaches its maximum at 420 nm for two-state model. The measured APS at a few wavelengths is also shown in this figure. Although data at longer wavelengths would be more conclusive, the available data are more consistent with the values and the trend of the two-state unfolding model.