Spatial effects on the speed and reliability of protein-DNA search

Abstract

Strong experimental and theoretical evidence shows that transcription factors (TFs) and other specific DNA-binding proteins find their sites using a two-mode search: alternating between three-dimensional (3D) diffusion through the cell and one-dimensional (1D) sliding along the DNA. We show that, due to the 1D component of the search process, the search time of a TF can depend on the initial position of the TF. We formalize this effect by discriminating between two types of searches: global and local. Using analytical calculations and simulations, we estimate how close a TF and binding site need to be to make a local search likely. We then use our model to interpret the wide range of experimental measurements of this parameter. We also show that local and global searches differ significantly in average search time and the variability of search time. These results lead to a number of biological implications, including suggestions of how prokaryotes achieve rapid gene regulation and the relationship between the search mechanism and noise in gene expression. Lastly, we propose a number of experiments to verify the existence and quantify the extent of spatial effects on the TF search process in prokaryotes.

Figure 1.

(A) We defined two types of searches: local searches in which the TF finds its binding site quickly using only hops and slides, and global searches in which the TF finds its binding site using hops, jumps and slides. In this illustration, the black oval is the TF, the gray line is the DNA and the cyan rectangle is the binding site. (B) In our model, we consider three types of movements that a TF can make with respect to DNA. Slides are rounds of 1D diffusion where the TF remains in constant contact with the DNA for a length of s bp. Hops and jumps are both types of 3D diffusion. Hops are short, and the dissociation and association sites on the DNA are close (linearly) and correlated. Jumps are long, and the dissociation and association sites may be quite distant along the DNA, though close in 3D space. (C) During a search, the TF alternates between 3D and 1D movements until it finds its site. At the end of a slide, the TF dissociates from the DNA, with probability p_hop takes a hop and associates to the same strand of DNA, and with probability p_jump = 1 – p_hop jumps to a new strand of DNA.

Figure 2.

(A) DNA exists in a compacted form in vivo, as illustrated on the top. To model the relative frequency and properties of hops and jumps, we looked at a 2D cross section of the DNA, imagining the DNA strands to be approximately straight rods on the short length scales we are dealing with. We defined hops as excursions that begin and end on the same strand of DNA in the cross section, shown with the dotted line, and jumps as excursions that begin and end on different DNA strands, shown with the dashed line. (B) Using a lattice model of the cross section, we calculated the probability of hops versus jumps from simulation, using 10⁶ runs. In E. coli, the approximate number of DNA strands in the lattice is 1500, which leads to p_hop = 0.83, but p_hop is relatively robust to changes in the DNA density. (C) Using the results of the lattice simulation, we calculated the distribution of the displacement along the DNA strand that takes place during each hop.

Figure 3.

(A) The probability of a local search depends on the effective sliding length, s_e, of the TF and the initial distance between the TF and its binding site d. Here we show the relationship for several values of = 10⁻⁶, 10⁻⁵, 10⁻⁴ and 10⁻³ M corresponding to s_e = 70 (circles), 210 (squares), 660 (diamonds), 2100 (triangles) bp, respectively. The solid line represents the analytical result and the markers represent the simulated result (n_trials = 1000/condition). (B) The average search time t_s depends on several parameters—here we plot it as a function of d for several values of the copy number n = 5, 10 and 20 copies/cell; = 10^–5 M. As n increases, the probability of a local search increases and the global search time (the plateau) decreases. For small n, the difference in t_s for small and large d is particularly striking. We simulated 5000 runs at each distance d. (C) The reliability of the search also depends on d. Here we plot the distribution of t_s for d = 50, 200 and 2000 bp for a single TF. In the box and whisker plots, the box has lines at the lower quartile, median and upper quartile values. The whiskers extend from the box to 1.5 times the interquartile range, the difference between the lower and upper quartiles. Data points beyond the whiskers were excluded.

Figure 4.

The global search time for a single TF depends non-monotically on its affinity for non-specific DNA, measured by the dissociation constant, . The search time is minimized when is equal to the concentration of non-specific DNA, [D] = 10⁻² M. However, the estimated range of is 10⁻⁶ to 10⁻³ M. See Supplementary Data, Section 1.4.1, for details.