Writhing Geometry of Open DNA

12 Pages·0.28 MB·English

by  V. Rossetto

Writhing Geometry of Open DNA
V. Rossetto, A.C. Maggs
Laboratoire de Physico-Chimie Th´eorique, UMR CNRS-ESPCI 7083,
10 rue Vauquelin, F-75231 Paris Cedex 05, France
Motivated by recent experiments on DNA torsion-force-extension characteristics we consider the
writhinggeometryofopenstiffmolecules. Weexhibitacyclicmotionwhichallows arbitrarilylarge
twistingoftheendofamoleculeviaanactivatedprocess. Thisprocessissuppressedforforceslarger
3 thanfemto-Newtonswhichallowsustoshowthatexperimentsaresensitivetoageneralizationofthe
0 C˘alug˘areanu-White formula for thewrithe. Using numerical methods we compare this formulation
0
of thewrithe with recent analytic calculations.
2

n
a I. INTRODUCTION results. We show that a strong correction to scaling can
J beexpectedduetotheformationofrareloops,whichgive

4 however exceptionally large contributions to the writhe.
Recent experiments in which DNA molecules are ma-
1 We find that the window of forces in which simple an-
nipulated [1, 2, 3] with the help of magnetic beads
] have led to a renewed interest in the statistical me- alytic theories, based on the Monge representation, are
h chanicalpropertiesofthe torsionallystiffwormlikechain valid is rather small.
p Insinglemoleculeexperiments,self-avoidanceconfines
[4, 5, 6, 7, 8]. In the experiments a bead is attached to
- the polymerto a singleinvariantknot. Analytic theories
o the end of a long DNA molecule; the other end remains
are unable to estimate the error in writhe due to sum-
i stuck to a surface. The bead is held in a magnetic trap
b ming over both knotted and unknotted configurations.
which allows the simultaneous application of a force, F
.
s and couple, Γ; one measures the mean distance of the We investigate this question numerically.
c
bead from the surface as a function of F and Γ. In an
i
s ingenioustheoreticalpaperitwasshown[8]thatthesta-
y
tisticalmechanicsofthisproblemarerelatedtotheprob- II. WRITHE AND LINKING NUMBER
h
p lem of a quantum particle in a magnetic field. However,
[ a crucialassumption was made in the formulationof the A. Closed curves

writhing geometry of the polymer.
1
v The concept of writhe was originally defined in the Thelinkingnumber k ofaclosedribbonisaninteger
8 context of studies on closed ribbons where it forms one topological invariant [9L, 10]. It can be decomposed into
2 partofatopologicalinvariant,thelinkingnumber[9,10]. two parts the twist, w and the writhe, r:
0 In this paper we show how the writhe can be usefully T W
1 generalized to study the geometry (and the mechanical k = r+ w. (1)
0 response functions) of open DNA molecules. This gener- L W T
3
0 alization introduces end corrections to the C˘alug˘areanu- Thisdecompositionisusefulbecausethewritheisafunc-
/ White formula for the writhe. We then perform simula- tion of the centerline r(s) of the ribbon.
s tions for the writhe distribution of an open chain which
c
ysi wexepecroimmpeanrtes iwnvitohlvtihneg manaanlyiptuiclatthioenoroiefsD. NWAewsihtohwbethadats Wr = 41π ds ds′ rr((ss))−rr((ss′′))3·drd(ss)×drd(ss′′). (2)
h inunconfinedgeometrieshaveunboundedfluctuationsin Z Z | − |
p the measured torsional angle. Under an exterior torque The linking number is invariant under deformations of
v: a bead can rotate an arbitrarily large angle. When the the shape which do not introduce self intersections. If
i DNA is under tension this rotation is an activated pro- during a deformation the centerline r(s) crosses itself
X cess with jumps of 4π in the mean angle. there is a discontinuity of 2 in k and thus, as w is
r Contrary to the calculation performed by Bouchiat continuous, a discontinuity of 2Lin r. Fuller [1T2, 13]
a W
and M´ezard [8] we find that this unbounded response is showed the integral of equation (2) could be simplified
not removed by the introduction of an intermediate cut and introduced the expression
off. However applications of tensions larger than femto-
Newtons suffice to render the problem finite in practice 1 eˆ (t t˙)
[11]. We find that their regularized expressions strongly WrF = 2π 1z+· t×eˆ ds (3)
overestimate the magnitude of writhe fluctuations for Z · z
tensions larger than femto-Newtons. where eˆ is the directionat both extremities of the open
z
Inordertointerpretexperimentsinwhichthemolecule chain. In this simplification information is lost so that
is under strong tension Moroz and Nelson used a Monge r and rF are related by the equation
W W
representationtoperformacalculationofwrithefluctua-
tions [6]. We numericallyinvestigatethe validity oftheir r rF =0 mod2 (4)
W −W
2
For notational convenience let us introduced the angles,
χ = 2π r and χ = 2π rF. A much more di-
C F
− W − W
rect approachto Fuller’s result is possible [14] by noting
thatthe writheofastiffpolymeris closelyrelatedtothe
geometricanholonomiesdiscoveredbyBerry[15]inwave
phenomena.
Equation(2)hasasimplegeometricinterpretation[16].
The projected path of a chain on a plane with normal u
can intersect itself. Each crossing is assigned a number,
1,accordingtothehandednessoftheintersection. The
±sumofthesenumbersn(u)isthewrithenumberfordirec-
tionuandtheexpression,eq.(2),isequaltotheaverage
of n(u) over all directions.
B. Bead rotation is given by an extended definition
of writhe
FIG. 1: A flat ribbon with zero linking number.
The above definition of the writhe, with its emphasis
on it being part of a topologicalinvarianthides, to some
sections go to infinity. In this limit the contribution to
degree,theinterpretationofχ asanangleofrotationin
C the integral of the section vanishes. Thus the experi-
many experimental situations. We shall now show that C
mentalrotationangleisgivenbytheχ fortheextended
C
an extended definition of the writhe, based on equation
constructwiththetwostraightlinesectionsextendingto
(2) is the writhe contribution to the bead rotation that
infinity [31].
is measured in DNA twisting experiments. Rather simi-
Consider generating an ensemble of chains with some
lar arguments have also been given in [7] for a polymer
arbitrary algorithm. The distribution of the configura-
betweentwoplanes,we givehearanextendedderivation
tionsofthechainsisindependentofthedynamicprocess
to point out the exact limits of the result.
creatingthemiftheyaresubjecttoBoltzmannstatistics.
Consider the planar ribbon in figure 1. The linking
If we now choose unknotted configurations and demand
number, k, is zero thus
that they be created via a process which preserves the
L
linking number we conclude that we can uniquely deter-
r+ w =0. (5)
W T mine the rotation angle from the writhe of the extended
We shall use this closed ribbon as a reference configu- construct. A crucial part of this argument is the restric-
ration to calculate the writhe of an open filament for tion of the construct of figure 2 to the sector k = 0.
L
which the initial and final tangents are parallel i.e. for We discuss the importance of this restriction in the next
t(0) = t(L), where L is the polymer length. We do this section.
using the geometry of figure 2. The polymer which is
P
alsoaribbonis embeddedinaconstructionconsistingof
twolongstraightsections 1, 2andaclosingloop . We C. Choice of linking sector
S S C
shallapplyeq.(2)to the constructionsoffigures1 and2
then take the length of the straight sections to infinity. In the deformation from figure 1 to figure 2 we wish
There are beads attached to each end of the polymer, to conserve the linking number k. To do so we must
L
joining on to the straight sections. These two beads are firstly forbid self crossing of the polymer configuration.
ourexperimentalreference. Weshallholdthelowerbead In any experiment with DNA in the absence of topology
1stationaryandlettheupperbead 2rotateduetothe changingenzymesthisrestrictioninreasonable. However
B B
writheofthepolymer. Thefinalpartofourconstruction this condition is insufficient to conserve the topology of
is the “twist absorber” . This is imagined as being a the entire construction. We must also forbid the cross-
T
joint,orsectionofthestraightribbonwhichtwistsfreely ing of the real polymer with the imaginary line from the
to absorb the writhe generated by the polymer. Let the end of the chain to infinity. In a recent numerical paper
twist of the polymer be zero so that all twist appears in [7] this wasachievedby graftingthe ends of the polymer
. to an external surface rather than beads. Experimen-
T
Start from the reference state of figure 1 and deform tally we would stop this crossing by introducing a steric
continuouslytoanarbitrarystatefigure2withoutgener- hindrance: Afinefiberintheneighborhoodofthemanip-
ating a selfintersectionofthe construct. The writhe can ulatingbeadwouldworkperfectly. Weshowbelow,how-
be calculated with the classic double-integral. Clearly ever,thateventhislevelofsterichindranceisnotneeded
since r+ w = 0 the writhe which is generated goes in the experiments as they are presently performed.
W T
into twisting the region of the chain. The total twist- Ifthechainisallowedto bendbackinsuchawaythat
T
ing angle is just 2π r = χ . Now let the straight is passes through the line to infinity there is a discon-
C
− W
3
against the magnetic force a distance comparable to its
diameter in order to force a loop over the point of at-
S
2 tachment. Formicronsizedbeadsthisintroducesachar-
acteristic scale of the force of k T/µm fN. For forces
B
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)T of more than a few femto-Newtons the∼process is expo-
B nentiallyrareandsuppressed. Thenaturaltimescalefor
2
attempts at crossing the barrier is given by the Zimm
time of the bead τ d3η/k T with d the bead size,
C ∼ B
P which is seconds.
This force is extremely low, the widely studied
crossover from random coil to semiflexible behavior in
B
1 the force-length characteristics of DNA occur at a force
scale of k T/ℓ with ℓ 53nm the persistence length
B p p
≈
S 1 of DNA. This intrinsic force scale for DNA much higher
than that which is needed to conserve linking number
with a 1µm bead. We conclude that most experiments
with tense DNA are performed in a regime where a high
FIG.2: Deformtheribbonoffigure1asinthediagram: The
energy barrier leads to conservation of the linking num-
initialandfinaltangentsofthepolymersection areparallel.
P ber ofthe constructof figure2. However,there is always
Long arms 1 and 2 attach the polymer to a closing loop
C. The expeSrimentaSlly important reference beads 1 and 2 a small probability of passageby this barrier so that un-
are attached to the two ends of the polymer. ThBe twistBis der torsion the true steady state of a DNA molecule is
absorbedexclusivelyinthesection . Notethebeadsarenot a state of cyclic motion in which the bead rotates by a
T
in general aligned vertically. series of activated jumps.
Because of these considerations we assume that all
topological invariants are conserved during experiments,
including linking number and the knot configuration of
the extended construct of figure 2.
D. Analytic expressions for the extended writhe
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) From the above discussion we are lead to the calcu-
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
lation of the writhe of the extended construct, figure 2.
FIG. 3: Demonstration of a cyclic motion of DNAleading to The double integral, eq. (2), on the interior of the chain
introductionofatwistof4π. Inthefirstandfourthfigurethe is identical to that commonly used for closed DNA, we
shape of the molecule is identical. Between the second and denote the corresponding angle by χ .
int
thirdfiguretheverticalextensionofthebeadpasses through The integral from one point on the linear extension to
themoleculeleadingtoadiscontinuityinthelinkingnumber.
infinityplusthesecondpointonthepolymercanbesim-
The tricky three dimensional geometry is most easily under-
plified as follows: The integral is evaluated by noticing
stood by repeating this sequence with the help of a belt or
that it is the spherical area, swept out by the vector
ribbon.
r(s) r(s′)
ˆe(s,s′)= − (6)
r(s) r(s′)
tinuity of ∆ k = 2. This allows one to construct a
| − |
L ±
cycle of motions in which the shape of the filament un- as s and s′ vary. Consider, now, a chain, r(s). Place
dergoes a cycle coming exactly back to its original shape
the origin at r(0). Let us now place s in the interior of
while the beadundergoesarotationof4π aboutthe ver- ′
the polymer and s on the extension of the polymer to
tical axis,figure 3. This result is the originofa common
infinity which is directed in the direction eˆ . Consider,
amphitheater demonstrationofthe importance of spinor now, s′ = 0. The curve uˆ(s) = ˆe(s,0) isza spherical
representationsoftherotationgroup: Ifoneholdsaplate ′
curve. As we now let s vary from 0 to infinity we sweep
horizontally in the palm of one’s hand one can spin it
out the area between uˆ and the point eˆ . This spherical
abouta verticalaxisby performing a suitable contortion z
area can be written as
of the arm. Againstall intuition the plate can be turned
an arbitrarilylarge angle. Photographsof Feynman per- uˆ duˆ/ds
formingthisdemonstrationaretobe foundin[17]. Each χend =eˆz· ds 1×+uˆ eˆ . (7)
cycle of the arm again gives rise to a rotation of 4π of Z · z
the plate. thenχ =χ +χ . Asimilarresulthasbeenfound[8]
C end int
This unbounded torsional fluctuation is an activated bydirectintegrationofeq.(2). Weareratherremarkably
process: WhentheDNAisundertensionthebeadmoves back to a problem in statistical mechanics which is very
4
closethatoftheFullerformulation,eq.(3),ofthewrithe where 2 is the Laplacian operator on the sphere. Here
fluctuations. We converteq. (7) to spherical coordinates P(t,s)∇is the probability of finding the chain oriented in
and find: the direction t at the point s. As a function of s the
vector t “diffuses” with diffusion coefficient which varies
dφ
χend = ds(1 cosθ) (8) as 1/ℓp.
− ds
Z LetusnowconsiderthewrithecalculatedwithFuller’s
The singularityat1+uˆ eˆ =0 correspondsto the limit formulaforsuch“diffusing”paths. Thereisasingularity
z
cosθ = 1. · in equation (3) near t = eˆz. In the neighborhood of
− this direction the integralm−easures twice the winding of
the random walk about the point eˆ . The winding of
z
−
arandomwalkhas singularpropertiesin the twodimen-
E. Twisting
sional plane and on a sphere [18]. In particular there is
logarithmic divergence in the winding properties in the
Experiments are sensitive to the sum of the writhing
continuum limit. It is this winding number divergence
and twisting fluctuations in a polymer. If we now allow
that was picked up in the analytic calculation of [8] and
the excitation of the torsional modes in the polymer of
necessitated an intermediate scale cut off in the calcula-
figure 2 the total rotation between the two ends is the
tion.
sum of the writhing angle χ and rotation due to in-
C
ternally excited twisting motions. In order to compare One of the principle conclusions of this paper is that
the twisting fluctuations to writhing fluctuations, we de- thesewindingnumbersingularitiesarenotpresentinthe
fine ℓ = C/k T the torsional length, where C is the distributionofthewrithedeterminedexperimentally: As
t B
torsional modulus [8]. This twisting mode is unmodified shownaboveinthecaseofconstrainedlinkingnumberit
bythewrithinggeometry,atleastinthesimplestmodels istheC˘alug˘areanu-Whiteexpressionforthewrithewhich
of DNA elasticity. The mean square twisting angle, χ givesthe exacttwisting angle. The Fuller expressiondif-
T
is given by χ2 = L fers by an arbitrary factor of 4nπ due to the winding
h Ti ℓt
number singularities not present in the original formu-
lation. We shall demonstrate numerically that it is the
passage from (2) to (3) which introduces these singular-
III. ARTEFACTS IN THE FULLER
FORMULATION OF WRITHE ities. Use of (3) will be shown to lead to a substantial
error in the calculation of χ with an overestimate by a
C
factor of 2.5 for a discretization corresponding to DNA.
Recentanalyticwork[6, 8]hasbeenbasedonthe sim-
plified formulation of eq. (3). It is much easier to treat
analytically than the full double integral in equation (2)
due to a direct mapping onto quantum mechanics: The
bending energy of a stiff beam in the slender body ap-
proximation of elasticity is given by IV. NUMERICAL METHODS
A dt(s) 2
E = ds , (9) Given the difficulty of treating the full C˘alug˘areanu-
2 ds
Z (cid:18) (cid:19) White expression for the writhe analytically we decided
to proceed by numerical exploration of the distributions
whereAisthebendingmoduluslinkedtothepersistence
of writhe implied by the C˘alug˘areanu-White and Fuller
length by ℓ =A/k T
p B formalisms. In particular we look for the Cauchy tail
To calculate the partition function one must now sum
predictedanalytically[8]in the writhe distributionfunc-
over all paths
tion. The existence of such a tail implies the absence
of a reasonable continuum limit of the wormlike chain
= e−E/kBT (10)
Z and continuous evolution of the response functions as a
pXaths function ofa microscopiccut off. We shallconclude that
the C˘alug˘areanu-White formulation remains finite even
The sum for the partition function is clearly closely re-
in the continuum limit.
lated to path integrals studied in quantum mechanics.
Formally the energy in eq. (9) looks like the kinetic en- Inournumericalinvestigationsweshallbeparticularly
ergyofafreeparticlemovingonasphere. Fromthesum interested in the fluctuations in the writhe of the open
of paths in equation (10) one derives a Fokker-Planck chain,as afunction ofthe tension. Due to the algorithm
equationwhich is entirely analogousto the Schroedinger used in generating the chains we are unable to generate
equation for a particle on a sphere: chains in an ensemble with an imposed torsionalcouple.
Our results are thus always for Γ = 0. The torsional
∂P(t,s) = 1 2P(t,s) , (11) fluctuationare,asusual,relatedtothelinearresponseof
∂s 2ℓ ∇ the chain via the fluctuation-dissipation theorem.
p
5
A. Numerical Calculation of Writhe The ensemble of k = 0 chains may contain knots.
L
While this ensemble may appear physically “unreason-
able”onemustnotforgetthatknotsarerareinthechains
A number of methods are available for the calculation
thatweshallstudy. Ithasbeennoted[20]thatforaflex-
of the writhe of a discretized polymer [19]. We calculate
ible chain unknotted configurations dominate the statis-
the discretized versions of the integrals of eq. (2) and
tics of chains even several hundred Kuhn lengths long.
equation (3) by recognizing that they are both areas on
For the chain lengths that we work with in this paper
a unit sphere. In the Fuller formulation it is the area
enclosed by the curve t(s). For the discretize chain the the contamination coming from such knotted configura-
tangentcurve becomes a seriesof link directions,t , cor- tionsshouldbeweak. Attheendofthepaperwepresent
i
a partial investigation of the influence of knots. We find
responding to points on a sphere. One connects these
thattheydo notmodify ourconclusionsasto the nature
points by geodesics and then sum over the area of the
of the continuum limit for writhing chains. The errors
triangles formed by two successive tangent vectors and
eˆ . due to the use of a knotted ensemble are much smaller
z
thanthedifferencesbetweentheFullerandC˘alug˘areanu-
χF = (eˆz,tn,tn+1) (12) White formulation of the writhe.
A
n
X
where is the area of the spherical triangle defined by
A
the three vectors.
We havealreadynotedthatthe fullexpression,eq.(2) C. Chain generation
corresponds to the area swept out by eˆ(s,s′). For two
linksformingadiscretechainthisdefinesasphericalrect-
In order to use the expressions for the writhe given
angle. We calculate its area by decomposing it into two
above we are interested in chains in which the initial
spherical triangles.
and final tangents are parallel (though the writhe for
Inbothcalculationswecalculatetheareaofaspherical
non parallel configurations does have a simple general-
triangle by using l’Huilier’s expression
ization [14]) but for which there is no constraint on the
final position of the chain. Rather than using a conven-
=4arctan( ( tan((a+b+c)/4) (13)
A × tionalMonte-Carloalgorithmtogeneratechainsweused
tan((c a+b)/4)
a simple “growth algorithm”.
− ×
tan((c+a b)/4)
− ×
tan((a+b c)/4))1/2)
−
where,a,bandcarethelengthsofthesidesofaspherical
triangle. Forourpurposesthetrianglehastobeoriented, 1. Zero force
the area can be either positive or negative.
A useful cross check in the programming is that for
WewishtogrowchainsoflengthL,persistencelength
any polymer the modulo relation of eq. (4) must be sat-
ℓ using a series of links of length b. In the absence of
isfied despite very different intermediate results in the p
tension we generate chains by starting from a single link
calculation. Numericallywefoundthatthe equalityheld in the eˆ direction at the origin. We then successively
to within 10−10, when working in double precision when z
add links to the chain with small random angle incre-
theextendedC˘alug˘areanu-Whitedefinitionofwrithewas
compared with the Fuller formulation. ment α0 ∼ b/ℓp to produce a single realization of an
equilibrated semiflexible chain. It is almost certain that
Since the double integral of equation (2) is reduced to p
this chain does not satisfy the boundary conditions on
a double summation this step takes O(N2) operations
the tangentthus we continuegrowinguntil the finaltan-
fora chainofN links. Forthe longchainsstudiedinour
gent is parallel to the initial tangent to within an angle
simulationsitisbyfarthesloweststepinthecalculation.
smallcomparedwithα0. Wekeepthepolymerinouren-
semble if the length of the polymer is less than 1.05 L,
×
otherwisethewholeconfigurationisrejectedandthepro-
B. Link Sector Choice cess restarted from the first link. The configurations are
then used to calculate the writhing distributions. The
The generation of long unknotted chains is numeri- curves that we generate are somewhat “imperfect” since
cally difficult; In our simulations we used anensemble of they are due to a mixture of lengths. This admixture of
chains with only the constraint k = 0 which, as shown chain lengths plays no role, however, in our analysis of
L
above,is the minimum constraintneeded for unbounded the asymptotic distribution of writhe. There is no self
torsional response. Our result can then be directly com- avoidance in this code; it can be shown from a Flory ar-
paredwithexistinganalytictheorieswhichdonotimpose gumentthatselfavoidanceisaweakeffectinsemiflexible
any topological constraints. chains of moderate length.
6
2. Finite force V. ASYMPTOTICS OF WRITHE IN OPEN
FLEXIBLE CHAINS
In the presence of an external force the algorithm is
slightly more complicated. We proceed by noting that Before presenting our numerical results on semiflexi-
the partition function of a chain under tension can be ble chains we wish to explore the scaling behavior of the
expressedinaverysimilarmanner tothe partitionfunc- writhe of a polymer described as a freely jointed chain
tion of a flexible polymer in an external potential [21]. with N links. This allows us to understand the length
We proceed by simulating the equation scales and the structures which are important in deter-
mining the writhe of a long molecule. Many of these
∂Z 1 results are already known for closed chains however we
= 2Z+fcos(θ)Z , (14)
∂s 2ℓ ∇ wish to demonstrate that the end corrections in eq. (7)
p
are subdominant in the limit of very long chains.
where θ is the angle between the direction of the force
and the local tangent to the polymer, and f = F/k T.
B
Z(s,θ) corresponds the number of configurations in A. Scaling arguments for flexible chains
whichthe chainpoints in the directionθ after a distance
s. Weshallstartwiththeinterpretationofthewritheasa
Asproposedin[22]weintroduceapoolofseveralchains signedareasweptoutbythevectorˆe(s,s′): Considertwo
which we grow simultaneously. As each link is added linksiandjoflengthaseparatedbyadistanceR . This
ij
there is angular diffusion as described above and a sec- area scales as A (a/R )2 when R a. When
ij ij ij
∼ ± ≫
ond process of birth or death of chains in the pool in a R the area is bounded above by 2π. Clearly when
ij
≪
ordertoaccountforthe force. Iffcos(θ) ispositivethen one averagesover random walks the integral in equation
it is considered to be a growth rate for reproduction of (2)giveszero. Themeansquaredwrithecanbeestimated
chains in the pool. If fcos(θ) is negative the chain is as
stochastically destroyed with the appropriate probabil-
ity. We alsomanagethe totalpoolsize asin[22]. Atthe hχ2Ci∼ A2ijP(Rij) (15)
end of a pool growth, we destroy the chains that do not pairs
X
satisfythe conditionforthe tangentvectorto be parallel
to within α0 at both extremities. where P(Rij) is the pair distribution function for the
polymer. For a three dimensionalGaussianpolymer this
There are severalsourcesof errorpossible with the al-
function scales as
gorithm. Themostdifficulttoevaluateistheeffectofthe
finite pool size. The result of a single run is an ensem-
P(R ) 1/R (16)
ble of severalconfigurationstogether with a totalweight ij ∼ ij
coming from the management of the pool size. For suf-
for lengths smaller than the radius of gyration, R
ficiently large pool sizes this weight is the same for each √aL of the polymer. Approximating the sum by anGin∼-
realization of the growth process. For small pool sizes,
tegral we find the internal contribution to the writhe,
however, this weight undergoes important fluctuations.
To calculate a correlation function from an ensemble of
L RG a4 a L
poolswechosetoselectasinglechainfromeachpooland χ2 R2dR (17)
int ∼ a4 R4R ∼ a
performed a simple average over at least 10,000 pools. Za
(cid:10) (cid:11)
Since we had no, a priory method of estimating errors
This integral converges at large distances, but is diver-
from this procedure we experimented with the pool size
gent at small distances. We conclude that the writhe in-
for several different values of the force. We found that
tegralisdominatedbythecutoffscalea. Forasemiflex-
evenwhenvaryingthepoolssizefromaslowas20chains
ible polymerthe cut–offa correspondstothe persistence
to 2000 chains the estimates of the mean square writhe
length ℓ . It is thus the structure at this length scale
werestablewithinafewpercent. Inourproductionruns p
which dominates the writhe of the molecule. The aver-
we chose a value of 50 chains per pool. An alternative
age crossing number is defined in a very similar manner
procedure would weight each pool according to the true
to the writhe except the unsigned area A is summed
variation of the total weight of each simulation. | ij|
over rather than the signed area. This has a non-zero
A systematic difference between a discretized chain
meanandscalesinthefollowingwayinaGaussianpoly-
and a continuous curve also occurs. Our principle aim
mer
is to understand experiments on DNA[1, 2] therefore we
choose ℓp = 53nm. We chose for b the half pitch of a χX L a2 a R2dR LlogL. (18)
single helix, 1.8nm, yielding ℓp 30b. This is also com- ∼ a4 R2R ∼ a a
parabletothediameterofthemo≃lecule. Inordertostudy Z
(cid:10) (cid:11)
the convergenceof the writhe to the continuum limit we leadingtoalogarithmicdivergence. Thusinthecrossing
shallalsoperformsomesimulationswith ℓ /b 30(see properties of an arbitrary projection of the polymer we
p
≫
section VIC). expect all length scales are important.
7
Finally, eq. (7), the end correction to the writhe is a
2
sum of random areas A a/R where R is now the
i i i
∼ ±
distance between the end of the polymer and the single 1.8
link i. The corresponding estimate of the mean square
1.6
writhe is thus
1.4
1 a2 a L
χ2 R2dR log . (19) 1.2
end ∼ a3 Z R2R ∼ a 1/2)d 1
giving a (cid:10)logari(cid:11)thmic contribution with the structure of 2χ(en0.8
the whole molecule being important.
When self avoidance is introduced in the problem we 0.6
use the result that P(R) 1/R4/3[23] to show that the 0.4
∼
integralinequation(17)isstilldominatedbyshortlength
0.2
scales. The results for the end correction and average
crossing number are however modified. They too be- 0
101 102 103 104 105 106
come sensitive to structure at short wavelengths in the L
polymer. Thusfromeq.(17)andeq.(19)theendcorrec-
tions remain small for long chains and are subdominant, FIG.4: Variation ofendcorrection, χ2 ,with Lshow-
χ2 χ2 . ingscaling identical to that of thewindhingenodfiarandom walk
h inti≫h endi p
Given the importance of this end correction in the in- aboutaninfiniteline. Freelyjointedchainswithupto320,000
terpretation of the experiments we now present a more links.
rigorousstudy of the endcorrectionand confirmits sub-
dominant nature compared with the internal contribu-
tions to the writhe. the internalcontributionswhich vary as χ2int ∼(L/ℓp)
for L ℓ .
p
≫ (cid:10) (cid:11)
Despite the similarity between eq. (7) and eq. (3) we
B. Magnitude of End Corrections find very different scaling for χ2 Llog(ℓ /b) and
F ∼ p
χ2 log2(L/ℓ ). At first sight this might seem
In order to further study the scaling behavior of the ratehnedr su∼rprising, hopwever in th(cid:10)e F(cid:11)uller formulation one
endcorrectionsweexaminethelimitL/ℓplarge. Wethus a(cid:10)verag(cid:11)es over realizations of two dimensional random
study the problem of a freely jointed chain rather than walks in the surface of the sphere whereas in equation
thesemiflexiblechainanddisproveargumentsof[24]that (7)weaverageoverthreedimensionalrandomwalkspro-
the dominant singularitiesin the writhe of a semiflexible jected ontoa sphere. The statisticalweightsaredifferent
chain come from ends due to the formal analogies be- even if the functions are similar.
tween eq. (3) and eq. (7). Numerical results (not shown
here)onsemiflexiblechainsleadtothesameconclusions.
Let us now make a hypothesis that the asymptotics of
χ are dominated by the largest contributions to the VI. WRITHE DISTRIBUTION OF
end
integrand of eq. (8) then SEMIFLEXIBLE POLYMERS
dφ
χ ds (20) A. Short Molecules
end
∼ ds
Z
which is the winding number of the polymer about eˆ . ForshortfilamentsoflengthL ℓ thedistributionof
z p
≪
The winding number of a semiflexible random walk, W, writhe calculated with the extended C˘alug˘areanu-White
about an infinite line scales as formulaandtheFullerformulaareindistinguishable;am-
biguitiesduetowindingaboutthepoleareexponentially
W2 log2(L/ℓp) (21) rare. Thewrithedistributionofaopenpolymerwithpar-
∼
allel tangents at each end in the limit L/ℓ 1 is given
where L is the length of chain and ℓp the persistence by the L´evy [14, 25] formula for the distrpib≪ution of the
length. The Cauchy singularity appearing in this prob-
area enclosed by random walk in a plane
lem is regularized by the stiffness of the chain, ℓ .
p
To check this hypothesis (and to improve on the very
ℓ 1
rough scaling analysis given above, eq. (19)) we gener- P(χ )= p . (22)
ated, figure 4, a large number of random walks for a C 2Lcosh2(χCℓp/L)
freely jointed chain. We then plot χ2 as a func-
tion of logL and look for a straight lihnee.ndWie conclude By generating an ensemble of 106 short chains and bin-
p
that the end correctionsto the writhe arecomparable to ning the writhe we verified that our code was able to
χ2 =log2(L/ℓ ). Theyarenegligiblecomparedwith reproduce this result.
end p
(cid:10) (cid:11)
8
B. Long molecules, zero tension
2
Wehavecharacterizedtheevolutionofthewritheprop- 10
fiergtuierseo5ftaopch,aiχn2as aasfuanfcutniocntioonf iotsf lLenfgotrh,ℓL.fixWede.pFlootr, 2>C s l o p e = 1
C p χ
smallL,eq.(22),wehave χ2 L2andforlongchains, <
we have χ2C (cid:10)∼L(cid:11), eq. (18(cid:10)).C(cid:11) ∼ 100 ope=2 (i)
As a second characteristic of the writhe dis- sl
(cid:10) (cid:11)
tribution, figure 5 bottom, we consider ρ4 =
(x x )4 / (x x )2 2 related to the kurtosis, cal- -2
−h i −h i 10
culated for a probability distribution p(x). If p is Gaus-
(cid:10) (cid:11) (cid:10) (cid:11)
sian, then ρ4 = 3. For the distribution eq. (22) ρ4 =
21/5. We see numerically that ρ4 = 21/5 for small L 10-1 100 L / l 101 102
p
and tends slowly to 3. However, there is a large peak
around L/ℓ = 2. The very strong non-monotonic be-
p 8
havior in figure 5 bottom is most striking; we shall now
explain the origin of this feature.
7
Let’scallback-facingsegments thesectionsofthechain
along which t eˆ < 0. We define, n , the number of
z S
· 6
such segments for a chain. If we compute ρ4 with only
(ii)
configurationsforwhichn =0thepeakdisappears. We
S 4
understandthatthepeakisduetotheformationofloops ρ 5
in the chain which can, sometimes, completely dominate
the writhing properties of a chain. Such important fea- 4
ture in the distribution are clearly missed in any Monge
description of a chain. 3
From these simulations, we also estimated the coeffi- 10-1 100 L / l 101 102 103
cient of proportionality K between χ2 and L/ℓ for p
C p
verylongchainswhenthedistributionofwrithehascon-
2
verged to a Gaussian: (cid:10) (cid:11) FIG. 5: Top: χC as a function of L. Bottom: ρ4 as a
function of L/ℓp. When L/ℓp 0 we find ρ4 21/5. There
(cid:10) (cid:11) → ≃
L is a peak around L/ℓp 2, due to the presence of a small
χ2 K K =2.85 0.03 (23) number of loops (typica≃lly one or two), which enlarge the
C ≃ ℓ ±
p probability distribution of the writhe. For larger values of
the coefficie(cid:10)nt is(cid:11)similar in magnitude to that found [26, L/ℓp ρ4 → 3 corresponding to a Gaussian distribution. ♦
corresponds to chains with nS = 0. 100,000 chains for each
27]forclosedchains,eventhoughourmodelforthechain
point, ℓp = 1000b for short chains and ℓp = 50b for long
is somewhat different.
chains.
C. Convergence of the writhe distribution between χ2 and χ2 for DNA value of ℓ = 30b is
F C p
2.4. Wealsoseethatthedifferencebetweenachainwith
(cid:10) (cid:11) (cid:10) (cid:11)
In order to characterizethe asymptotic distributionof 30 links per persistence length with the continuum limit
the writhe, and study the importance of the Cauchy tail is small (about 3%).
in Fuller formulation, we have performed simulations on A divergence of the angular fluctuations implies the
a series of chains of length L =8ℓ . With chains of this breakdown of linear response in the continuum limit.
p
lengthknots remainrather rarewhilst the energetic bar- Since, however, χ2 converges to a finite value we con-
h Ci
rierneededforachainorientedinthedirectioneˆ towind clude that a microscopic cutoff is not needed to render a
z
about the direction eˆ is only a few k T. We are thus torsionally stiff chain finite.
z B
−
sensitivetothe windingsingularitiesofthe Fullerformu-
lation. In our simulations we vary the discretization so
that there are L/ℓ = 10, 30, 100, 300, 900, 2700 links D. Long tense molecules
p
per persistence length and we generate 200000 indepen-
dent configurationsfor eachvalue ofL/ell . We plotthe Until now we have ignored the effect of tension on the
p
varianceofχ andχ asafunctionofthe discretization configuration of the DNA, except to remark that even
F C
infigure6. We observeacontinuousevolutionshowinga very low tensions justify the use of a conserved linking
logarithmic divergence for the Fuller formulated writhe, number in the interpretation of the experiments. In this
andaconvergenceto astable value forthe C˘alug˘areanu- section we indicate how the writhe of DNA varies in the
White formforquite moderatevalues ofL/ℓ . Theratio presence of external forces.
p
9
50 1
L10
DNA
/

40 <χF2> > lp
2
F
χ
0
>30 <10
2
χ
<
20
-1
10
10
<χ 2>
C
0
1 2 3
10 10 l /b 10 -2
p 10
10-1 f l 100 101 102
p
FIG. 6: Variance of the writhe computed with two differ-
ent formulations as a function of discretization. χF evolves FIG. 7: χ2F ℓp/L as a function of the scaled tension. At
logarithmicly,whilstχC convergestoaconstant. DNAcorre- large forces the result converges to eq. (26). At low tensions
sponds toℓp/b≃30, for which χ2F / χ2C =2.4. L/ℓp =8, the writhe(cid:10)is i(cid:11)ndependent of the force. Simulations for ℓp =
200000 chains perpoint. The st(cid:10)atis(cid:11)tica(cid:10)l err(cid:11)oris smaller than 250b. Curves for L/ℓp = 0.25, 0.5, 1, 2, 4, 8, 16, 32 (bottom
symbols’ size. to top). The straight line is eq. (26).
When amolecule is under hightensionthe molecule is
usingthe Fullerexpressionforthe writhe. Athighforces
largelyalignedparallelto the externalforce. We canuse
we see thatthe curvesconvergetowardsthe law eq. (26)
a simplified, quadratic form for the Hamiltonian [6]
and that at low forces the curves saturate as expected
1 from the above scaling argument. Somewhat surprising
E = 2 K(∂t⊥)2+Γ(t⊥∧t˙⊥·eˆz)2+kBTft2⊥ ds however is the rapid crossover which occurs for forces
Z n o (24) comparable to f 1/ℓp when L/ℓp 1. In the neigh-
where t⊥ describes the transverse fluctuations in the di- borhoodofthisfo∼rcethereisapronou≫nced“shoulder”on
rection of the molecule. We can find the mean squared the curve. Convergence to the long chain limit is rather
writhing angle using the usual methods of equilibrium slow. It is not until L 60ℓp that we see a saturation
∼
statistical mechanics. in writhing curves. This slow convergencecan be under-
stood rather easily by noting that any chain within ℓ
∂2log p
χ2 = Z (25) of the surface of the polymer coil is in a region of lower
F ∂ (βΓ)2
than average density.
(cid:10) (cid:11)
A short calculation gives We have performed a series of simulations on differ-
ent levels of discretizationofthe polymers. We find that
1 1 L as the discretization becomes coarser the large shoulder
χ2 = . (26)
F 4sfℓp ℓp dominates over the law in 1/√f for the mean square
writhe. This is illustrated in figure 8 where we use 30
(cid:10) (cid:11)
Thisexpressioncanonlybeexpectedtobevalidforforces links per persistence length for the discrete chain. The
such that fℓp >1. chainoflengthL=ℓpstilldisplaysaregimeinagreement
To estimate the writhe of a molecule at low forces we with eq. (26). However with longer chains the regime in
return to the remark above that the internal contribu- 1/√f isoverwhelmedbythecrossovertothelowtension
tion to the writhe is dominated by structure occurring regime. Eq. (26)substantiallyunderestimatesthewrithe
at the scale ℓp. Under low tensile forces the structure of fluctuations in the domain out to fℓp = 10 correspond-
a polymer is unchanged out to a length scale ℓ = 1/f. ing to very large forces of 1pN. At such forces other
f ∼
We conclude that under low tension, when ℓ ℓ the correctionscome into play,including the chiralnature of
f p
≫
writhe of a molecule becomes independent of its degree the DNA chain.
of elongation and thus the tension, f. It is only un- It is interesting to note that the Fuller formulation
der the highest forces when the semiflexible nature of gives results which are useful over a larger window of
the molecule is sampled that we see an evolution of the forces. The Fuller and C˘alug˘areanu-White curves are
writhe with force. very similar down to tensions fℓ 2; analytic calcu-
p
∼
Tovalidateourcodeforgeneratingtensemoleculeswe lations based on the Fuller formulation can be expected
performed a series of simulations with a chain of per- to be useful for tensions larger than 0.2pN. At very
sistence length ℓ = 250b, results are shown on figure 7 lowforcesthe fluctuationsareratherdifferent. Theratio
p
10
L L
/ p F=0.1 pN / p F=0.1 pN 100 <χC2>total lp /L
> l > l <χ 2> l /L
2 F 2 C L C nS=0 p
χ<100 <χT2> χ<100 <χT2> 2> l /pC eq. (26) <<χχCC22>>nnSS==12 llpp //LL
χ
<
-1
10
eq. (26) eq. (26)
-1 -1
10 10
-1 f l 0 1
10 p 10 10
10-1 10f0 l 101 102 10-1 100f l101 102 FIG. 9: χ2C ℓp/L as a function of fℓp. Circles all chains.
p p Triangles, averaged from no back-facing segment. We have
(cid:10) (cid:11)
plottedthecurvesfornS =1, andnS =2, (+). Eachpoint
×
FIG.8: χ2F ℓp/L(left)and χ2C ℓp/L(right)asafunction c3o0mb.pEuqteudatfiroonm(32060),0so0l0idcolinnfiegurationsoflength8ℓp withℓp =
of the scaled tension. For longer chains the expected regime
(cid:10) (cid:11) (cid:10) (cid:11)
in 1/√f is hidden by a large shoulder from the crossover to
the low force regime. Simulations for ℓp = 30b. Curves for
L/ℓp = 1, 2, 4, 8, 16 and 32 (bottom to top). The straight
boldlineisequation(26). Atzeroforce,theratio χ2 / χ2 until now leads to an ensemble containing both knotted
F C
tends to 2.5 as L/ℓp grows. The horizontal dashed line is and unknotted chains, even if k = 0. We now present
(cid:10) (cid:11) (cid:10) (cid:11) L
the amplitude of fluctuation due to twist fluctuations when a preliminary investigation on the influence of knots on
ℓt/ℓp = 1.5. The vertical dashed line corresponds to F = thedistributionofwrithe. Thisquestionisrelatedtothe
0.1pN for DNA. problem of the closure of the chain: If the polymer is
allowed to pass through the line to infinity, a knot may
appear, change or disappear. We thus continue with the
χ2 / χ2 is about 2.5; use of the Fuller formulation
F C assumption that the tension of the DNA is sufficiently
strongly over estimates the torsional response functions.
high that the extended construct of figure 2 remains in
(cid:10) (cid:11) (cid:10) (cid:11)
the same state throughout an experiment.
In order to select the knotted chains, we have com-
E. Origin of the shoulder
puted the Jones polynomial V [28, 29]. The choice of
theJonespolynomialisjustifiedbytheJonesconjecture,
Tounderstandtheoriginoftheshoulderwhichappears that is V = 1 if and only if K is the trivial knot. We
K
in figure 6 we have classified the different configurations haveusedthealgorithmofKauffman[29,30]tocalculate
as a function of the number of back-facingsegments and V andperformtheclassificationofchainsinfunctionof
K
computed χ2C for subsets with nS fixed. The results their knots. Recently it has been noted that probabil-
are shown in figure 9 for ℓ /b = 30. When fℓ 1 we ity of knot formation is small [20] in short chains. Our
(cid:10) (cid:11) p p ∼
see that the curves for different values of n separate. simulations confirm this point : for chains of 8 persis-
S
n =0 curves do not exhibit a shoulder. tence lengths the proportion of knotted chain is around
S
The originof the shoulder in figure 8 is essentially the (5 1) 10−4 when f =0.
± ×
same as the origin of the peak in figure 5. Under a large To interpret a micromanipulation experiment one
force, the configurations with back-facing segments are shouldperformaveragesoveranensemble ofchains with
statistically rare because they have a large energy cost. the same given knot. In practice one hopes the experi-
Their contribution to χ2 is, however, important. ment is performed with the trivial knot. The probabil-
C
ity distribution of the writhe angle χ has to be mod-
C
(cid:10) (cid:11)
ified, because we do not count the knotted configura-
VII. WRITHE AND KNOTS tions. We have computed χ2 , where the label “tk”
C tk
stands for “trivial knot” and compared it to χ2 , for
(cid:10) (cid:11) C
Clearly, self avoidance of the chain implies that topo- different lengths with no force. Results are plotted on
(cid:10) (cid:11)
logicalinvariantsareconstrained[24],including k. An- figure 10. The knot configurations that our algorithm
other topological invariant that should be constrLained is generated lead to an overestimated value for χ2 . The
C
theknotconfiguration. Thealgorithmthatwehaveused correction is about 7.5% for the longest chains that
− (cid:10) (cid:11)