Observation of a Power-Law Memory Kernel for Fluctuations within a Single Protein Molecule

The fluctuation of the distance between a fluorescein-tyrosine pair within a single protein complex was directly monitored in real time by photoinduced electron transfer and found to be a stationary, time-reversible, and non-Markovian Gaussian process. Within the generalized Langevin equation formalism, we experimentally determine the memory kernel $K(t)$, which is proportional to the autocorrelation function of the random fluctuating force. $K(t)$ is a power-law decay, $t^{-0.51\pm 0.07}$ in a broad range of time scales (${10}^{-3}–10\mathrm{s}$). Such a long-time memory effect could have implications for protein functions.

Figure 1

(a) Schematic of the structure of the FL and anti-FL complex, adapted from Ref. 6. ${\mathrm{Tyr}}^{37}$ and FL, ET donor and acceptor, are highlighted. (b) Monoexponential fluorescence lifetime decay for a single FL molecule. Multiexponential fluorescence decay for the FL and anti-FL complex at both ensemble and single-molecule levels. The instrumental response function with 60 ps FWHM. a.u., arbitrary units.

Figure 2

(a) Segment of the single-molecule $x(t)$ trajectory with the corresponding probability density function $P(x)$ converted from ${\gamma }^{-1}(t)$ using Eq. (1). The dashed line is the Gaussian fit for $P(x)$. (b) Potential of mean force $U(x)$ obtained from $U(x)=-k_{B}T\ln [P(x)]$. The dashed line is the best fit to a harmonic potential $U(x)=k_{B}T{x}^2/2\theta$, where $\theta =0.22{\AA }^2$.

Figure 3

(a) $⟨x^3(0)x(t)⟩$ vs $⟨x(0)x^3(t)⟩$ for the same single FL and anti-FL complex for various $t$. The diagonal dashed line is the prediction of time reversibility. (b) $⟨x(0)x(3t)⟩⟨x(0)x(t)⟩+⟨x(0)x(t){⟩}^2+⟨x(0)x(2t){⟩}^2$ vs $⟨x(0)x(t)x(2t)x(3t)⟩$ of the same complex. The diagonal dashed line is the prediction for a Gaussian process.

Figure 4

Autocorrelation function of distance fluctuation $C_x(t)$ (open circles, average of 13 molecules under the same experimental condition), determined with high time resolution using Eq. (3), with $C_x(0)=k_{B}T/m{\omega }^2=\theta =0.22{\AA }^2$. The solid line is a fit to $C_x(t)=C_x(0)e^{t/t_0}\mathrm{erfc}(\sqrt{t/t_0)}$ with parameter $\zeta /m{\omega }^2=0.7{\mathrm{s}}^{0.5}$. The error bounds (dashed line) were estimated by the method described in Ref. 17.

Figure 5

Normalized $\tilde{K}(s)$ calculated from $C_x(t)$ in Fig. 4 using Eq. (8) (open circles). The full line is the fit of $s^{-0.49}$. Its inverse Laplace transform gives $K(t)\propto t^{-0.51}$. The dashed lines are error bounds carried over from the error bounds of $C_x(t)$ in Fig. 4.