Transcript Slide 1
Isabel K. Darcy
Mathematics Department
Applied Mathematical and Computational Sciences (AMCS)
University of Iowa
http://www.math.uiowa.edu/~idarcy
Hyeyoung Moon, Michigan
Joint
Rob Scharein, Hypnagogic Software
with: Guanyu Wang, University of Iowa
Danielle Washburn, University of Iowa
This work was partially supported by the Joint DMS/NIGMS Initiative to Support
Research in the Area of Mathematical Biology (NSF 0800285). ©2008 I.K. Darcy. All rights reserved
Mathematical Model
Protein =
DNA =
=
=
=
C. Ernst, D. W. Sumners, A calculus for rational tangles:
applications to DNA recombination, Math. Proc. Camb. Phil.
Soc. 108 (1990), 489-515.
protein = three dimensional ball
protein-bound DNA = strings.
Protein-DNA complex
Heichman and Johnson
Slide (modified) from Soojeong Kim
Tangle = 3-dimensional ball containing strings
where the endpoints of the strings are fixed on the
boundary of the ball.
=
Protein = 3-dimensional ball
DNA = strings
≠
Tangle = 3-dimensional ball containing strings
where the endpoints of the strings are fixed on the
boundary of the ball.
=
≠
Protein = 3-dimensional ball
DNA = strings
For geometry: see 12-12:30 -Mary Therese Padberg,
Exploring the conformations of protein-bound DNA: adding
geometry to known topology, Wednesday, March 14, 12-12:30pm
and poster .
Topoisomerase II performing a crossing change on DNA:
Cellular roles of DNA topoisomerases: a molecular perspective, James C. Wang,
Nature Reviews Molecular Cell Biology 3, 430-440 (June 2002)
Topoisomerases are involved in
• Replication
• Transcription
• Unknotting, unlinking, supercoiling.
• Targets of many anti-cancer drugs.
Topoisomerases are proteins which cut one segment of
DNA allowing a second DNA segment to pass through
before resealing the break.
• Crossing Change
• Knot distance
Let K1 and K 2 be knots.
d(K1,K2 ) theminimumnumber of crossingchanges
needed to convertK1 into K 2 where theminimum
is takenover all diagrams of K1.
• Unknotting number
Let K be a knot.
T henu ( K ) d ( K ,01 ) where is 01 theunknot.
Example
01
31
Unknottingnumber of 31 u(31 ) 1
Figure: courtesy of Hyeyoung Moon
• There are undetermined values in the knot distance
table.
• For example,
51
31
01
41
T hus, d (51 ,41 ) 3.
1
1
d (51 ,41 ) | (51 ) (41 ) | | 4 0 | 2.
2
2
Slide
courtesy
of
Hyeyoung
Moon
Knot distance tabulation
• The distances between two knots up to mirror
images are tabulated.
http://math.uiowa.edu/~idarcy/TAB/tabnov.pdf
Slide courtesy of Hyeyoung Moon
Knot distance tabulation
T heknotdistanceshas been tabulated for rationalknots,
some non - rationalknotsand compositeof rationalknots
up to13 crossingusing thefollowingmathematic
al theories[DS] :
1) Classification of distanceone rationalknots([DS1][T o]),
2) T riangleinequality,
1
3) d ( K1 , K 2 ) | ( K1 ) ( K 2 ) |
2
where ( K ) is thesignatureof K ([Mk]),
Knot distance tabulation
4) Linkingform requirements on |H1(M k )|
whereM k is thedouble strandedcoverof S 3 over K
& H1(M k ) is thefirst homologygroup of that.([Mk]),
5) Homologyrequirements on H1(M k ) ([DS1]),
6) Unknottin
g number one knotsare prime([Sc][Zh]).
7) [D, Moon] Jones polynomial
Tangle
Equations
Determining
upper bounds
Determining
upper bounds
Rational Tangles
Rational tangles alternate between vertical crossings & horizontal crossings.
k horizontal crossings are right-handed if k > 0
k horizontal crossings are left-handed if k < 0
k vertical crossings are left-handed if k > 0
k vertical crossings are right-handed if k < 0
Note that if k > 0, then the slope of the overcrossing strand is negative,
while if k < 0, then the slope of the overcrossing strand is positive.
By convention, the rational tangle notation always ends with the number of
horizontal crossings.
Rational tangles can be classified with fractions.
A knot/link is rational if it can be formed from a
rational tangle via numerator closure.
N(2/7) = N(2/1)
Note 7 – 1 = 6 = 2(3)
when B = c/d, E = f/g, and |cg – df| > 1
TopoICE in
Rob Scharein’s
KnotPlot.com
Cover: Visual presentation of knot distance metric created using the software
TopoICE-X within KnotPlot. A pair of knots in this graph is connected by an edge if
they can be converted into one another via a single intersegmental passage. This
graph shows all mathematically possible topoisomerase reaction pathways involving
small crossing knots. D, Scharein, Stasiak. (Nucleic Acids Res., 2008; 36: 3515–
3521).
All possible topoisomerase-mediated reaction pathways from the unknot to 5.1 involving
rational knots with less than 14 crossings.
Darcy I K et al. Nucl. Acids Res. 2008;36:3515-3521
© 2008 The Author(s)
Tangle table
is joint
work with
Rob
Scharein,
Danielle
Washburn,
Guanyu
Wang,
Melanie
DeVries,
et. al.
A tangle which is not generalized Montisinos
A tangle which is not generalized Montisinos
A tangle which is not generalized Montisinos
Table of 4-crossing parity zero 2-string tangles.
D, Melanie DeVries, Danielle Washburn, Guanyu Wang, Rob Scharein, et al.
Parity 0
Parity ∞
Parity 1
Table of 4-crossing parity infinity 2-string tangles.
.
Parity ∞
parity one 2-string tangles.
Parity 1
Table of parity zero tangles
4 crossings: 6 tangles
5 crossings: 44 tangles
6 crossings: 228 tangles
7 crossings: 1430 tangles
8 crossings: 8868 tangles
9 crossings: 59878 tangles
Note the table
currently contains
many repeats
Crossing Sign Determination
Right-hand Rule
Right-handed Crossing
+1
Left-handed Crossing
-1
L
positive crossing
L
negative crossing
Signed crossing changes
L L is called a crossingchange.
L L is called a crossingchange.
Signed knot distances
d (K1,K2 ) is theminimum number of crossingchanges
needed to convert K1 into K2 where only crossingchanges
are allowed.
d (K1,K2 ) is theminimumnumber of crossingchanges
needed to convertK1 into K2 where only crossingchanges
are allowed.
Crossing Sign Determination
Right-hand Rule
Right-handed Crossing
+1
Left-handed Crossing
-1
TopoICE-R
+1 tangle
corresponds
to a negative
crossing since
h + q + p(b+1)
is odd
Recombination:
from the wall of the Pisa Cathedral. Photo courtesy of Rob Scharein
Montesinos knot/link
A Montesinosknot/linkis N(
and
a1
a
r e) where e is an integraltangle
b1
br
ai
is a rationaltanglefor i 1, ,r and r 3. Here, we assume ai and bi
bi
are relativelyprimeand 0 ai bi . T hatis,
a1
ai
is not an integraltangle.
bi
ar
b1
e
br
Solving tangle equations
Theorem [Hyeyoung Moon, D]
For s,t 3,
as
0
a1
N(U ) N( e1 )
1
b1
bs
zt
x
z1
and N(U ) N( e2 )
y
v1
vt
where ai ,bi ,e1 are integersand 0 ai bi for1 i s ,
x,y,zj ,v j ,e2 are integersand 0 z j v j for1 j t
and U (
c
c1
n ) (h1, ,hm ) is a generalized M - tangle.
d1
dn
Solving tangle equations
if and onlyif for s,t 3 and for some 1 j1 s,
(1) if m 1, thenU (
a j1
b j1
a j1 1
b j1 1
a j1 2
b j1 2
a j1 1
b j1 1
) (e1 )and
a j1 a j1 1
a j1 2 a j1 1 x
zt
z1
N( e2 ) N(
e1 )
v1
vt
b j1 b j1 1
b j1 2 b j1 1 y
(2) if m 3 and hm 0, thenU (
where h1 e1 and N(
N(
a j1
b j1
a j1 1
b j1 1
a j1
b j1
a j1 1
b j1 1
z
z1
t e2 )
v1
vt
a j1 2
b j1 2
a j1 1
b j1 1
x
e1 ).
h2 x y
a j1 2
b j1 2
a j1 1
b j1 1
) (h1 , h2 ,0 )
Solving tangle equations
(3) if m 3, hj 0 for all1 j m and hm 0 or
if m 3, hj 0 for all a j m, then
U (
a j1
b j1
a j1 1
b j1 1
a j1 3
b j1 3
a j1 2
b j1 2
) (h1 , , hm )
E[hm , ,h3 ] a j1 1
zt
z1
where h1 e1 and
and N( e2 )
E[hm , ,h2 ] b j1 1
v1
vt
N(
a j1
b j1
a j1 1
b j1 1
a j1 3
b j1 3
a j1 2
b j1 2
xE[h3 , ,hm 1 ] yE[h3 , ,hm ]
e1 ).
xE[h2 , ,hm 1 ] yE[h2 , ,hm ]
Note thatif h2 0 in (2), thesolutionis thesame as thesolutionin (1).
Solving tangle equations
Theorem [Hyeyoung Moon, D]
For t 3,
N(U
z
0
a
x
z
) N( ) and N(U ) N( 1 t e2 )
1
b
y
v1
vt
where a,b,x,y,zj ,v j ,e2 are integersand 0 z j v j for1 j t
and U is a generalized M - tangle.
if and onlyif for t 3 ,
U (
c1 pa c1b
pa c1b c1
) (h,0 )and (
) (h,0 ) for all integers
d1 d1b qa
d1b qa d1
c1 ,d1,p and q such thatd1 p-qc1 1 and 0 c1 d1 where h is an integer
x
z1 z 2 z3
c
pa c1b
x
such that
is not an integer.In thiscase, N( ) N( 1
).
hx y
v1 v2 v3
d1 d1b qa hx y
Note that thechoiceof c1 and p such thatd1 p-qc1 1 has no effecton U .
Solving tangle equations
Theorem 2.3 [Hyeyoung Moon, D]
For s 3,
N(U
a
0
a
x
z
) N( 1 s e1 ) and N(U ) N( )
1
b1
bs
y
v
where ai ,bi ,e1 ,x,y,z,vare int egersand 0 ai bi for1 i s ,
and U is a generalized M - t angle.
if and onlyif for s 3 ,
U (
c1 pz c1v
x
pz c1v c1
x
) (h,0 ) ( ) and (
) (h,0 ) ( )
d1 d1v qz
y
d1v qz d1
y
for all int egersc1 ,d1,p and q such t hatd1 p-qc1 1 where h is an int eger
x
is not an int egerfor y ' such t hat yy' 1 1 mod x.
'
hx y
a
a a
c
pz c1v
x
In t hiscase, N( 1 2 3 e1 ) N( 1
).
b1 b2 b3
d1 d1v qz hx y '
such t hat
Not e t hat t hechoiceof c1 and p such t hatd1 p-qc1 1 has no effecton U .