Dynamics of Small Loops in DNA Molecules Alexei A. Podtelezhnikov and Alexander V. Vologodskii* Department of Chemistry, New York University, New York, New York 10003 Received October 25, 1999; Revised Manuscript Received January 24, 2000 ABSTRACT: The kinetics and thermodynamics of loop formation by short segments of double-stranded DNA was studied by computer simulation. The DNA molecule was modeled as a discrete wormlike chain. Brownian dynamics was used to simulate the dynamic properties of the chain. Since the average time of loop formation, τ a, grows sharply when the loop size drops below DNA persistence length, we were unable to simulate the process directly for such small loops. Instead, we used the relationship between the equilibrium probability of loop formation, P, τa, and the average time of loop decay, τd. The values of P and τd were simulated directly. A new Monte Carlo algorithm was developed allowing efficient calculation of P for small DNA loops. The algorithm is also applicable to more complex models of a polymer chain, particularly to DNA models with intrinsic curvature. We also considered loop formation by a segment of a DNA molecule and found that the values of τd and τa are weakly affected by the total chain size. Our results showed that the formation of small loops is a very slow process: for loops less than 50 nm in size τa can be comparable to the lifetime of the cell. I. Introduction The formation of loops by DNA molecules is an important step in many key biological processes. 1,2 The thermodynamics of the process have been carefully studied both experimentally and theoretically. Shimada and Yamakawa obtained accurate theoretical results for the probability of loop formation by a wormlike chain, usually used to describe DNA conformational proper- ties. 3 These results are in agreement with the experi- mental data of Baldwin and co-workers. 4,5 Because it is more difficult to obtain theoretical results for intrinsi- cally curved DNA molecules (however, see ref 6), computational approaches based on Monte Carlo simu- lations were developed. 7,8 On the other hand, the kinetics of loop formation has not been studied extensively. It is difficult to investigate the process experimentally, since the rate of cyclization of DNA molecules by joining their cohesive ends is not limited by diffusion. Thus, the cyclization kinetics gives no information about the kinetics of loop formation. Another experimental approach to the problem based on competition between two reactions of site-specific recombination was used recently. 9,10 It seems, however, that in this system the enzymatic kinetics rather than the kinetics of juxtaposition of the specific sites is the rate-limiting stage of the reactions. 10,11 Theoretical approaches gave important results for the rate of loop formation in long polymer chains, 12 but they cannot be applied to small loops since they are based on Gaussian statistics of the chain conformations. In this situation, Brownian dynamics (BD) simulations 13 provide a unique way to study the question. During the past decade, it has been shown that the BD approach is capable of describing the actual times of large-scale DNA motion. 14-17 The method was used recently to study the kinetics of DNA looping for linear DNA molecules 18 and for segment juxtaposition in supercoiled circular DNA. 19 The power of modern computers does not allow, how- ever, the simulation of slow processes in DNA dynamics, like the formation of very small or very large DNA loops. Here we developed a special approach to estimate the rate of loop formation for such cases. We found that the formation of small DNA loops, 50-100 base pairs in length, is an extremely slow process with a character- istic time that can exceed the lifetime of the cell. II. Methods of Computations A. DNA Model. Our DNA model combines the features of the wormlike chain and bead-spring mod- els. 14,20 A DNA molecule of the length L was modeled as a chain consisting of N straight segments of equal equilibrium length l 0 . The bending energy, E b , was specified by the angular displacements θ i of segment i + 1 relative to segment i: where R is the gas constant and T is the absolute temperature. The bending rigidity constant R is defined so that k straight segments of the model chain cor- respond to one Kuhn statistical length of DNA. The exact relationship between k and R is described in ref 20; for k . 1 there is an approximate equation R= k/4. 21 The replacement of the continuous wormlike chain with a semiflexible chain consisting of hinged rigid segments is an approximation that improves as k increases. To mimic hydrodynamic properties of DNA, we at- tached virtual beads to each vertex of the model chain. We chose l 0 so that it equals to the diameter, d, of touching beads, d ) l 0 ) 3.18 nm. 22 Considering that DNA Kuhn statistical length is equal to 100 nm, 23 this choice of l 0 corresponds to k ) 31.45 (R) 7.775). This k guarantees good approximation of the continuous worm- like chain. 24 We used the Rotne-Prager approximation for the tensor of hydrodynamic interaction. 25 We also introduced the stretching elasticity to the model chain to facilitate the dynamic simulations. The stretching energy, E s , was computed as where l i is the length of the segment i. The stretching rigidity  was set equal to 50, and this allowed us to E b )RRT ∑ i)1 N-1 θ i 2 (1) E s ) RT l 0 2 ∑ i)1 N (l i - l 0 ) 2 (2) 2767 Macromolecules 2000, 33, 2767-2771 10.1021/ma991781v CCC: $19.00 © 2000 American Chemical Society Published on Web 03/15/2000