"Linking numbers and nucleosomes"

Date:
1972-1977
Reference:
PP/CRI/H/6/12/2
Catalogue details

Licence: In copyright

Credit: "Linking numbers and nucleosomes". Source: Wellcome Collection.

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    1332 f. brock fuller limit one obtains the equation (2) Wr(X) = Lk(X, X, + Y.) —- 0. 2 it Here 0 is a real number depending on the choice of Y, which, reduced modulo 2tc, gives the angle 0. iii. proof of the theorem Let X 0 , Xi be two space curves as in the theorem. Let JV be a unit vector perpendicular to both unit tangents T 0 (a) and 2\(a). Let (X 0 , U 0 ) and (X 1} Uj) be the twist-free strips with the initial normals U 9 {a) — U x (a\ = N. Then, from equation 2, we find |sinT:(irr(X 0 ) - TFriXJ)| =|sin-i-(0 o - 9 X )\. By using trigonometric identities we obtain |sinl(0 o - 8 x )j = |-([(ï7 0 (ô)- U x (b)).2?Y + [(F 9 (6) - V x (b))-N]*)™, whence, by the triangle inequality, ] s i n i.(0 o _0 1 )| < l(|í7o(&) - U l (b)\ + |F 0 (ft) — Fj(6)|). Again, by the triangle inequality : I F 0 - Fj| = I T 0 X U 0 - T x X U x \ < I U 0 - TJ X \ + I T 0 — T x \. Combining' all the above, we obtain : (3) IsinTrOFríXJ-TFr^ K! U 0 (b)-U x (b)\+ 11 T 0 (b) - T x (b)\. Let W(t) = Í7 0 (í) — U x {t) for a^.t¿Cb. ^Ye estimate W(t) by esti mating its components on the (T 0 , U 0 , F 0 )-frame. The identities and inequalities used in the remaining part of the proof involve only repeated use of the fact that (T 0 , U 0 , F 0 ) and (T v U 1 , I',) are orthonormal frames d rr
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