Topology with Biological Applications
John M. Sullivan
Institut fur¨ Mathematik, Technische Universitat¨ Berlin Berlin Mathematical School
Geometry and Shape Analysis in Biological Sciences IMS, NUS, Singapore 2017 June 13 Berlin opportunities
International graduate school Courses in English at three universities www.math-berlin.de
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 2 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions Geometric energies for surfaces
Symmetric quadratic function of principal curvatures ZZ E := a + bK + c(H H )2 dA − 0
Gauß–Bonnet says RR K = const. thus irrelevant for variational problems If symmetric and fixed area
1 ZZ W := H2 dA 4π elastic bending energy for surfaces Scale-invariant, even Mobius-invariant¨
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 3 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions Lipid bilayer membranes
Lipid vesicles as models of cell membranes Hydrophobic tails hidden by hydrophilic heads Fluid surface of lipid (surfactant) molecules: free to shear
Images from NASA Ames and Vrije Univ. Amsterdam Minimize W (or perhaps RR (H H )2 if sides differ) − 0 John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 4 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions Physical constraints
Fixed volume enclosed Fixed surface area Fixed ∆A = RR H dA
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 5 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions Toroidal vesicles
Studied by Xavier Michalet (now UCLA) Agree with W-minimizing simulations
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 6 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions Higher-genus vesicles
Again studied by Michalet Demonstrate Mobius¨ invariance
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 7 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions Lipid membranes as tubes
Work with Sahraoui Cha¨ıeb (now KAUST)
Pearling by changing H0 or ∆A
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 8 / 39 Bending Energy: Lipid Vesicles and Sphere Eversions CMC trinoid pump
3-parameter family of 3-ended CMC surfaces
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 9 / 39 Sphere Eversion Minimax Sphere Eversion
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 10 / 39 Sphere Eversion Sphere eversion
Turn a sphere inside out Mathematical rules Not too hard (embedded) Not too easy (hole or crease) Possible [Smale 1959] But no explicit eversion for many years [Phillips 1966] Must have quadruple point [BanMax 1981] Simplest sequence of events [Morin 1992] Usually work from half-way model Suffices to simplify this to round sphere
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 11 / 39 Sphere Eversion Minimax eversion
Energy k for surface with k-tuple point ≥ Spheres critical for W known [Bryant] Lowest saddle at W = 4 Use this as halfway model for eversion [Kusner]
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 12 / 39 Ropelength Geometric Knot Theory
Geometric properties of knotted space curve determined by knot type or implied by knottedness (e.g. Fary/Milnor:´ TC > 2πbr 4π) ≥ Optimal shape for a given knot usually by minimizing geometric energy
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 13 / 39 Ropelength Ropelength
Definition Thickness of space curve = 2 reach = diameter of largest embedded normal tube Ropelength = length / thickness
Positive thickness implies C1,1 Definition Gehring thickness = minimum distance between components
works with Milnor’s link homotopy
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 14 / 39 Ropelength Ropelength
Inventiones 150 (2002) pp 257–286, arXiv:math.GT/0103224 with Jason Cantarella, Rob Kusner Results Minimizers exist for any link type Some known from sharp lower bounds Simple chain = connect sum of Hopf links Middle components stadium curves: not C2
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 15 / 39 Ropelength Minimizers
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 16 / 39 Ropelength Lower bounds
Geom. & Topol. 10 (2006) pp 1–26, arXiv:math.DG/0408026 with Elizabeth Denne and Yuanan Diao
Theorem K knotted = ropelength 15.66 (within 5% for trefoil) ⇒ ≥
Proof uses essential alternating quadrisecants:
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 17 / 39 Ropelength Quadrisecant
Line intersecting a curve four times Every knot has one (Pannwitz – 1933 Berlin)
Three order types
simple flipped alternating
Theorem (Denne thesis) Every knot has an essential alternating quadrisecant
(Essential means no disk in R3 r K spans secant plus arc of K.)
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 18 / 39 Ropelength criticality Criticality
Balance Criterion: tension vs. contact force Characterizes ropelength-critical links by force balance
S
c c C1 1 2 C2
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 19 / 39 Ropelength criticality Criticality papers
Gehring case – no curvature bound Geom. & Topol. 10 (2006) pp 2055–2115, arXiv:math.DG/0402212 with Jason Cantarella, Joe Fu, Rob Kusner, Nancy Wrinkle
Ropelength case – with curvature bound Geom. & Topol. 18 (2014) pp 1973–2043, arXiv:1102.3234 with Cantarella, Fu, Kusner
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 20 / 39 Ropelength criticality Kuhn–Tucker
Minimization problem n Given f , gi : R R, (i = 1,..., m), minimize f (p) s.t. gi(p) 0. → ≥ Definition p is a constrained critical point for minimizing f if for any tangent vector v at p with f , v < 0 we have g , v < 0 for some active g . ∇ ∇ i i p is balanced if f = nonnegative combination of active g . ∇ ∇ i Theorem (Modified Kuhn–Tucker) p is constrained-critical balanced. ⇐⇒
Only involves derivatives of f and gi at p (linear functions). Linear algebra Farkas alternative ∼ John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 21 / 39 Ropelength criticality Infinite dimensions
Strategy: combine Clark’s theorem on derivatives of min-functions with our version of K–T that applies to: X vector space (of variations), Y compact space (of active constraints), C(Y) = continuous functions , C∗(Y) = Radon measures . { } { } Theorem (CFKSW’06) For any linear f : X R and g: X C(Y), t.f.a.e.: → → Balanced: nonneg. Radon meas. µ on Y s.t. f (ξ) = R g(ξ) dµ. ∃ Y Strongly critical: ε > 0 s.t. ∃ f (ξ) = 1 = y (gξ)y ε. − ⇒ ∃ ≤ −
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 22 / 39 Ropelength criticality The clasp
Clasp: one rope attached to ceiling, one to floor Again with semicircles?
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 23 / 39 Ropelength criticality The Gehring clasp
Gehring clasp has unbounded curvature (is C1,2/3 and W2,3−ε) Half a percent shorter than naive clasp
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 24 / 39 Ropelength criticality The Gehring clasp
Gehring clasp has unbounded curvature (is C1,2/3 and W2,3−ε) Half a percent shorter than naive clasp
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 25 / 39 Ropelength criticality Example Tight Link
Critical Borromean rings – IMU logo maximal (pyritohedral) symmetry, each component planar piecewise smooth (42 pieces in total) some described by elliptic integrals
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 26 / 39 Ropelength criticality Borromean Rings
Uses clasp arcs and circles; 0.08% shorter than circular Curvature < 2 everywhere = also ropelength-critical ⇒
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 27 / 39 Ropelength criticality The tight clasp
Tight clasp slightly longer Kink (arc of max curvature) at tip
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 28 / 39 Ropelength criticality The tight clasp
Tight clasp slightly longer Kink (arc of max curvature) at tip
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 29 / 39 Ropelength criticality Other ways to find critical points
Symmetric criticality for tight knots Cantarella, Ellis, Fu, Mastin: JKTR, 2014
Gordian split link Coward, Hass: Pacific J, 2015
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 30 / 39 Ropelength criticality Biological applications
Knotted DNA Great source of motivation for geometric knot theory Are tight shapes correlated with ensemble average shapes?
Image from Andrzej Stasiak, EPFL
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 31 / 39 Ropelength criticality Periodic links
Links in 3-torus Closed and infinite components when lifted to R3 Not enumerated yet Can extend ropelength criticality theory Work with Myf Evans ++
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 32 / 39 Ropelength criticality Periodic links
Singly periodic case Links in solid torus S1 D2 × Diagrams in annulus Enumeration of small knots extended to links by Franziska Schlosser¨ Stress/strain relationship as we vary periodicity
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 33 / 39
T G
oue1 (2014) 18 Volume , opology & eometry da,w hw(hoe 9 htaynnrva nthsrplnt at ropelength has knot these we nontrivial Combining any and 14). that number, (Proposition 19) number crossing least (Theorem bridge the show the equals we least this ideas, at is that it conjecture prove we knot: cone allel has knot nontrivial 18, any number which Theorem least from In point at ideas. a additional angle of two existence on the rely prove and we intricate more are mates terms in 21) (Theorem differ- bounds a ropelength need the better we of get zero, number we linking here with approach: convert links ent to For surface bound. inequality cone length isoperimetric a a the of to area this and the component, estimate given for to the sharp argument calibration spanning never a is using bound by this it tain that believe We 11. Theorem ber minimizers ropelength prob- that link show Gehring the examples be to new not built solution need Our link the Hopf was [ES,Oss,Gag]. the 1. which was lem Figure circles, link in tight round shown a two chain of from simple example the known construct previously as to only such us The links, allowing cases, tight simple of many families in infinite sharp is bound to This al. linked et 10). Cantarella rem is least Jason component at surrounding one is curve if shortest length thickness, span- its unit a of then and link others, tube a normal For the surface. of ning intersections considering by proved length, be not need minimizers ropelength that shows it ropelength 10; has Theorem configuration by This tight ends). is the and at circles (with curves stadium 1. Fig. 2 idi n oto n-nh(imtr rope? (diameter) one-inch under of ropelength foot has one in knot rope- tied any has whether trefoil of tight tion the that around suggest length [Pie] algorithm SONO Pieranski’s was [LSDR] bound Ropelength criticality
et foecmoeti nttikesln a oa ikn num- linking total has link unit-thickness a in component one if Next, rope- for bounds lower new several are paper the of results main The nttikeskoshv iia oe onso egh u h esti- the but length, on bounds lower similar have knots Unit-thickness n h egho h ern ls adtegnrccap hc nldsaGhigac is arc) Gehring a includes which clasp, generic the (and clasp Gehring the of length the
4 ipecanof chain simple A
ihteohrcmoet,te t eghi tleast at is length its then components, other the with Singly periodic links smttccosn number crossing asymptotic exactly, computed be can clasps transitional and kinked the of length excess the While ⇡
fako,wihmaue o aytmsi rse vrapar- a over crosses it times many how measures which knot, a of
+2 osrit oteecs eghms emntnclyincreasing. monotonically be must length excess the so constraint,
nrae,w r teghnn h curvature the strengthening are we increases, As constraint. thickness any of absence
C
⇡
2 the in length infimal the be would which tetrahedron, bounding the of inradius the 32 p
4
n edntb unique. be not need and ,
ls ob h ifrnebtentelnt ftecapadfu times four and clasp the of length the between difference the be to clasp our of / ; ⌧ ⇡ . 2
.
66
nScin5 eitouethe introduce we 5, Section In .
needn fapriua onigttaern edfiethe define we tetrahedron, bounding particular a of independent xeslength excess / ; ⌧ ⇡ `.
u mrvdetmt tl evsoe h l ques- old the open leaves still estimate improved Our . k naway a in , different but same the with clasps various of length the compare To 5 ⌧ 6 ⇡
. 83 clasps tight the of Geometry 7.6 2 ⇡ ig,tecnetsmof sum connect the rings, ⇡ n
15
⇡ ijitui-aisdssi h ln (Theo- plane the in disks unit-radius disjoint up. show to narrow too is regions
. ⇠
ewe h ikadGehring and kink the between 71 length of segment shorter much 000224 : 0 a 21
2
optreprmns[DP using [SDKP] experiments Computer . The wide. pixel one about border black thin a as visible barely is sections
⇡ ⇠ fFeda n e[FH]. He and Freedman of .
ewe h ern n shoulder and Gehring the between length of segment longer 003878 : 0 45 b
+ h etpeiul nw lower known previously best The . The black. in included are segments straight two The colors. white and blue
P
2022 alternating in shown are clasp the of J arcs Cantarella, kinked and J H G Fu, Gehring, R B Kusner shoulder, and“tail,” J M Sullivan
n
2
ls.Fo ett ih,testraight the right, to left From clasp. the show figures These 8: Figure / ; 1
where , . 1 k 24
1 ⌧ hti:Cnako be knot a Can is: that , arcsin oflns a ebitfrom built be can links, Hopf P n aallovercrossing parallel stelnt fthe of length the is
2 John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 34 / 39 n> ⇡ +2 (4 C ⇡ 1 2 . ⇡ 4) + eob- We . p Figure 4: The clasp is the simplest configuration of two interlooped arcs. On n k
by , the left, we see the basic clasp where the endpoints are constrained to lie in
n parallel planes. On the right, we have the angled clasp where the four ends 8
of the rope make an angle of arcsin ⌧ with the horizontal. We will study
hscmltstepofo hoe ..Apcueo h ls per nFgr 8. Figure in –critical appears clasp clasp configurations the of picture for A varying 7.8. values Theorem of of ⌧ and proof the . completes This
npitcniin fTerm41 r aifida well. as satisfied are 4.13 Theorem of conditions endpoint
z
Ä snra otecntan lnsa h npit fteac othe so arc, the of endpoints the at planes constraint the to normal is before, As In the clasps we discuss next, the ends of each arc — attached the boundaryC planes —
are straight segments. Clearly we could extend these to be infinite rays and talk about a ern rsoe h aac qainb construction. by equation balance the obey arcs Gehring
D
complete clasp. It would be balanced by the same compactly0 supported strut and kink ote byteblneeuto swl.Frhr the Further, well. as equation balance the obey they so , have and force strut no 0 T
u hsi reb 2a.Tesrih emnsbear segments straight The (27a). by true is this But . arcsin measures used for the compact clasp. is kink the of angle / 2 = ⌧ .
2038 7 Sullivan The M J tight and Kusner clasp B R Fu, G H J Cantarella, J
Our next example is a variation on the “simple clasp” which we considered previously in [4, Section 9]. This clasp is a system of two interlooped ropes as in Figure 4 (left), one anchored to the floor and one to the ceiling. We studied the problem of minimizing the total length subject to the Gehring condition that the two strands are everywhere separated by at least unit distance, that is, that the link-thickness is at least 1. In fact, we considered the entire family of “⌧ –clasp” problems, 0 ⌧ 1, in which the four ends of the two ropes are no longer vertical but make an angle of arcsin ⌧ with the horizontal. (Thus the case ⌧ 1 is the basic clasp described above.) In each case we D described in detail a critical configuration (a “Gehring clasp”) that we conjectured to be minimizing. Surprisingly, for ⌧ 1 the Gehring clasp is a C 1 curve with unbounded D curvature (that is, not C 1;1 ). Here we consider the analogous problem in the more physically realistic setting of the present paper where the constraint is Thi 1. Where the Gehring ⌧ –clasp would have curvature greater than 1= , our –critical ⌧ –clasp now has a kinked arc. Note that the struts in these critical clasps always connect one component to the other. Thus
eometry & opology, Volume 18 (2014) G T Ropelength criticality Rainbow loom bands
“Brunnian link making device and kit”
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 35 / 39 Ropelength criticality Doubly periodic links
Links in thickened torus T2 I × Links in thickened surfaces equivalent to Kaufmann’s virtual knots
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 36 / 39 Ropelength criticality Triply periodic links
Chiral rod packing Π+ becomes close to helical
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 37 / 39 Ropelength criticality Periodic entangled structures
Entanglement at mesoscale Important in physics of soft materials Can lead to exotic macroscopic properties Like negative Poisson ratio
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 38 / 39 Ropelength criticality Periodic entangled structures
Keratin filaments in skin cells under expansion (Evans)
John M. Sullivan (TU Berlin) Topology with Biological Applications 2017 June 13 39 / 39