Dynamical transition in proteins and non-Gaussian behavior of low-frequency modes in self-consistent normal mode analysis

Self-consistent normal mode analysis (SCNMA) is applied to heme c type cytochrome f to study temperature-dependent protein motion. Classical normal mode analysis assumes harmonic behavior and the protein mean-square displacement has a linear dependence on temperature. This is only consistent with low-temperature experimental results. To connect the protein vibrational motions between low and physiological temperatures, we have incorporated a fitted set of anharmonic potentials into SCNMA. In addition, quantum harmonic-oscillator theory has been used to calculate the displacement distribution for individual vibrational modes. We find that the modes involving soft bonds exhibit significant non-Gaussian dynamics at physiological temperature, which suggests that it may be the cause of the non-Gaussian behavior of the protein motions probed by elastic incoherent neutron scattering. The combined theory displays a dynamical transition caused by the softening of few “torsional” modes in the low-frequency regime ($<50{\text{cm}}^{-1}$ or $<6\text{meV}$ or $>0.6\text{ps}$). These modes change from Gaussian to a classical distribution upon heating. Our theory provides an alternative way to understand the microscopic origin of the protein dynamical transition.

Figure 1

Characterization of the single frequency displacement distribution at 300 K in (a) the displacement is similar to the classical distribution, (c) the displacement distribution is similar to a Gaussian, and (b) is a cross between the two. The horizontal axis is displacement variable $u$.

Figure 2

(Color online) Comparison of the experimental (from Ref. [43]) and theoretical (from classical NMA) cytochrome f iron VDOSs. Solid line: experiment; dotted line: theory.

Figure 3

(Color online) The iron MSD vs temperature plot for various heme proteins. Line: iron MSD of cyt f by classical NMA. Cross and dashed line: iron MSD of cyt f from SCNMA with Gaussian distribution approximation. Star: iron MSD of cyt f from our SCNMA by implementing non-Gaussian displacement distribution. Diamond: iron MSD of myoglobin by Mössbauer absorption measurement from Ref. [4]. Upper triangle: iron MSD of cyt c by Mössbauer absorption measurement from Ref. [5].

Figure 4

The cyt f iron in-plane and out-of-plane MSDs from SCNMA. Dashed line: iron in-plane MSD from classical NMA; circle: iron in-plane motion from SCNMA; solid line: iron out-of-plane motion from classical NMA; star: iron out-of-plane motion from SCNMA.

Figure 5

(Color online) A plot of $⟨{\sum }_{i=1}^n{m}_i{r}_i^2⟩$ as a function of temperature $\left(T\right)$ and (low) frequency $\omega$ from Eq. (1). $u_0$ is one atomic weight.

Figure 6

(Color online) Cyt f iron MSD from SCNMA in three frequency regimes. (a) Star: $<20{\text{cm}}^{-1}$ (2.5 meV), (b) diamond: $>20{\text{cm}}^{-1}$ (2.5 meV) and $<50{\text{cm}}^{-1}$ (6 meV), and (c) circle: $>50{\text{cm}}^{-1}$ (6 meV).