Role of protein fluctuation correlations in electron transfer in photosynthetic complexes

We consider the dependence of the electron transfer in photosynthetic complexes on correlation properties of random fluctuations of the protein environment. The electron subsystem is modeled by a finite network of connected electron (exciton) sites. The fluctuations of the protein environment are modeled by random telegraph processes, which act either collectively (correlated) or independently (uncorrelated) on the electron sites. We derived an exact closed system of first-order linear differential equations with constant coefficients, for the average density matrix elements and for their first moments. Under some conditions, we obtained analytic expressions for the electron transfer rates and found the range of parameters for their applicability by comparing with the exact numerical simulations. We also compared the correlated and uncorrelated regimes and demonstrated numerically that the uncorrelated fluctuations of the protein environment can, under some conditions, either increase or decrease the electron transfer rates.

Figure 1

The two-level donor-acceptor system interacting with two uncorrelated noisy environments ${\xi }_1\left(t\right)$ and ${\xi }_2\left(t\right)$; ${\lambda }_n^{\left(a\right)}$ are the constants of interaction. The superscript $a=1,2$ indicates the noisy environment, and the subscript $n=1,2$ indicates the electron site.

Figure 2

Dependence of the rate $\Gamma$ in Eq. (26) on the noise amplitude $d$ and the correlation rate $\gamma$. Parameters: $V_{12}=5$, $\varepsilon =30$.

Figure 3

Asymptotic rate $\Gamma$ of Eq. (23) vs the dimensionless amplitudes of two uncorrelated noises ${\mu }_1$ and ${\mu }_2$. Parameters: $V_{12}=3$, $\varepsilon =30$. Top: ${\gamma }_1=5$, ${\gamma }_2=15$. Bottom: ${\gamma }_1={\gamma }_2=10$.

Figure 4

Strongly coupled dimer (${\mu }_d=1$). Time dependence (in picoseconds) of the density matrix components: ${\rho }_{11}\left(t\right)$ (blue) and ${\rho }_{22}\left(t\right)$ (red). Parameters: $V_{12}=30$, ${\varepsilon }_1=60,{\varepsilon }_2=30$, ${\gamma }_1=10$, ${\gamma }_2=15$. (a) $d_1^{\left(1\right)}=10$, $d_2^{\left(1\right)}=10$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=0$; (b) $d_1^{\left(1\right)}=10$, $d_2^{\left(1\right)}=0$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=10$. Initial conditions: ${\rho }_{11}\left(0\right)=1$, ${\rho }_{22}\left(0\right)=0$.

Figure 5

Time dependence (in picoseconds) of the density matrix components: ${\rho }_{11}\left(t\right)$ (blue) and ${\rho }_{22}\left(t\right)$ (red). Parameters: $V_{12}=10$, ${\varepsilon }_1=60,{\varepsilon }_2=30$, ${\gamma }_1=10$, ${\gamma }_2=15$. (a) $d_1^{\left(1\right)}=10$, $d_2^{\left(1\right)}=10$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=0$, (b) $d_1^{\left(1\right)}=10$, $d_2^{\left(1\right)}=0$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=10$. Initial conditions: ${\rho }_{11}\left(0\right)=1$, ${\rho }_{22}\left(0\right)={\rho }_{12}\left(0\right)=0$.

Figure 6

Weakly coupled dimer (${\mu }_d=0.1$). Time dependence (in picoseconds) of the density matrix components: ${\rho }_{11}\left(t\right)$ (blue) and ${\rho }_{22}\left(t\right)$ (red). Parameters: $V_{12}=3$, ${\varepsilon }_1=60,{\varepsilon }_2=30$, ${\gamma }_1=10$, ${\gamma }_2=15$. (a) $d_1^{\left(1\right)}=10$, $d_2^{\left(1\right)}=10$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=0$; (b) $d_1^{\left(1\right)}=10$, $d_2^{\left(1\right)}=0$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=10$. Initial conditions: ${\rho }_{11}\left(0\right)=1$, ${\rho }_{22}\left(0\right)={\rho }_{12}\left(0\right)=0$.

Figure 7

Weakly coupled dimer (${\mu }_d=0.1$). Time dependence (in picoseconds) of the density matrix components: ${\rho }_{11}\left(t\right)$ (blue and green curves) and ${\rho }_{22}\left(t\right)$ (red and orange curves). Choice of parameters: $V_{12}=3$, ${\varepsilon }_1=60,{\varepsilon }_2=30$, ${\gamma }_1=5$, ${\gamma }_2=15$. Blue and red curves: $d_1^{\left(1\right)}=20$, $d_2^{\left(1\right)}=-10$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=0$, $\Gamma =3.6$ $({\mu }_1=1.5;{\mu }_2=0)$. Green and orange curves: $d_1^{\left(1\right)}=20$, $d_2^{\left(1\right)}=0$, $d_1^{\left(2\right)}=0$, $d_2^{\left(2\right)}=-10$, $\Gamma =0.73$ $({\mu }_1=1;{\mu }_2=0.5)$. Solid curves correspond to the solutions of the exact Eqs. (7, 8, 9). Dashed curves correspond to the approximate solutions given by Eqs. (21) and (22). Initial conditions: ${\rho }_{11}\left(0\right)=1$, ${\rho }_{22}\left(0\right)={\rho }_{12}\left(0\right)=0$.