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

Zeba Wunderlich 2 Leonid A. Mirny 0 1 0 Department of Physics, Massachusetts Institute of Technology , Cambridge, MA 02139, USA 1 Harvard-MIT Division of Health Sciences and Technology 2 Biophysics Program, Harvard University , Cambridge, MA , 02138 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 threedimensional (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. - INTRODUCTION ProteinDNA interactions are vitally important for every cell. Transcription factors (TFs) are proteins that interact with specific DNA sequences to regulate gene expression. The targeting of TFs to their sites is a passive process; therefore, it seems natural to assume that TFs simply diffuse through the nucleus (in eukaryotes) or cell (in prokaryotes) until they find their sites. In the 1970s, this assumption was challenged by the observation that, in vitro, the prokaryotic TF LacI is able to find its binding site 100 times faster than expected by three-dimensional (3D) diffusion in the solvent (1). This led to the suggestion of a facilitated diffusion mechanism in which TFs alternate between 3D diffusion, jumping, through the volume of the cell and one-dimensional (1D) sliding along the DNA to rapidly locate their binding sites (24). This hypothesis was corroborated by several pieces of evidencemost strikingly several single molecule studies in which the authors visualized individual proteins sliding along DNA (57). Several groups have also mathematically modeled this process and shown it to be a plausible way of making the search significantly faster than 3D diffusion alone (3,4,811). Several aspects of facilitated diffusion, however, remain puzzling, e.g. the effect of the DNA sequence composition and conformational transitions in the protein on the rate of sliding (10,12) and role of the DNA conformation (11). Here we consider how spatial effects influence the search process. Specifically, we ask whether and how search time depends on the initial distance between the protein and the target site. The distance dependence of the TF search process has not been considered before because the rate of a bimolecular reaction in 3D is distance-independent (13). Therefore, the time it takes for a protein diffusing in 3D to find its target does not depend on the initial distance between the two, as long as this distance is greater than the size of the target. In contrast, the time of search in two dimensions (2D) (e.g. on a membrane) or in 1D (e.g. along DNA or along a filament) is distance-dependent (13). Therefore, we ask: can the 1D component of facilitated diffusion make search much faster for a protein that starts a small distance from its target site? Here we use simulations and analytical estimates to demonstrate that TF search time indeed depends on the initial position of the TF with respect to its binding site. We show that the trajectories can be naturally separated into fast local and slow global searches (Figure 1A). We find that if a TF starts sufficiently closeless than 1000 base pairs (bp) for our model organism Escherichia coli to its binding site, a local search is likely. While studying how spatial effects contribute to the search process, we observe that upon dissociation from Global Jump Transcription Factor DNA DNA, a protein is likely to quickly re-associate near is dissociation point, thus making a short-range hop, rather than a long-range jump (Figure 1B). We examine how these two types of spatial excursions influence the search process, allowing us to reconcile the widely ranging experimental measurements of the sliding length (6,7,14,15). Finally, we show that the strong non-specific binding of TFs to DNA makes global search rather slow, thus making local search appreciably faster. Moreover, local searches have significantly smaller variance in the search time, making them an attractive mechanism to deliver DNA-binding proteins to their targets quickly and reliably. There are a number of biological implications of these spatial effects. Since transcription and translation are coupled in bacteria, proteins are produced near the location of their genes. Therefore, TFs whose genes are co-localized with their binding sites are likely to use a local search mechanism. The efficiency of local search provides a physical justification for the observed co-localization of TF genes and their binding sites in prokaryotic genomes (1618). We also propose a number of experiments to test the mechanism and its predictions. MATERIALS AND METHODS Characterizing hops using simulations To include hops in the search model, we needed to estimate the relative frequency of hops and jumps and the displacement due to hops. Assuming that DNA could be treated as straight rods on the length scale of a hop, we considered the problem in a cylindrical geometry and simplified it further to a 2D geometry (Figure 2A). In the 2D cross-section, DNA strands are represented as absorbing circles. To simulate diffusion in 2D, we discretized the cross section into a 1 mm2 square lattice with 1 nm spacing and randomly distributed DNA strands, each with an absorbing radius of 2 nm. We simulated a TF trajectory as a random walk on the lattice, starting from its dissociation from DNA and ending with its association to DNA. Trajectories that started and ended on the same DNA strand were called hops; otherwise they were jumps (Figure 2A). From these trajectories, we calculated the DNA probability of a hop as a function of the number DNA strands in the lattice (Figure 2B). Using the length of the hop trajectories, we also calculated the displacement along the DNA strand during a hop for lattices with 1500 strands, the approximate density of DNA in E. coli. We assumed that, in the 3D geometry, two-thirds of the random walk steps were in the 2D plane and one-third were in the z-directionalong the DNA. Therefore, given the l (...truncated)