Effect of supercoiling on formation of protein-mediated DNA loops

P. K. Purohit and P. C. Nelson

Phys. Rev. E 74, 061907 – Published 21 December 2006; Erratum Phys. Rev. E 75, 039903 (2007)

Abstract

DNA loop formation is one of several mechanisms used by organisms to regulate genes. The free energy of forming a loop is an important factor in determining whether the associated gene is switched on or off. In this paper we use an elastic rod model of DNA to determine the free energy of forming short (50–100 basepair), protein mediated DNA loops. Superhelical stress in the DNA of living cells is a critical factor determining the energetics of loop formation, and we explicitly account for it in our calculations. The repressor protein itself is regarded as a rigid coupler; its geometry enters the problem through the boundary conditions it applies on the DNA. We show that a theory with these ingredients is sufficient to explain certain features observed in modulation of in vivo gene activity as a function of the distance between operator sites for the lac repressor. We also use our theory to make quantitative predictions for the dependence of looping on superhelical stress, which may be testable both in vivo and in single-molecule experiments such as the tethered particle assay and the magnetic bead assay.

Received 4 August 2006

DOI:https://doi.org/10.1103/PhysRevE.74.061907

Erratum

Erratum: Effect of supercoiling on formation of protein-mediated DNA loops [Phys. Rev. E 74, 061907 (2006)]

P. K. Purohit and P. C. Nelson

Phys. Rev. E 75, 039903 (2007)

Authors & Affiliations

P. K. Purohit^*

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

P. C. Nelson^†

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*Electronic address: purohit@seas.upenn.edu
^†Electronic address: nelson@physics.upenn.edu

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Vol. 74, Iss. 6 — December 2006

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Figure 1
(Color online) (a) Repression of a gene product controlled by the lac repressor in E. coli cells. The data are from Ref. 8, Fig. 3(a); see that paper for an explanation of the units on the vertical axis. The horizontal axis gives the distance along the DNA between the centers of the two operators, each 21 basepairs long. In this paper we will express operator spacing by a number $L$ that equals this number minus 21 bp (Fig. 2). Other experiments have obtained similar curves using operators located on a plasmid [9]. (b) Looping free energy inferred from the data in (a), showing a roughly periodic modulation with operator spacing (from Ref. 10, Fig. 3). The maxima of this function correspond to poor looping efficiency, that is, to the minima in panel (a). There is a slight minimum in the lower envelope of this function around 70–80 bp, corresponding to our $L \approx 50 - 60$ bp. A similar function emerges from the more detailed analysis of Garcia et al. [11].Reuse & Permissions
Figure 2
(a) Crystallographically derived structure of the lac repressor (LacI, dark gray) bound to two operator segments of DNA (black) [14]. The light gray curve represents a DNA conformation interpolating between the operator segments, obtained in Ref. 15. (b) Cartoon of the class of loops we will study. The DNA is considered free in the region between the two exit points $s_{i}$ and $s_{f}$ . These exit points are located at $\pm 10.5$ bp from the operator centers. Within the binding sites themselves, the protein may induce kinks in the DNA, as shown. (c) Definition of the initial and final tangent vectors ${\hat{t}}_{i}$ , ${\hat{t}}_{f}$ , the separation vector $a$ , and the angle $θ_{a}$ characterizing our idealized DNA-protein complex. $a$ is the vector joining the two exit points, located at arclengths $s_{i}$ and $s_{f}$ . The arclength separation between exit points is $L = s_{f} - s_{i}$ . The vectors ${\hat{t}}_{i}$ , ${\hat{t}}_{f}$ , and $a$ are all assumed to be be coplanar, and the angle $θ_{a}$ from $a$ to $- {\hat{t}}_{f}$ is equal to that from $- a$ to ${\hat{t}}_{i}$ (the “planar, symmetric coupler” idealization). In the example shown, $θ_{a} > 90 °$ . Although the coupler is planar, the loop itself will not in general be so, as illustrated here. (d) The ⋆-loop configuration corresponding to (c) (see text Sec. 3C). This loop is always planar.Reuse & Permissions
Figure 3
(Color online) (a) Equilibria between the unlooped state $U$ and various looped states $n$ , $n^{'}$ . We have omitted the regulatory proteins, indicating the operator sites by tick marks. We wish to treat the system as a small subsystem of interest (inside dashed line), thermodynamically coupled to a large reservoir (outside). Each looped state differs from the others by a $2 π$ rotation of one strand of the DNA about its axis at its binding site. Thus, although the total linking number is the same for every state shown, nevertheless the set of looped states divides into classes labeled by an integer. The two looped states shown have the same elastic energy inside the dashed lines, but quite different total free energy changes because of the torsional stress arising from supercoiling outside the dashed lines. (b) One way to distinguish topological classes of looped states is to choose a standard reference arc $C$ that completes each of the looped configurations, then compute the linking numbers of the resulting closed loops.Reuse & Permissions
Figure 4
(Color online) The geometry of our 2D exercise. The contour length of the DNA in the loop is $L$ . The size of the protein complex is represented by $a$ ; its geometry is summarized by the parameter $θ_{a}$ . $a$ and $θ_{a}$ determine the boundary conditions for the boundary value problem for the geometry of the loop. The element at arclength $s$ from the center exerts a force $F$ on the element at $s + d s$ . Due to the assumed symmetry, $F$ points along the negative $z$ axis as shown. $F$ is also the sideways force exerted on the ends of the rod by the protein complex.Reuse & Permissions
Figure 5
(Color online) (a) The elastic energy of DNA loops as calculated from an elastic rod model of DNA. We have assumed $θ_{a} = 71 °$ and $a = 4.0 nm$ with $ξ_{p} = 50 nm$ , $ξ_{t} = 18 nm$ , and $M_{ext} = - 1.0 k_{B} T$ . The graph shows the quantity defined in Eq. (6), plus an arbitrary linear function of $L$ , because such a function was dropped in our derivation of Eq. (6). The exchange of stability between various topoisomers at the peaks of the modulations (40,52,63,74 bp) has been emphasized by plotting the energy of the two competing topoisomers with different symbols (circles and stars). The continuous line connects the lowest energy topoisomers at each value of the length $L$ of the loop. (b) The black curve plots the elastic energy of a planar loop without any twist, as calculated using Eq. (19). This curve would be appropriate for looping with nicked DNA.Reuse & Permissions
Figure 6
(Color online) Predicting the effect of changing the sign and magnitude of $M_{ext}$ . Again we took $θ_{a} = 71 °$ and $a = 4.0 nm$ with $ξ_{b} = 50 nm$ , $ξ_{t w} = 18 nm$ and plotted the energy for three values of $M_{ext}$ . All three panels have the same, arbitrary, linear function of $L$ added to the looping free energy change. The sign of the assumed torque $M_{ext}$ in panel (c) is the opposite of that in panel (a). The minima and maxima in panel (c) are shifted by about 4 bp as compared to those in panel (a).Reuse & Permissions

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