First passage time of $N$ excluded-volume particles on a line

Igor M. Sokolov, Ralf Metzler, Kiran Pant, and Mark C. Williams

Phys. Rev. E 72, 041102 – Published 17 October 2005

Abstract

Motivated by recent single-molecule studies of proteins sliding on a DNA molecule, we explore the targeting dynamics of $N$ particles (“proteins”) sliding diffusively along a line (“DNA”) in search of their target site (specific target sequence). At lower particle densities, one observes an expected reduction of the mean first passage time proportional to $N^{- 2}$ , with corrections at higher concentrations. We explicitly take adsorption and desorption effects, to and from the DNA, into account. For this general case, we also consider finite-size effects when the continuum approximation based on the number density of particles breaks down. Moreover, we address the first-passage-time problem of a tagged particle diffusing among other particles.

Received 12 June 2005

DOI:https://doi.org/10.1103/PhysRevE.72.041102

Authors & Affiliations

Igor M. Sokolov^*

Institut für Physik, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany

Ralf Metzler^†

NORDITA, Nordic Institute for Theoretical Physics, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark

Kiran Pant and Mark C. Williams^‡

Department of Physics, Northeastern University, 111 Dana Research Center, Boston, Massachusetts 02115, USA

^*Electronic address: igor.sokolov@physik.hu-berlin.de
^†Electronic address: metz@nordita.dk
^‡Electronic address: mark@neu.edu

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Vol. 72, Iss. 4 — October 2005

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Images

Figure 1
Mean first passage time $T (n_{0})$ in a one-sided system, as a function of the density $n_{0}$ of excluding walkers that cannot occupy the same lattice site. The target is placed at $x = 0$ . The maximum density is $n_{0} = 30 %$ , whereby each particle occupies one lattice site. The dashed line corresponds to the exact result $T (n_{0}) = π ∕ n_{0}^{2}$ from Eq. (17), with the dimensionless diffusion coefficient $D_{1 D} = 1 ∕ 2$ . The simulation data agree nicely with the dilute limit, with only a slight deviation for larger densities. Each data point corresponds to $10^{5}$ runs, except for $10^{3}$ realizations for the lowest density. Note the comparatively small error bars.Reuse & Permissions
Figure 2
Behavior of the mean first passage time $T (N)$ as a function of the number $N$ of TF’s attached to a DNA of length $1000$ according to Eq. (3), for the dilute case (solid line), TF size $λ = 1$ (long-dashed line), and $λ = 10$ (short-dashed line). Excluded-volume effects reduce the target search time $T (N)$ .Reuse & Permissions
Figure 3
Survival probability of the target site—i.e., the probability that no TF has reached the specific binding site—according to Eqs. (15, 21). This case corresponds to vanishing desorption, $k_{1} = 0$ . The plot parameters are indicated in the figure.Reuse & Permissions
Figure 4
Survival probability $S (t)$ from Eq. (18) for finite desorption rate $k_{1}$ . Note the logarithmic ordinate; the plot parameters are indicated in the figure. The incomplete decay in the case of vanishing adsorption, $k_{0} = 0$ , is distinct.Reuse & Permissions
Figure 5
Dimensional binding rate $k_{a}$ in 1/s as a function of protein concentration $C$ in nM, converted from Fig. 1 for parameters corresponding to $100 mM$ salt. The fitted 1D diffusion constant for sliding along the dsDNA is $D_{1 D} = 3.3 \times 10^{- 9} {cm}^{2} ∕ \sec$ , located nicely within the experimental value $10^{- 8} - 10^{- 9} {cm}^{2} ∕ \sec$ [9].Reuse & Permissions
Figure 6
Mean first passage time $T (L)$ of target search in a pronouncedly finite system of length $L$ , corresponding to the one-sided system with target site at $x = 0$ and reflecting boundary condition at $x = L$ . We chose the parameters $D_{1 D} = 1 ∕ 2$ for the dimensionless diffusion constant, the initial protein density $n_{0} = 10 %$ , the adsorption rate $k_{0} = 10^{- 4}$ , and vanishing desorption rate $k_{1} = 0$ . The curved dashed line corresponds to Eq. (30), which approximates small systems, while the horizontal dashed line at $T = 168$ is determined by numerical integration of $S (t)$ , Eq. (21). Each data point represents $10^{5}$ runs. Again, note the rather small error bars.Reuse & Permissions

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