ANRV411-BB39-18 ARI 2 April 2010 11:20 Biophysics of Knotting Dario Meluzzi,1 Douglas E. Smith,2 and Gaurav Arya1 1Department of Nanoengineering and 2Department of Physics, University of California at San Diego, La Jolla, California 92093; email: [email protected], [email protected] Annu. Rev. Biophys. 2010. 39:349–66 by University of California - San Diego on 05/31/10. For personal use only. Key Words First published online as a Review in Advance on polymer physics, entanglement, DNA, proteins, topoisomerase Annu. Rev. Biophys. 2010.39:349-366. Downloaded from arjournals.annualreviews.org February 16, 2010 The Annual Review of Biophysics is online at Abstract biophys.annualreviews.org Knots appear in a wide variety of biophysical systems, ranging from This article’s doi: biopolymers, such as DNA and proteins, to macroscopic objects, such as 10.1146/annurev.biophys.093008.131412 umbilical cords and catheters. Although significant advancements have Copyright c 2010 by Annual Reviews. been made in the mathematical theory of knots and some progress has All rights reserved been made in the statistical mechanics of knots in idealized chains, the 1936-122X/10/0609-0349$20.00 mechanisms and dynamics of knotting in biophysical systems remain far from fully understood. We report on recent progress in the biophysics of knotting—the formation, characterization, and dynamics of knots in various biophysical contexts. 349 ANRV411-BB39-18 ARI 2 April 2010 11:20 variety of topics related to the biophysics of Contents knotting. The organization of these topics re- flects our attempt to address the following gen- INTRODUCTION .................. 350 eral questions: Where and how do knots form? TYPES OF KNOTS.................. 350 How likely are knots to form? What are some KNOTTING IN BIOPHYSICAL properties of knots and knotted systems? In SYSTEMS ........................ 352 what processes do knots play a role? When and PROBABILITIES OF how do knots disappear? In addressing these KNOTTING ..................... 354 questions, we aim for a qualitative presentation FEATURES OF KNOTTED of recent works, emphasizing the diversity of SYSTEMS ........................ 356 methods and results without delving extensively Size of Knots and Knotted into technical details. A more comprehensive Systems ........................ 356 treatment of specific topics can be found in Knot Localization.................. 357 books (1, 19) and in the various cited reviews. Strength and Stability of Knotted Systems ............. 357 DYNAMIC PROCESSES TYPES OF KNOTS INVOLVING KNOTS ............ 358 The ability to discern and classify different Knot Diffusion .................... 358 kinds of knots is an essential requirement for Electrophoresis .................... 359 understanding biophysical processes involving Unknotting ........................ 359 knots. The mathematical field of Knot theory CONCLUSION ..................... 361 offers powerful tools for detecting and classify- ing different knots (1). A knot is a topological state of a closed, nonintersecting curve. Two closed curves contain knots of the same type INTRODUCTION if one of the curves can be deformed in space Knots are fascinating topological objects that to match the other curve without temporarily have captured human imagination for centuries. opening either curve. In practice, a 3D knot- They find a plethora of useful applications, ted curve is mathematically analyzed by first from tying shoelaces to securing surgical su- projecting it onto a 2D plane and then exam- tures. But knots can also be a nuisance, crop- ining the points, known as crossings, where ping up in long hair, electrical cords, and other the curve crosses itself in the 2D projection inconvenient places. Equally important, knots (Figure 1a). Note that when we talk about are interesting subjects for scientific inquiry and knots in open curves, such as a linear string or by University of California - San Diego on 05/31/10. For personal use only. have attracted increasing attention from physi- DNA molecule, we are imagining that the ends cists and biophysicists: Various physically rel- of those curves are connected using a sensible, Annu. Rev. Biophys. 2010.39:349-366. Downloaded from arjournals.annualreviews.org evant systems have an undeniable capacity to well-defined procedure to yield corresponding become entangled. Notable examples include closed curves (Figure 1b). biopolymers such as DNA and proteins. An The absence of a knot is called the unknot understanding of these knots beyond the con- or trivial knot. It can always be rearranged to fines of mathematical topology and theoretical yield a projection with zero crossings. Knots, in physics is essential to bring about new discov- contrast, give rise to projections with nonzero eries and practical applications in biology and numbers of crossings. The minimum number Knot: a topological nanotechnology. of crossings, C, is an invariant for any arrange- state of a closed 3D Here we describe some recent experimen- ment of a closed string with a given knot. C curve, also a knot-like tal and theoretical efforts in the biophysics is often used to classify knots into different conformation of an open chain of knotting. We begin with a brief introduc- types. Specifically, each knot type is denoted tion to knot classification. We then explore a as CS, where S is a sequence number within 350 Meluzzi · Smith · Arya ANRV411-BB39-18 ARI 2 April 2010 11:20 a b Projection Open curve Closed curve c d 31 q–1 + q–3 – q–4 Trefoil 4 2 –1 –2 1 q – q + 1 –q + q Figure-eight SK SP 51 q–2 + q–4 – q–5 + q–6 – q–7 S 52 q–1 – q–2 + 2q–3 – q–4 + q–5 – q–6 K SP e f g Square Granny Slipknot D knot knot SK S K Figure 1 Knot types and features. (a) Knots formally exist only in 3D curves (left). Knot projections are 2D representations of knots (right). (b) Knot-like conformations in open curves are often encountered in biophysics (left). To analyze such knots, their loose ends must be connected, according to some procedure, to obtain a closed curve (right). (c) Projections of the four simplest nontrivial knot types, with the corresponding CS denominations and Jones polynomials (see text for definition of CS) (adapted from http://katlas.math. toronto.edu/wiki/Main Page). (d ) The size of a knot, SK , in a polymer may be less than the size of the polymer, SP, containing the knot. (e) In a slip link arrangement, entropic competition between the knotted loops causes the ring to squeeze one of the knots. The size of the latter can be deduced from the position of by University of California - San Diego on 05/31/10. For personal use only. the ring. Adapted from Reference 64. ( f ) The size of a tight knot can be estimated from the volume of the ∼ 2 1/3 enclosing ideal knot representation: SK (D L) , where D and L are the diameter and length of the outer Annu. Rev. Biophys. 2010.39:349-366. Downloaded from arjournals.annualreviews.org tube. Adapted from Reference 39. ( g) Square and granny knots can tie ropes together but unravel easily at the molecular scale. Slipknots in proteins have been studied to assess the effects of knots on stability. the family of knot types having the same C Simple knots can be distinguished visually (Figure 1c). Some common knots are also re- by comparison with published tables, but ex- ferred to by name: 31 and 41 are called trefoil tensively knotted systems require mathematical and figure-eight, respectively. The number of methods of knot classification. One ingenious different knot types having the same C increases strategy for classifying knots is to transform rapidly with C: There are only 3 knots with 6 a knot projection into a special polynomial crossings, but 1,388,705 knots with 16 cross- formula, which depends on the knot type but ings (42). The number C serves as a measure of not on any particular projection. Comparing knot complexity. this polynomial with those enumerated in www.annualreviews.org • Biophysics of Knotting 351 ANRV411-BB39-18 ARI 2 April 2010 11:20 knot tables enables the identification of the chain segments, is facilitated by enzymes such as knot type from a given projection. Examples topoisomerases and recombinases. Fundamen- of such polynomials include the Alexander, tal insights into the mechanisms of these and Jones polynomial: a mathematical Jones, and HOMFLY polynomials (1). V.F.R. other enzymes have resulted from detailed anal- expression that can be Jones was awarded the famous Fields Medal in yses of knots in DNA (13, 59, 98, 115). computed by analyzing mathematics in 1990 for his groundbreaking DNA topoisomerases are classified as type I the crossings in discovery of the Jones polynomial. These or type II (93). Type I DNA topoisomerases any particular 2D pro- polynomials occasionally fail to distinguish temporarily break a single DNA strand and jection of a knot, and serves as fingerprint different knots and become computationally allow it to pass through the complementary to uniquely identify prohibitive with projections of many crossings, strand (7). Knotted dsDNA results when cir- the type of the knot but they are invaluable tools for analyzing the cular dsDNA is nicked or gapped and the en- dsDNA: vast majority of simpler knots. zyme breaks a strand at a location opposite double-stranded DNA the nick (23). In contrast, type II topoisom- DNA erases temporarily break both strands in one topoisomerases: KNOTTING IN BIOPHYSICAL segment of dsDNA, allowing one segment to enzymes that allow SYSTEMS pass through another intact segment before the single or double Knots can form via two general mechanisms: strands are chemically rejoined (72, 93). Type strands of DNA to pass through other single threading of loose ends or breaking and re- II DNA topoisomerases introduce knots into or double strands of joining of segments. Linear double-stranded supercoiled circular DNA in vitro (114), pro- DNA to change the DNA (dsDNA) molecules undergoing random viding a way to assess the DNA supercoiling topology of a closed cyclization in solution exemplify the first mech- activity of other enzymes, such as condensins dsDNA molecule anism.