Diffusion in correlated random potentials, with applications to DNA

Many biological processes involve one-dimensional diffusion over a correlated inhomogeneous energy landscape with a correlation length ${\xi }_c$. Typical examples are specific protein target location on DNA, nucleosome repositioning, or DNA translocation through a nanopore, in all cases with ${\xi }_c\approx 10\text{nm}$. We investigate such transport processes by the mean first passage time (MFPT) formalism, and find diffusion times which exhibit strong sample to sample fluctuations. For a displacement $N$, the average MFPT is diffusive, while its standard deviation over the ensemble of energy profiles scales as $N^{3∕2}$ with a large prefactor. Fluctuations are thus dominant for displacements smaller than a characteristic $N_c⪢{\xi }_c$: typical values are much less than the mean, and governed by an anomalous diffusion rule. Potential biological consequences of such random walks, composed of rapid scans in the vicinity of favorable energy valleys and occasional jumps to further valleys, is discussed.

Figure 1

(a) Prokaryotic transcription factor sliding; (b) nucleosome repositioning.

Figure 2

(a) Energy of local elastic deformation and (b) potential profile correlator, as calculated from the data supplied by the server BEND.IT for a segment of $E.coli$ genome. The deformed DNA sequence is assumed to be of length $L=15\text{bp}$.

Figure 3

(Color online) ssDNA transport through the nanopore; on the right: charge density $q\left(x\right)$ and correlator $g\left(r\right)=⟨{\left[\delta U\left(x\right)-\delta U(x+r)\right]}^2⟩∕(2⟨\delta U^2\left(x\right)⟩)$ as a function of the coordinate $r$.

Figure 4

(Color online) Mean first passage times: typical versus average. Thick solid lines are the result of averaging over 1000 realizations of potential profiles $\left(\beta \sigma =1.0\right)$: (a) correlated profile with ${\xi }_c=40.0$; (b) uncorrelated profile.

Figure 5

MFPT standard deviation for $\beta \sigma =1.0$ for correlated and uncorrelated potential profiles.

Figure 6

Median versus disorder-averaged [solid lines calculated from Eq. (23)] MFPT. Median values were calculated for 1000 realizations of potential profiles: (a) Correlated potential profile with ${\xi }_c=20.0$; (b) uncorrelated potential profile.

Figure 7

Typical MFPT for $N⪡N_c$ at various values of $\beta \sigma$: (a) Uncorrelated potential profile; (b) correlated potential profile with ${\xi }_c=10$. Solid lines are the analytical estimates from Eqs. (31, 32).

Figure 8

Probability density functions for MFPT calculated for $\mathrm{100\hspace{0.17em}000}$ uncorrelated profile realizations at $\beta \sigma =2$.

Figure 9

Random walk in (a) uncorrelated, and (b) correlated with ${\xi }_c=20.0$, potential energy profiles.

Figure 10

Pseudoenergy probability density for a profile of length $L=\mathrm{10\hspace{0.17em}000}$, with $\sigma =1.0$, ${\xi }_c=20.0$. Insets: (a) Typical potential profile; (b) potential profile correllator $g\left(r\right)=1∕2⟨{\left[U\left(x\right)-U(x+r)\right]}^2⟩$; the averaging was performed over 1000 profile realizations.