Formation of loops in DNA under tension

We study the formation of loops along a DNA molecule under applied tension, as might occur in single-DNA micromanipulation experiments with proteins which are able to simultaneously bind two DNA sites. We consider the case of “bare” DNA in the loop, which forms a “teardrop” shape, and the case where a single DNA-bending protein produces a “kink” in the middle of the loop; the presence of a right-angle kink in the loop reduces its bending energy by a factor of 3. Using the bending energy plus an estimate of the free energies associated with fluctuations and the elasticity of the extended nonlooped DNA, we obtain a probability distribution for loops as a function of loop size and force. Force strongly suppresses formation of all loops, but suppresses large loops more severely than small ones. This quenching effect of force is reduced in the presence of a kink in the loop. We also calculate the speed at which length is absorbed into loops between arbitrary positions along the DNA (i.e., for non-sequence-specific loop forming proteins). The speed of retraction of the molecule decays as a stretched exponential function of the force with characteristic force scales depending on the geometry of the loops.

Figure 1

Specific and nonspecific loops. (a) DNA chain without any loops. (b) A specific loop of length $l$ formed between the two sequences marked in bold. (c) Nonspecific loops that are formed between any two points along the DNA chain.

Figure 2

(a) The teardrop loop; angle at the apex of the loop is $\pi$. (b) The kinked loop; angle at the apex of the loop is $\beta$.

Figure 3

(a) The teardrop loop, $\beta =\pi \mathrm{rad}$, and (b) the kinked loop, $\beta =2.3562\mathrm{rad}$. These are the exact shapes of the loops.

Figure 4

Zero-force probability distribution for the length of the loop for the circular loop, the teardrop loop, and the kinked loop. Note that the most probable loop size shifts to shorter distances along the loop and the peak height increases from (a) to (c).

Figure 5

Notation for the lengths in the scaled coordinates. Note that all quantities are specified for one half of the loop. Also shown is the “reaction distance” $\delta$, which is the distance with in which the ends of the loop must approach each other in a segment volume to form a loop. The “opening angle” of the loop is $2\left(90°-\alpha \right)$. The ends of the polymer are pulled in through a distance of $ϵ$ by the force.

Figure 6

Teardrop loop: Probability distribution for the length of the tear-drop loop as a function of the number of base pairs along the chain for different forces [24].

Figure 7

Kinked loop: Probability distribution for the length of the kinked loop as a function of the number of base pairs along the chain for forces between 0.01 and $1\mathrm{pN}$.

Figure 8

Most probable loop size for (a) the teardrop loop and (b) the kinked loop as a function of the applied tension in piconewtons.

Figure 9

Logarithm of the probability at the peak as a function of the applied tension in piconewtons: (a) teardrop loop and (b) kinked loop. The interpolation between the low and high force regimes is plotted in dotted lines.

Figure 10

Plot of the logarithm of the rate at which length is absorbed into nonspecific loops. The $Y$ axis is ${\log }_{10}\left[{\int }_0^{L}dllp\left(l\right)\right]$; the integral is proportional to the rate at which length is retracted during the formation of the first few loops along a long DNA molecule. Shown as an inset is a sketch of the process by which the DNA molecule is folded into nonspecific loops of different sizes when it is being stretched by force. The quantity $v$ is the integral shown above and $f$ is the applied tension.

Figure 11

Probability of forming a teardrop loop of a specific length as a function of the applied tension in piconewtons. The curves cross each other around $0.3\mathrm{pN}$, above which the smallest loop is the most probable. The interpolation between the low and high force regimes is plotted in dotted lines.

Figure 12

Probability of forming a kinked loop of a specific length as a function of the applied tension in piconewtons. The interpolation between the low and high force regimes is plotted in dotted lines.

Figure 13

Trajectory of the teardrop loop in the presence of applied tension in the asymptotic limit.

Figure 14

Trajectory of the kinked loop in the presence of applied tension in the asymptotic limit. Note the large amount of length associated with the bend to meet the loop.