Distance versus energy fluctuations and electron transfer in single protein molecules

Stochastic nature due to distance and energy fluctuations of single protein molecules involved in electron-transfer (ET) reactions is studied. Distance fluctuations have been assumed previously for causing the slow fluctuations in the ET rates between a donor-acceptor pair constrained to a native protein. Although the observed $t^{-1∕2}$ power law can be derived using Langevin dynamics with a simple chain model, some discrepancies exist. The friction coefficient and the Rouse segment time constant deduced from experimental data are several orders of magnitude too large, even though the extracted force constant is reasonable. Therefore, questions are raised about the distance-fluctuation mechanism and the activationless ET hypothesis. As an alternative mechanism, we considered fluctuations in activation energy and analyzed the data from two different single protein experiments to determine spectral distribution of energy fluctuations.

Figure 1

(Color online) (a) Ribbon diagram of a FL and anti-FL protein, showing a dimeric structure with a donor Tyr (tyrosine) and an acceptor FL (fluorescein). (b) Ball-stick diagram of the same protein, showing that the native protein is highly compact and the donor-acceptor pair is located near the edge of the protein structure.

Figure 2

(Color online) (a) Backbone structure (pKa plot) of the FL and anti-FL protein, showing the position of the donor Tyr and an acceptor FL. The edge-to-edge distance between the donor and the acceptor is $3.7\mathrm{\AA }$. (b) Schematic representation of a protein molecule by an ideal Rouse chain in 3D space, containing a donor $\left(D\right)$ and an acceptor $\left(A\right)$ with nearest neighbors coupled by a spring.

Figure 3

(Color online) Distance ACF $C_Q\left(t\right)∕C_Q\left(0\right)$ of Eq. (5) for various chain-segment units $N$ with a distributed $\Gamma$. At $t∕{\tau }_c⪡1$ when time $t$ is shorter than the correlation time ${\tau }_c$, $C_Q\left(t\right)$ remains flat and changes to $t^{-1∕2}$ power law at $t∕{\tau }_c>1$, and breaks down much later to decay single exponentially with a rate dictated by the chain length $N$.

Figure 4

(Color online) (a) ACF of fluorescence-lifetime fluctuations, $C_2\left(t\right)$, (solid curve), as compared to the room-temperature experimental data of Fre protein with a donor Tyr and an acceptor FAD. (b) Distance-fluctuation ACF ${\beta }_B^2{C}_Q\left(t\right)$ (solid curve) based on the distribution Rouse model of Eq. (5). The dash curve represents the Mittag-Leffler function with $H=3∕4$ from the fractional Brownian noise (FBN) model. The time unit is s.

Figure 5

(Color online) Distance-fluctuation ACF ${\beta }_B^2{C}_Q\left(t\right)$ (solid curve) from Eq. (5) and the FBN model vs the experimental data of FL and anti-FL complex with a donor Tyr and an acceptor FL. The time scale for $t$ is second.

Figure 6

(Color online) Distance-fluctuation ACF $C_Q\left(t\right)∕C_Q\left(0\right)$ versus a number of pairs of cross links among beads other than the nearest neighbor couplings. A chain of $250\text{units}$ is considered with an ensemble average over randomly chosen cross links. The presence of cross links could severely damage the simple $t^{-1∕2}$ power-law behavior of an ideal Rouse chain.

Figure 7

(Color online) (a) Energy-fluctuation ACF ${\beta }_B^2{C}_Q\left(t\right)$ of fluorescence-lifetime fluctuations data of Fre protein with a Tyr-FAD pair. (b) Energy-fluctuation ACF ${\beta }_B^2{C}_Q\left(t\right)$ for FL and anti-FL complexes with Tyr-FL pair. Data were fitted by Eq. (10) of the energy-fluctuation model using the spectral density function of Eq. (16), i.e., a special case with $\alpha \sim \frac{1}{2}$ and $\xi =-\frac{1}{2}$ for Eq. (13). It is found that $J_0\sim 8\mathrm{meV}$ for both fits and ${\Omega }_c\sim 16{\mathrm{s}}^{-1}$ for Fig. 7a and ${\Omega }_c\sim 1{\mathrm{s}}^{-1}$ for Fig. 7b. The unit for $t$ is s.