Isabel K. Darcy. Joint with:

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1 Isel K. Drcy Joint with: Mthemtics Deprtment Applied Mthemticl nd Computtionl Sciences (AMCS) University of Iow Hyeyoung Moon, Michign Ro Schrein, Hypngogic Softwre Gunyu Wng, University of Iow Dnielle Wshurn, University of Iow This work ws prtilly supported y the Joint DMS/NIGMS Inititive to Support Reserch in the Are of Mthemticl Biology (NSF ). 008 I.K. Drcy. All rights reserved

2 Mthemticl Model Protein = = DNA = = =

3 C. Ernst, D. W. Sumners, A clculus for rtionl tngles: pplictions to DNA recomintion, Mth. Proc. Cm. Phil. Soc. 08 (990), protein = three dimensionl ll protein-ound DNA = strings. Slide (modified) from Sooeong Kim Protein-DNA complex Heichmn nd Johnson

4 Tngle = 3-dimensionl ll contining strings where the endpoints of the strings re fixed on the oundry of the ll. = Protein = 3-dimensionl ll DNA = strings

5 Tngle = 3-dimensionl ll contining strings where the endpoints of the strings re fixed on the oundry of the ll. = Protein = 3-dimensionl ll DNA = strings For geometry: see -:30 -Mry Therese Pderg, Exploring the conformtions of protein-ound DNA: dding geometry to known topology, Wednesdy, Mrch 4, -:30pm nd poster.

6 Topoisomerse II performing crossing chnge on DNA: Cellulr roles of DNA topoisomerses: moleculr perspective, Jmes C. Wng, Nture Reviews Moleculr Cell Biology 3, (June 00)

7 Topoisomerses re involved in Repliction Trnscription Unknotting, unlinking, supercoiling. Trgets of mny nti-cncer drugs.

8 Topoisomerses re proteins which cut one segment of DNA llowing second DNA segment to pss through efore reseling the rek.

9 Crossing Chnge Knot distnce Let K d(k,k nd K ) the minimum numer of crossing chnges needed to convert e knots. into is tken over ll digrms of K K K where the minimum. Unknotting numer Let K e knot. Then u( K) d( K,0 ) where is 0 the unknot.

10 Exmple 3 0 Unknotting numer of 3 u(3 ) Figure: courtesy of Hyeyoung Moon

11 There re undetermined vlues in the knot distnce tle. For exmple, Thus, d(5,4 ) 3. d( 5,4 ) (5) (4) 4 0. Slide courtesy of Hyeyoung Moon

12 Knot distnce tultion The distnces etween two knots up to mirror imges re tulted. Slide courtesy of Hyeyoung Moon

13 Knot distnce tultion The knot distnces hs een tulted for rtionl knots, some non - rtionl knots nd compositeof rtionl knots up to3 crossing using the following mthemticl theories [DS] : ) ) Clssifiction of Tringle inequlity, distnce one rtionl 3) d( K, K) ( K) ( K) where ( K) is the signture of K ([Mk]), knots ([DS][To]),

14 Knot distnce tultion 4) 5) 6) Linking form requirementson where M & H (M k k is the doule strnded cover of S ) is the first homology group of tht.([mk]), Homology requirementson H (M ) ([DS]), over K Unknotting numer one knots re prime ([Sc][Zh]). H (M k k ) 3 7) [D, Moon] Jones polynomil

15 Tngle Equtions

16 Determining upper ounds

17 Determining upper ounds

18 Rtionl Tngles Rtionl tngles lternte etween verticl crossings & horizontl crossings. k horizontl crossings re right-hnded if k > 0 k horizontl crossings re left-hnded if k < 0 k verticl crossings re left-hnded if k > 0 k verticl crossings re right-hnded if k < 0 Note tht if k > 0, then the slope of the overcrossing strnd is negtive, while if k < 0, then the slope of the overcrossing strnd is positive. By convention, the rtionl tngle nottion lwys ends with the numer of horizontl crossings.

19 Rtionl tngles cn e clssified with frctions.

20 A knot/link is rtionl if it cn e formed from rtionl tngle vi numertor closure. N(/7) = N(/) Note 7 = 6 = (3)

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22 when B = c/d, E = f/g, nd cg df >

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24 TopoICE in Ro Schrein s KnotPlot.com Cover: Visul presenttion of knot distnce metric creted using the softwre TopoICE-X within KnotPlot. A pir of knots in this grph is connected y n edge if they cn e converted into one nother vi single intersegmentl pssge. This grph shows ll mthemticlly possile topoisomerse rection pthwys involving smll crossing knots. D, Schrein, Stsik. (Nucleic Acids Res., 008; 36: ).

25 All possile topoisomerse-medited rection pthwys from the unknot to 5. involving rtionl knots with less thn 4 crossings. Drcy I K et l. Nucl. Acids Res. 008;36: The Author(s)

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27 Tngle tle is oint work with Ro Schrein, Dnielle Wshurn, Gunyu Wng, Melnie DeVries, et. l.

28 A tngle which is not generlized Montisinos

29 A tngle which is not generlized Montisinos

30 A tngle which is not generlized Montisinos

31 Tle of 4-crossing prity zero -string tngles. D, Melnie DeVries, Dnielle Wshurn, Gunyu Wng, Ro Schrein, et l. Prity 0 Prity Prity

32 Tle of 4-crossing prity infinity -string tngles.. Prity

33 prity one -string tngles. Prity

34 Tle of prity zero tngles 4 crossings: 6 tngles 5 crossings: 44 tngles Note the tle currently contins mny repets 6 crossings: 8 tngles 7 crossings: 430 tngles 8 crossings: 8868 tngles 9 crossings: tngles

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37 Crossing Sign Determintion Right-hnd Rule Right-hnded Crossing + Left-hnded Crossing -

38 L positive crossing L negtive crossing Signed crossing chnges L L L L is is clled clled crossing chnge. crossing chnge.

39 Signed knot distnces d (K,K needed toconvert re llowed. ) is the minimum numer of crossing chnges K into K where only crossing chnges d (K,K needed toconvert re llowed. ) is the minimum numer of crossing chnges K into K where only crossing chnges

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41 Crossing Sign Determintion Right-hnd Rule Right-hnded Crossing + Left-hnded Crossing -

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43 TopoICE-R + tngle corresponds to negtive crossing since h + q + p(+) is odd

44 Recomintion:

45 from the wll of the Pis Cthedrl. Photo courtesy of Ro Schrein

46 Montesinos knot/link A Montesinosknot/link is N( nd i i is rtionl tngle for i,,r re reltively prime nd 0 i r r nd r i.tht is, e) where e is n integrl tngle i i 3. Here, we ssume is not n integrl tngle. i nd i r r e

47 Solving tngle equtions Theorem [Hyeyoung Moon, D] For s,t 3, nd where N(U N(U nd 0,,e i U i x,y,z x y ( ) ),v re integers c d,e N( N( z v nd 0 re integers c d n n z v t t s s e nd 0 e ) ) ) (h,,h i m z i ) is for v i for s, generlized M - tngle. t

48 Solving tngle equtions. ) nd where,0, then 0, 3 nd () if ) nd then, () if, 3 nd for some for if nd only if ) e y x h x N( e v z v z N( e h ) h (h ) ( U h m ) e y x N( e v z v z N( ) (e ) ( U m s s,t t t m t t

49 Solving tngle equtions the solution in (). thesme s the solution is 0 in (), if Note tht. ) nd nd where,, then, 0 for ll 3 if 0 or nd 0 for ll 3 (3) if h ) e ],h, ye[h ],h, xe[h ],h, ye[h ],h, xe[h N( e v z v z N( ],h, E[h ],h, E[h e h ) h (h ) ( U m, h m h m, h m m m m m t t m m m m

50 Solving tngle equtions Theorem [Hyeyoung Moon, D] For t where if U c nd 3, nd only if ( N(U c d such tht 0 for is Note tht thechoice x hx ),, x,y,z y 3, ) (h,0 ) nd ( not n integer. of nd,d,p nd q such tht d p-qc re integers nd 0 z U is generlized M - tngle. p d N(,v t c q,e ) c nd N(U p d x y ) c q nd 0 c In this cse, p such tht d N( c d z v v d for ) (h,0 ) for ll integers N( p-qc zt v where z v z v t e h is n integer z v 3 3 t ) N( c d hs no effect on U. ) p d c q x hx y ).

51 Solving tngle equtions Theorem.3 [Hyeyoung Moon, D] For s where if U 3, nd nd only if ( N(U such tht In this cse, 0,,e c d for for ll integers c i is Note tht thechoice i x hx ) N(,x,y,z,v re integers nd 0 y ' N( 3, U is generlized M - tngle. s pz cv d v qz ) (h,0 ) ( not n integer for of c 3 3 nd e e ) nd N(U x y ) ) nd (,d,p nd q such tht d p-qc s s y N( p such tht d ' such tht c d pz cv d v qz i i for where p-qc x y yy pz cv d v qz ) c d ' N( i ) (h,0 ) ( z v ) h is n integer mod x hx y s, ' ). x. x y hs no effect on U. )