The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon

C. K. Au1

Abstract: Ribbons may be used for the model- (2006)]. Bending due to applied forces in straight ing of DNAs and proteins. The topology of a rib- carbon nanotube can be easily happen while in- bon can be described by the linking number, while trinsically coiled tubes have been observed as its geometry is represented by the writhe and the well. A curve has only length; it has no width twist. These quantities are integrals and are re- or thickness. If endowed with a width in a cer- lated by the Cˇalugˇareanu’s theorem from knot the- tain direction, a curve becomes a ribbon.Ifthe ory. This theorem also describes the relationship width is uniform in all directions, then a tube between the various conformations. The heart of results. Ribbons serve as a convenient way to the Cˇalugˇareanu’s theorem rests in the Gauss In- model macromolecules such as DNA and pro- tegral. Due to the large number of molecules, the tein [Carson and Bugg (1986), (1987), (2000)]. topology and the geometry of a ribbon model can The functions of these macromolecules are de- be very complicated. As a result, these integrals termined by their form, or geometry, while their are commonly evaluated by numerical methods. inter-connectedness, or topology, is important to The writhe of a ribbon is usually computed by many biological processes [Cozarelli, Boles and mapping its self-crossing onto a unit sphere. The White (1990), Fuller (1971)] such as protein fold- twist of a ribbon is mainly due to the geometric re- ing, replication and transcription. In the parlance lationship between its central spine and boundary. of biology, the chemical constituents, in a linear This article offers a geometric interpretation of the string of symbols, are called the “primary” repre- twist of a ribbon in terms of crossing between the sentation. The spatial arrangement of balls-and- central spine and boundary of the ribbon. This ap- sticks, as in the celebrated double-helix model for proach facilitates the calculation of the twist of a the DNA, is called the “secondary” representa- ribbon. Some basics in visibility will enhance the tion. Upon a closer examination, the balls at the intuition. two ends of a stick follow certain geometric mo- tifs. Keyword: Linking number, writhe, twist, Thus, the two helical clusters of amino acids in Gauss Integral, Cˇalugˇareanu’s theorem. the DNA, for example, can be abstracted as two intertwining ribbons. 1 Introduction Curvature and torsion characterize how a curve L Coiling and twisting are common physical phe- changes with respect to the Frenet-Serret frame on nomena of ﬁlament deformation. It can be found the curve itself. But suppose the frame of refer- in example from everyday life such as ribbon and ence is on another curve K. Then, two new quan- telephone cords as well as the nano-scale DNA tities, writhe and twist, arise. In a closed end rib- [Steffen, Murphy, Lathrop, Opel, Tolleri and Hat- bon, let there be a central “spine” K. Also, let one ﬁeld (2002)] and protein molecules [Karcher, Lee, of the two boundary curves of the ribbon be L. Kaazempur-Mofrad and Kamm (2006)], and car- Then, the “tangling” between curve K and curve bon nanotube [Park, Cho, Kim, Jun and Im L gives an indication of how complex the ribbon is, spatially. Intuitively, as writhe and twist are 1 School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore. Email: analogous to curvature and torsion, they describe [email protected] the geometry of a ribbon. Yet, because curve K 152 Copyright c 2008 Tech Science Press CMES, vol.29, no.3, pp.151-162, 2008

“threads” curve L,orviceversa,theremustbe end space curves K and L is given by another parameter that characterizes the “knotti- 1 (dp×dq) · pq ness” of the tangle – called linking number,a Lk(K,L)= (1) π × 3 topological quantity. Indeed, the Cˇalugˇareanu’s 4 K L |pq| theorem [Cˇalugˇareanu’s (1959)] gives the connec- ( , ) tion. Similarly, the writhe Wr K K [Dennis and Han- nay (2005)] of a closed end curve K is The topology of two separate curves (of zero width and thickness) is easier to visualize than 1 (dp×dq) · pq Wr(K,K)= (2) that of a single ribbon (with one curve as its 4π K×K |pq|3 “spine” and another as its boundary). Consider the two interlocking closed end curves K and L, (dp×dq)·pq where the expression × 3 is termed ( , ) K L |pq| the linking number Lk K L remains the same Gauss Integral. even more geometry is added to the two curves – by giving each more writhe and twist. For two 2.1 Crossing number as complex measure open end curves, their linking number would necessarily be Lk(K,L) = 0, as one can “slip” out of The amount of entanglement of two space curves the entanglement by the other. Similarly, for two can be characterized by the number of crossings. curves, one closed and the other open, the linking The intuition is clear: the more crossings there number Lk(K,L)= 0; they are not linked. are, the more entangled the two curves must be in space. Yet, the number of crossings depends Fundamentally, the two quantities, linking num- on the direction of the projection. In other words, ber Lk(K,L) and writhe Wr(K,K) are rooted in over all possible orientations, there must be a di- the evaluation of the so-called Gauss Integral. rection which gives the maximal number of cross- Due to the large number of molecules, the topol- ings; corresponding, there must be one that yields ogy and the geometry of a ribbon model can be the minimal. A weighted average with crossing very complicated. As a result, difﬁculties arise number as the weight over all angles can be cal- in evaluating the integral. Several approaches for culated. computing the writhe of a polygonal curve [Ban- choff (1976), Cimasoni (2001), Agarwal, Edels- The crossing of two space curves is determined brunner and Wang (2002), Dennis and Hannay by the visibility. A point q is visible to point p (2005), Klenin and Langowski (2000)] are dis- if there is a line of sight pq connecting the points cussed. Different from the evaluation of writhe p and q. Two space curves cross each other if a and linking number of a ribbon which only in- point on one curve is visible to a point on the other volve the ribbon central spine and boundary, com- curve. putation of twist of a ribbon needs to consider the Consider two space curve segments, dq and dp,a material between two curves. This paper reviews point p on segment dp, as the source for viewing the technique of interpreting the writhe in terms the target segment dq. As the point p moves along of spherical area and offers a similar geometric in- segment dp,segmentdq is visible to point p if the terpretation to the twist which enables an approxi- line of sight pq intersects the planar quadrilateral = (dp×dq)·pq mation formula. Based on these interpretations, it area dA |pq| which is perpendicular to the also explains the algebraic property of summing line of sight pq as depicted in ﬁgure 1. The solid up two real numbers of writhe and twist always angle dΩ includes all lines of sight along which yields a linking number integer in the Cãlugãre- the segment dq crosses dp is given by anu’s theorm. ( × ) · Ω = dA = dp dq pq d 2 3 (3) 2Reviews |pq| |pq| Mathematically, the linking number Lk(K,L) Hence, the solid angle equals to the Gauss Inte- [Dennis and Hannay (2005)] between two closed gral. The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon 153

K where r = 1 is the radius of the unit sphere. d: p

dA d: (grey) v

K’ q dp u dq line of sight pq (white) dq D Figure 1: Planar quadrilateral and solid angle crossing map F K (pq) with area dS Figure 2: Spherical quadrilateral and planar When two curves are oriented, in having a “head” quadrilateral and a “tail”, then the notion of signed crossing arises due to the cross product dp×dq in the planar quadrilateral dA. Given two skewed curve segments dq and dp, the crossing is said to be 2.2 Computation of linking number and writhe positive, if the tip of the arrow for the curve seg- The linking number Lk(K,L) between the two ment in front (ordered along the direction of line curve segments K and L is the average value of of sight, from the source to the view plane) is to be the signed crossing number over every possible rotated counter-clockwise with an angle less than line of sight: π – in order to be aligned with the curve segment 1 in the back; conversely, the crossing is negative. Lk(K,L)= dΩ (5) 4π Mapping the solid angle in equation (3) onto a unit sphere is a common approach to evaluate the which yields the expression as defined by equa- ( , ) Gauss Integral. tion (1). Similarly, the writhe Wr K K of a self intersect curve K is the average value of the Definition A crossing map χ is a map of the K signed crossing number over every possible line line of sight from curve segment K to L on a of sight: unit sphere S (these two curve segments K and L can be either from a single curve or two dis- 1 Wr(K,K)= dΩ (6) tinct curves), hence χK : K×L → Ssuch that v = 4π χ( , )= pq ∀ ∈ ∀ ∈ ∈ p q |pq| , p K , q L and v S. which gives equation (2) for the writheWr(K,K). Figure 2 shows the relationship between the signed area of the planar quadrilateral dA (as de- Discretizing a curve into a series of line segments picted in figure 2) and that of the spherical quadri- facilitates the computation of Gauss Integral nu- lateral dS on a unit sphere S. Since both areas merically. Figure 3(a) shows two linear segments dS and dA subtend the same solid angle dΩ in a p jp j+1 and qkqk+1. The segments cross each sphere: other along the line of sight pq. dA dS Specifically, the point p can move along the seg- dΩ = = (4) |pq|2 r2 ment p jp j+1 while the point q goes from qk to 154 Copyright c 2008 Tech Science Press CMES, vol.29, no.3, pp.151-162, 2008

p j number given by equation (1) can be rewritten as

n n ( , )= 1 Lk p Lk K L π ∑ ∑ S jk (8) 4 j=1, k=1, n+1=1n+1=1

p j1 Lk = ∀ ∈ ∀ ∈ where S jk S jk, p j K, qk L. ( , ) q k Similarly, the writhe Wr K K expressed by equation (2) is rewritten as q k 1 q Wr(K,K)= (a) Two linear segments n n 1 p j+1 qk+1 (dp×dq) · pq ∑ ∑ (9) p q π | |3 j1 k 1 4 j=1 k=1 p j qk pq j=k n+1=1 j=k±1 : jk n+1=1 which implies

1 n n p q Wr(K,K)= ∑ ∑ SWr (10) j1 k π jk 4 j=1 k=1 j=k n+1=1 j=k±1 + = S jk n 1 1

Wr where S = S jk, ∀p j ∈ K and ∀qk ∈ K. p q jk j k 1 Two special cases for the indices deserve some elaboration. When j = k ± 1, the quadrilateral produced on the unit sphere is the great circle in p q j k the plane containing the two segments; its area is nil or S = 0 ( j = k ± 1).Butwhenj = k,the (b) The crossing jk area degenerates to a point (with zero area) on the Figure 3: Angle ranges as curved quadrilateral unit sphere.

3 Computation of twist qk+1. The extreme solid angle Ω jk of an area The twist Tw(K,L) of a ribbon is due to the ge- swept out on the unit sphere must be bounded ometric relationship between its central spine K by the four segments: p jqk, p jqk+1, p j+1qk and and the boundary L.LetL(s)=K(s)+u(s), p j+1qk+1. where u(s) is a vector perpendicular to the unit The spherical quadrilateral crossing map with sur- tangent vector t of curve K pointing from curve face area S jk, illustrated in ﬁgure 3(b), gives the K to curve L and s is their common parameter. ( , ) entire range of solid angle Ω jk along which the The twist Tw K L [Dennis and Hannay (2005)] two line segments cross when projected. is From equation (3) and (4), ( , )= 1 ( × ) du · Tw K L π t u ds (11) 2 K ds p j+1 qk+1 ( × ) · = Ω = dp dq pq S jk jk 3 (7) 3.1 Twist of a ribbon p j qk |pq| The computation of the twist Tw(K,L) of a rib- where p j ∈ K and qk ∈ L. Hence, the linking bon is different from that of its linking number The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon 155

Lk(K,L) and writhe Wr(K,K), which only in- curve K volve the curve K (the central spine of a rib- u bon) and curve L (one of the boundaries of a ribbon). Since the material between the central spine (curve K) and the boundary (curve L) also con- tribute to the twist, it must be considered in the curve L computation. A ribbon is characterized by two curves K = K(s) and L(s)=K(s)+u(s)(∀s ∈ R)asshowninﬁg- (a) A ribbon ure 4(a). Curve K represents the central spine of the ribbon while its boundary is the curve L. Let K = p p ···p and L = q q ···q be two se- 1 2 n 1 2 n p j1 p j p j1 ries of line segments. A discrete ribbon model is shown in Figure 4(b) which relates all the points on curve K to those on curve L.When the two free ends of the ribbon are connected into a closed knot, then K = p1p2 ···pnp1 and L = q1q2 ···qnq1. In a discrete ribbon model, the integrals of linking number Lk and writhe Wr expressed in equa- q j1 q j q j1 tion (1) and (2) can be computed more efﬁciently, and with nearly equal effectiveness, by consider- (b) A discrete ribbon model ing their geometric meanings. A similar approach Figure 4: A ribbon and its model is employed to compute the twist Tw. Rewriting equation (11) as K=K(s) Tw(K,L)=∑ Tw (12) j j L(s)=K(s)+u(s) t p j1 v 2

q j1 with Tw = 1 p j+1 (t×u) du ds. Consider the D u j 2π p j ds j ribbon segment characterized by points p j, p j+1 v1 on curve K as shown in figure 5. Since curve K and L have the same parmeterization, the corresponding points q j and q j+1 are identified on curve L. The segment q j q j+1 twists around p j p j+1 with dihedral angle α j. Since the twist Tw in equation (11) can also been expressed in terms 13 of crossings , the vectors v1, v2, v5 and v6 shown v in figure 5 are the extreme lines of sight to have 5 p j crossings between the two curve segments q j × v6 The vectors t, u and t u forms a moving frame t u u along the curve K which is similar to Frenet du Figure 5: Segment on curve L twists around seg- frame. The vector ds measures how the curve L is moving with respect to curve K. Taking the dot ment on curve K product with t×u measures how much it is moving in a direction perpendicular to the direction of the curve. 156 Copyright c 2008 Tech Science Press CMES, vol.29, no.3, pp.151-162, 2008

3.2 Equations and mathematical expressions Let M be the series of line segments M = r1r2 ···rn. The spherical quadrilateral v1v2v5v6 The crossing map of segment q q + with re- j j 1 is defined as spect to segment p j p j+1 isshowninfigure6. Since curve K and L have the relationship L(s)= ( )+ ( ) χ ( ) v1v2v5v6 = v|v = χK(pr), K s u s , the crossing map K u is a portion of equator with dihedral angle α . The spherical j ∀p ∈ K, r ∈ M = L (14) region v1v2v5v6 is the crossing map χK(pq) with points p and q on segment p j p j+1 and segment q j Figure 7 depicts the crossing map with 1 > t > 0. q j+1 respectively. Due to the twist condition, the crossing maps are within the spherical lune with The shaded region within the spherical lune is the accumulated crossing map with respect to curve dihedral angle α j. K as the parameter t decreases from 1. As the parameter t starts changing from 1, the location crossing map F K (pq) of curve M slides away from curve L and moves towards curve K. This is equivalent to thicken- (shaded area) ing curve L (or adding material between curves L and M) as parameter t decreases from 1. In figure v2 7(a), the shaded region v1v2v5v6 (as shown in figure 5) is enlarged to v1v3v5v7 as the parameter t decreases from 1 to 0.5. This is the accumulated crossing map of the “thickened” curve segments L (with width t) with respect to curve segment K,

v1 v1v3v5v7 = v|v = χK(pr), D j t,1≥t>0.5 v5 ∀p ∈ K, r ∈ M =(1 −t)K+t · L (15)

v6 Hence, the complete spherical lune v1v4v5v8 is the crossing map of a ribbon with spine curve K crossing map F K (u) and boundary curve L Figure 6: Crossing maps of segments p j p j+1 and = | = χ ( ), q q + v1v4v5v8 v v K pr j j 1 t,1≥t≥0 ∀p ∈ K, r ∈ M =(1 −t)K+t · L (16) A parameter t (t ∈ [0,1] ⊂ R) is used to characterize the material between the curves. The material The twist Twj between segment p j p j+1 and seg- between curve K and L is expressed as curve M ment q j q j+1 is the average crossing over every on the ribbon parallel to K and L with a speciﬁc possible line of sight, hence parameter t: Ω j Twj = (17) M(s)=(1−t)K(s)+t ·L(s), ∀s ∈ R, 0 ≤ t ≤ 1 4π (13) From equation (4), the solid angle is area of the α When t =1,curveM coincides with curve L spherical lune which is 2 j, hence, which implies no material is between the curves α Tw = j (18) K and L. j 2π The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon 157

v Figure 8 shows two examples of computing the 4 twist and writhe of two conformations of an open v 3 end ribbon by considering the area of the crossing v 2 maps. In figure 8(a), a twisted ribbon and its discrete model with six segments are shown. Hence, there are six lunes (cyan region) on the unit sphere as the crossing map. Similarly, a discrete coiled v1 Dj ribbon of ten segments is employed to depict the crossing map in figure 8(b). The crossing map v 5 consists of two parts; one is on the left hand side of the northern hemisphere which the other part is v on the right hand side of the southern hemisphere. 6 v v 7 8 For these two simple conformations, a point in the map refers to a line of sight from the centre of the (a) Crossing map of the thickened curve L unit sphere to the point which gives one crossing (as t decreases from 1 to 0.5) with (the map shown in figure 8(a) gives a crossing be- respect to curve K. tween curves K and L while that in figure 8(b) yields a self crossing of curve K).

v 4 4Cˇalugˇareanu’s theorem v 3

v 2 The mathematics of knot theory for which Cˇalugˇareanu’s theorem is one of the pillars applying to closed end curves. It relates the linking number (the topology) with the writhe and v1 twist (the geometry) of a ribbon model. The link- Dj ing number of a closed end ribbon is an invariant v 5 while writhe and twist are interchangeable. Math- ematically,

v 6 Lk(K,L)=Wr(K,K)+Tw(K,L) (20) v 7 v 8

4.1 Algebraic property (b) Crossing map of the thickened curve L (as t decreases from 0.5 to 0) with Since curves K and L are closed end, their crossing maps cover the entire unit sphere. Hence the respect to curve K. n n ∑ ∑ Lk Figure 7: The crossing maps of a twisted ribbon expression S jk covers the whole unit j=1, k=1, n+1=1n+1=1 sphere. The linking number is obtained by di- viding it by the area of the unit sphere surface Hence, the twist of a ribbon is (which is 4π). Hence, the linking number of two n closed end curves is an integer number of times ( , )= 1 α Tw K L π ∑ j (19) that the unit sphere covered by the image of the 2 j=1 two curves under the mapping χK. On the other where hand, the geometry of a ribbon is described by its writhe and twist. Summing up the writhe (Wr) −1 q jp j×q jp j+1 q j+1p j×q j+1p j+1 α j = cos · and twist (Tw), which are not necessarily integers q jp j×q jp j+1 q j+1p j×q j+1p j+1 (indeed most often they are real numbers), always ∀p j ∈ K and ∀q j ∈ L. yields an integer (linking number Lk). 158 Copyright c 2008 Tech Science Press CMES, vol.29, no.3, pp.151-162, 2008

The three quantities Lk(K,L) · 4π, SWr and STw are all real numbers. Rearranging yields A twisted ribbon S S Lk(K,L)= Wr + Tw (23) 4π 4π

SWr Therefore, the two geometric quantities 4π STw (which is Wr)and 4π (which is Tw) on the right A discrete twisted ribbon Crossing map hand side of equation (20) are real numbers and the topological quantity Lk(K,L) on the left is an (a) A twisted ribbon and its crossing map integer.

4.2 Open end ribbon Whatever the linking number between two closed end curves might be, the change in the geometry A coiled ribbon as writhe and twist counter-balances each other. Algebraically, the derivative of a constant (for the linking number) is zero. Therefore, according to (20)

δWr(K,K)+δTw(K,L)=0 (24) A discrete coiled ribbon Crossing map When a system is closed, as in the case of a knot (b) A coiled ribbon and its crossing map (or with the total energy in a system), there tends Figure 8: Discrete ribbon models and their cross- to be a “conservation” theorem characterizing the ing maps behavior of the different aspects in the closed system. Yet, most proteins in their native forms are open end. To save Cˇalugˇareanu’s theorem, the notion of “nano-robotics” comes to rescue. It is obvious to explain the reason by considering The mechanics of a single bio-molecule at the their geometric interpretations. nanometer scale is an important pursuit as it Substituting equation (8), (10) and (19) into equa- provides insight into how a molecule functions tion (20) yields under external inﬂuences, as well as in reveal- ing the limitations of classical mechanics in the S = S +S (21) total Wr Tw Newtonian scale. The forces underlie the var- n n = ∑ ∑ Lk ∈ ious chemistries and molecular biology of ge- where Stotal S jk R is the total area μ j=1, k=1, netic material are of the order of N. Several n+1=1 n+1=1 techniques differing in force ranges are avail- of the crossing maps on the unit sphere, SWr = n n able: atomic force microscopy [Mehta, Rief, Spu- ∑ ∑ Wr ∈ S jk R is the area of the crossing dich, Smith and Simmons (1999)](AFM), optical j=1 k=1 j=k n+1=1 tweezers, and magnetic tweezers [Fritz, Baller, j=k±1 n+1=1 Lang, Rothuizen, Vettiger, Meyer, Güntherodt, n Gerber and Gimzewski (2000), Eggar (2000)]. maps due to the writhe and STw = 2 ∑ α j ∈ R is j=1 Mechanicstically, a molecule is “clamped” at one the area of the crossing maps due to the twist. end and on a substrate and is manipulated by the Rewriting equation (21) as other, free end. Both the AFM and optical tweezers are for applying pulling forces on the free end, Lk(K,L) · 4π = SWr +STw (22) in different force ranges. The forces range for The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon 159

AFM is greater than 1μN while that for the optical their spatial positions in each shown conformation tweezers is between 0.1μN to 150μN. Measuring are measured. Three canonical conformations of the rate and force at which a DNA is packed by the ribbon are illustrated in figure 10. Figure 10(a) a virus is an example of using the optical tweez- shows a twisted ribbon while a coiled ribbon is ers. Pulling or pushing is translational. If rota- shown in figure 10(b) and 10(c). Their writhe and tional input is required, such as in coiling a DNA twist are computed. and measuring the rate of uncoiling by an en- The writhe (Wr) and twist (Tw) of the conforma- zyme, the magnetic tweezers are used. The typ- tions shown in figure 10(a) and 10(b) conforms to ical force range for a magnetic tweezer is 0.01- equation (24). However, the conformation shown 100μN. In geometry, these nano-manipulation ap- in figure 10(c) (which is the same the conforma- paratuses effectively close the open end curve. tion shown in figure 9(f)) yields a decrement in writhe (a) A twisted ribbon with Wr=0, Tw=1 as (a) Twisting an open end shown in figure 9(a). ribbon by 2S. A coiled ribbon with Wr =∼ 0.95, Tw =∼ 0.05 as A B shown in figure 9(e). (b) End B of the ribbon Another conformation of a coiled ribbon with moves toward end A Wr =∼ 0.07, Tw =∼ 0.05 as shown in figure 9(f). which is fixed. A B

(d) Another supercoiling (a) A twisted ribbon with Wr=0, Tw=1 as shown in figure 9(a) conformation as one end gets closer to the A B other end. B (e) Another supercoiling A conformation as both ends meet. A B (b) A coiled ribbon with Wr # 0.95, Tw # 0.05 as shown in figure 9(e) (f) End B moves to the “other” side of end A.

B A

Figure 9: Various conformations due to the mo- B A tion of the ribbon ends (c) Another conformation of a coiled ribbon with Wr#0.07, Tw#0.05 as shown in figure 9(f) Twisting an open end ribbon by 2π as depicted in Figure 10: Canonical ribbon conformations figure 9(a) yields a conformation with planar central spine K. The self crossing map of this curve on the unit sphere is a great circle with zero area. When the ends of an open end ribbon is held (for Hence, the writhe of a twisted ribbon is zero and instance, the DNA or protein molecules held by the twist is 1. As one end of the ribbon moves to- the nano-manipulation apparatuses), it is equiva- wards the other end, the twisted ribbon turns into lent to have an imaginary ribbon connecting both a coiled ribbon as shown in figure 9(b) – 9(e). ends as illustrated in figure 11. A twisted open Twenty points are marked on the spline K and end ribbon is shown on the left hand side with two boundary curve L of a ribbon respectively and ends A and B. As end B of the ribbon moves to- 160 Copyright c 2008 Tech Science Press CMES, vol.29, no.3, pp.151-162, 2008 wards end A, it turns into a coiled ribbon as shown Tw(K,L)=Tw(Wr(Kr,Lr)) +Tw(Wr(Ki,Li)) on the right hand side with the twist converting (26) into writhe. where 1 Imaginary Tw(Wr(K ,L )) = ∑ α , A r r π j j part 2 ∀p j ∈ Kr and ∀q j ∈ Kr and 1 Tw(Wr(K ,L )) = ∑ α , i i 2π j j ∀p ∈ K and ∀q ∈ K B B j i j i A Therefore, substituting equation (25) and (26) into (24) and rearranging yields

δ ( ( , )) +δ ( ( , )) Wr Wr Kr Kr Tw Wr Kr Lr crossing real part of the ribbon +δWr(Wr(K ,K )) +δTw(Wr(K ,L )) i i i i = 0 +2δWr(Wr(K ,K )) i r Imaginary part of the ribbon Figure 11: Closing an open end ribbon (27)

Equation (27) can be categorized into two parts, Rewriting the closed end curve K = Kr ∪ Ki and the real part (which is equivalent to an open end L = Lr ∪Li where the subscripts r and i represent ribbon) and the imaginary part of a “closed” end the real part and the imaginary part of a “closed” ribbon. end ribbon respectively. The writhe Wr(K,K) When an open end ribbon is isotropic to a new and the twist Tw(K,L) are expressed as conformation, any change in twist will be exactly balanced by the change in its writhe provided Wr(K,K)=Wr(Wr(K ,K )) +Wr(Wr(K ,K )) r r i i the crossings due to either twist or writhe re- + ( ( , )) 2Wr Wr Ki Kr (25) main in the real part as shown in ﬁgure 8(a)-8(e). where This condition refers to δWr(Wr(Ki,Ki)) = 0, δWr(Wr(K ,K )) = 0andδTw(Wr(K ,L )) = 0 1 i r i i Wr(Wr(K ,K )) = ∑ ∑ SWr, in equation (27) which gives r r 4π j k jk ∀ ∈ , ∈ p j Kr q j Kr δWr(Wr(Kr,Kr)) +δTw(Wr(Kr,Lr)) = 0 (28) ; When the crossings (due to either twist or 1 Wr(Wr(K ,K )) = ∑ ∑ SWr, writhe) are moved to the imaginary part as i i π j k jk 4 shown in ﬁgure 8(f), the terms δWr(Wr(Ki,Ki)), ∀ ∈ , ∈ p j Ki q j Ki δWr(Wr(Ki,Kr)) and δTw(Wr(Ki,Li)) no and longer vanish and the changes in twist and writhe in the open end (the real part) ribbon do not 1 Wr(Wr(K ,K )) = ∑ ∑ SWr = Wr(K ,K ) counter balance each other. Hence, the writhe i r π j k jk r i 4 decreases as the crossing moves to the imaginary ∀ ∈ , ∈ p j Kr q j Ki part. The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon 161

5 Concluding remarks ics. Vol.5, no.2, pp.103-106. Carson, M.; Bugg, C. E. (2000): Early rib- The geometry and the topology of a ribbon are re- bon drawings of proteins. Natural Structural and lated to each other by their linkingnumber, writhe Molecular Biology, vol.7, no.8, pp.624-625. and twist. It turns out that these three quantities in the Cˇalugˇareanu’s theorem are rooted in the Cimasoni, D. (2001): Computing the writhe of notion of an area on a unit sphere, more gener- a knot. Journal of Knot Theory Ramifications, ally known as the Gauss Integral.Behindthese vol.10, pp.387-295. quantities are the crossings. The integrals can be Cozarelli, N. R.; Boles, T.; White, J. (1990): very complicated and are hard to compute for a DNA Topology and its biological effects, Cold complex ribbon conformation, since no analytic Spring Harbor Laboratory Press. expression is available for the curves. Simplifi- Dennis, M. R.; Hannay, J. H. (2005): Geome- cation is offered by having considered their geo- try of Cˇalugˇareanu’s Theorem. Proceedings of the metric interpretations. These geometric interpre- Royal Society A: Mathematical Physical and En- tations also reveal the reason for the property of gineering Sciences, vol.461, no.2062. pp.3245- the Cˇalugˇareanu’s theorem that adding two real 3254. numbers (writhe Wr and twist Tw) always yields Eggar, M. H. (2000): On White’s formula. Jour- an integer (linking number Lk). The change in nal of Knot Theory and its Ramification,vol.9, twist and writhe for an open end ribbon is also no.5, pp.611-615. discussed. Fritz, J.; Baller, M. K.; Lang, H. P.; Rothuizen, The Cˇalugˇareanu’s theorem accounts for the H.; Vettiger, P.; Meyer, E.; Güntherodt, H. J.; transformation between a twisted ribbon and a Gerber, C. H.; Gimzewski, J. K. (2000): Trans- coiled ribbon which happens in many biological lating biomolecular recognition into nanome- processes. To understand these processes better, chanics. Science, vol.288, no.5464. pp.316-318. many more issues deserve further exploration – one of which has to the forces and energies that Fuller, F. B. (1971): The writhing number of drive a molecule from one conformation to an- a space curve. Proceedings of the National other. Academy of Science, vol.68, pp.815-819. Karcher, H.; Lee, S. E.;Kaazempur-Mofrad, M. R.; Kamm, R. D. (2006): A coarse-grained 6 References model for force-induced protein deformation and Agarwal, P. K.; Edelsbrunner, H.; Wang, Y. kinetics. Biophysical Journal, vol. 90, no.8, (2002): Computing the writhing number of a pp.2686-2697. polygonal knot. Discrete & Computational Ge- Klenin, K.; Langowski, J. (2000): Computa- ometry, Vol.32, No.1, pp.37-53. tion of writhe in modeling of supercoiled DNA. Banchoff, T. (1976): Self linking numbers of Biopolymers, vol.54, pp.307-317. space polygons. Indiana University Mathemati- Mehta, A. D.; Rief, M.; Spudich, J. A.; Smith, cal Journal, vol.25, no.12, pp.1171-1188. D. A.; Simmons R. M. (1999):Single-molecule Cˇalugˇareanu, G. (1959): L’intégrale de Gauss biomechanics with optical methods. Science, et l’Analyse des nœuds tridimensionnels. Re- vol.283, no. 5408, pp.1689-1695. vue Roumaine de Mathématiques Pures et Ap- Park, J. Y.; Cho, Y. S.; Kim, S. Y.; Jun, S.; Im, pliquées, Vol.4, pp.5-20. S. (2006): A quasicontinuum method for defor- Carson, M.; Bugg, C. E. (1986): Algorithm for mations of carbon nanotubes. CMES: Computer ribbon models of proteins. Journal of Molecular Modeling in Engineering & Science, Vol.11, no.2, Graphics, vol.4, no.2, pp.121-122. pp.61-72. Carson, M.; Bugg, C. E. (1987): Ribbon models Steffen N. R.; Murphy, S. D.; Lathrop, R. H.; of macromolecules. Journal of Molecular Graph- Opel, M. L.; Tolleri, L.; Hatfield, G. W. (2002): 162 Copyright c 2008 Tech Science Press CMES, vol.29, no.3, pp.151-162, 2008

The role of DNA deformation energy at individual base steps for the identiﬁcation of DNA-protein binding sites. Genome Informatics, vol. 13, pp. 53-162.