The pulling force of a single DNA molecule condensed by spermidine

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by  R. Zhang

The pulling force of a single DNA molecule condensed by spermidine
R. Zhang and B. I. Shklovskii
Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455
(Dated: February 2, 2008)
In a recent experiment, a single DNA double helix is stretched and relaxed in the presence of
spermidine,ashortpositivepolyelectrolyte,andthepullingforceismeasuredasafunctionofDNA
extension. In a certain range of spermidine concentration, a force plateau appears whose value
4
showsmaximumasafunctionofspermidineconcentration. Wepresentaquantitativetheoryofthis
0
plateau force based on the theory of reentrant condensation and derive almost parabolic behavior
0
of the plateau force as a function of the logarithm of the spermidine concentration in the range of
2
condensation. Ourresult is in good agreement with experimental data.

n
a PACSnumbers: 87.15.La,61.41.+e,64.70.-p,87.15.He
J

9
DNAcondensationbystronglypositivelychargedpro- from 200 µM to 200 mM). Dual-trap tweezers are used

] teins,histones,is usedby natureforcompactionofDNA to stretch a single DNA molecule tethered between two
t
f in cell nucleus (DNA is strongly negatively charged). protein-coatedpolystyrenebeads. DNAisfirststretched
o
Positive proteins, protamines, are used for additional and then relaxed. During these processes, the pulling
s
. compactionof DNA in the sperm [1]. Gene therapy uses force f is measured. Experiments are repeated at vari-
t
a complexes of DNA with long positive polyelectrolytes or ousspermidine concentrations. It isobservedthatinthe
m
other macrocations. The net charge of these complexes intervals <s<s , there is a fairly largerange of DNA
c d
- can be positive and therefore they are not repelled by extension,x,whereforcef issignificantlylargerthanthe
d
negative cell membrane in the course of gene delivery. force described in the wormlike chain model [7]. More-
n
o Thus, condensation of DNA by positive macrocations is over, in this range of x, f is almost a constant (a force
c an extremely important physical phenomenon. It is in- plateau, see Fig. 2). The plateau value of the force, f ,
p
[ tensively studied in the model system of double helix is plotted as a function of s in Fig. 3.

3 DNA with spermidine, a flexible polymer with length 15
v ˚A and charge +3. It is known that in a dilute DNA f
1 solution at some concentration of spermidine, s = s ,
c
2
each long DNA molecule self-condenses into a toroid [2].
3
When s grows farther, at a much larger concentration,
0
1 s = sd, DNA dissolves back [3]. This phenomenon is
3 calledreentrantcondensationandhasgottheoreticalex-
0 planations [4, 5]. fp
/
t
a
m
0 x x
- L 0
d f f
n
o Bead Bead
FIG. 2: The pulling force f measured as afunction of exten-
c
: sionxisschematicallyshownbythesolidlineforsc <s<sd,
v when DNAiscondensed byspermidine. At x<x0,theforce
i
X fp is a constant whose value depends on s. The dashed line
x showsthewormlikechainbehavioroff intheabsenceofDNA
r
a condensation(s<sc ors>sd). Atx>x0,thecondensateis
completely eliminated by the pulling force and the wormlike
chain behavior is recovered.
FIG. 1: Schematic illustration of the experimental setup.
Partly condensed long DNAis pulled bytwo beads. In this paper, we suggest a quantitative theory of the
f (s) curve based on the theory of reentrant condensa-
p
Recently, a new single molecule technique has been tion[5]. Weassumethatthewholeprocessisslowenough
used to measure the force necessary to pull a single sothatthesystemisalwaysinequilibrium. Theninzero
DNAdoublehelixfromatoroidalcondensateofDNA[6] order approximation, the force needed to detach DNA
(Fig. 1). In the experiment, extremely small concentra- fromthe condensateis justthe freeenergydifferenceper
tion of DNA (2 nM of nucleotides) is dissolved in wa- unitlengthbetweenthefreecoilstateandthecondensed
ter with certain concentrations of spermidine (varying state of the DNA-spermidine complex. We call it f0.
2
Thepullingforcef measuredintheexperimentissome- per spermidine molecule is reduced [9]. This short range
p
what larger than f0 since the detached part of the DNA force of self-attraction leads to condensation of DNA if
molecule is stretched, and therefore loses entropy with the macroscopic Coulomb repulsion is not very strong.
respect of its free coil state. In the end of the paper, As a result, in the vicinity of the neutralizing concen-
we show that this effect adds a small correction to the tration s0 (sc < s0 < sd), DNA-spermidine complex is
pulling force, condensed into a toroid.
The condition of equilibrium between spermidine
k T molecules in the bulk and on the surface of DNA has
B
fp =f0 1+s lf0 ! (1) a form
when kBT/lf0 ≪ 1. Here l is the persistence length of µc+Zeφ=kBT ln(sv0), (2)
DNA. Inthis paper,we firstcalculatef0 andthen fp us-
where Z = +3 is the valence of spermidine, φ is the
ingEq.(1). OurresultisshowninFig3. Itdemonstrates
electric potential on the surface of the DNA-spermidine
reasonably good agreement with experimental data.
complex,andv0 isthenormalizingvolumeofspermidine.
The left hand side is the chemical potential of a spermi-
2.0 dine molecule in the complex, the right hand side is its
chemicalpotentialinthebulkofthesolution. Noticethat
the entropy of spermidine can be expressed through the
total concentration s because absorbtion of spermidine
N
byDNApracticallydoesnotchanges(theconcentration
p 1.0 ofDNA nucleotides is at least105 times smaller than s).
p
f In the vicinity of s0 (sc < s < sd), the net charge of
the complex is small and the spermidine concentration
on the DNA surface does not change much. Since µ is
c
0 determined by this concentration, it is approximately a
0.1 1 10 100 1000 constant. According to Eq. (2), when φ = 0, i.e., the
s mM complex is neutral, there is a simple relation between µ
c
and s0,
1 |µ |
c
FIG. 3: Comparison between thetheoretical result (thesolid s0 = exp − . (3)
line)andexperimentaldata(pointswitherrorbarstakenfrom v0 (cid:18) kBT(cid:19)
Fig. 2 of Ref. [6]) for the pulling force fp as a function of
spermidine concentration s. f0 is shown by thedashed line. Now we calculate the free energy of the charged com-
plexinitscoilstate. Forthispurpose,wetreattheDNA-
spermidine complex as a capacitor with the capacitance
Let us start from the theory of reentrant condensa-
per unit lengthC. Suppose s is suchthat the complex is
tionofDNAwithspermidinewithoutanyexternalforces.
In the solution, positively charged spermidine molecules overcharged (s0 < s < sd). Overcharging enhances the
Coulombself-energyandreducesboththecorrelationen-
are absorbed on the negatively charged surface of dou-
ergyandtheentropyofspermidinemolecules. Takingthe
ble helix DNA. Since spermidine molecules are strongly
free energy of a neutral coil as zero, and using Eq. (2),
charged(+3), they form a two dimensional strongly cor-
we get the free energy of the complex per unit length
relatedliquidonthesurfaceoftheDNAmolecule. When
anewspermidinemoleculeapproachesthe DNAhelix,it
1 Cφ Cφ 1
2 2
repelsalreadyabsorbedspermidinemoleculesandcreates f1 = Cφ + µc− kBT ln(sv0)=− Cφ , (4)
2 Ze Ze 2
an electrostatic image of itself, similar to the image on
a conventional metallic surface. Attraction to the image where the three terms allow for the three parts of the
leads to an additional negative chemical potential µc for free energy mentioned above, and Cφ/Ze is the number
spermidinemoleculesonthesurfaceofDNA.Asaresult, of spermidine molecules overcharging the complex. The
if s exceeds some concentration, s0, charge of the DNA- finalexpressionisthesameasthefreeenergyofacapaci-
spermidine complex changes sign and becomes positive, torkeptunderaconstantvoltageφ. Itiseasytoseethat
i.e.,spermidinemoleculesoverchargeDNA(seereviewof this expressionis alsotrue foranunderchargedcomplex.
the theory of charge inversion in Ref. [8]). SincetheDNA-spermidinecomplexcanbeconsideredas
Adsorbed layers of the correlated spermidine liquid a long cylinder, C is given by expression
alsoleadtoself-attractionofDNA.Inthespotwheretwo
turnsofDNAtoucheachother,thesurfacedensityofthe D
C = , (5)
correlated liquid is doubled and the correlation energy 2ln(1+r /R)
s
3
where D = 80 is the dielectric constant of water, rs is f0 goes to zero,
the Debye-Hu¨ckel screening radius, and R=10 ˚A is the
radius of the double helix cross section. 1 2|ε|Z2e2
Wefirstassumethatr ≫R. Inthiscase,thecomplex sc,d =s0exp ∓k T C−C′ , (8)
s B r !
in the condensed state is practically neutral. Indeed, if
the condensed complex were charged, the Coulomb-self wheretheupper(lower)signcorrespondstothefirst(sec-
energyofthe macroscopiccondensatewouldbetoolarge ond) subscript of s . Exactly at these two concentra-
c,d
to hold it. We define phenomenological parameter ε<0 tions, one can see transition from the coil state to the
as the free energy per unit length of the condensed com- condensed state or vice versa in light scattering experi-
plex calculated from the free energy of a neutral coil. It ments with solutions of DNA and spermidine [5].
includesthegainofcorrelationenergyinthespotswhere InordertocalculateC′,weassumethatthecondensate
two turns of DNA touch each other, and also the loss of is macroscopic and densely packed. Then
certain entropic elasticity of the coil in the condensate.
When the complex goes from the neutral condensed φ= ∞ ρe−r/rs4πr2dr = 4πrs2ρ, (9)
state to the chargedcoil state, the free energy increment 0 Dr D
Z
per unit length is
where ρ is the charge density of the condensate. The
1 charge of the complex per unit length is πR2ρ/α where
2
f0 =− Cφ −ε. (6) α = 0.91 is the filling factor for the hexagonal dense
2
packing of cylinders. This gives
ρπR2 DR2
f C′ = = . (10)
0 φ α 4αr2
ε s
′ ′
We see that when r ≫ R, we can drop C in C −C
s
′
because C ≪C, i.e., the condensate is almost neutral.
As we mentioned before, f0 is only the zero order ap-
proximation to the pulling force. The pulling force f is
p
givenby Eq.(1). We need four experimentalparameters
to calculate it. The spermidine concentrations s at
c,d
whichthe condensate dissolves,the averageDNA persis-
0 lns lns lns lns
c 0 d tence length l, and the average Debye-Hu¨ckel screening
radius r . Following Ref. [6], we take s = 0.35 mM,
s c
sd = 150 mM. Therefore s0 = 7.2 mM (Using Eq. (8)).
Weusel=500˚Acorrespondingtothepersistencelength
FIG. 4: The zero order approximation to the pulling force,
f0, as a function of lns given by an inverted parabola. The of a neutral DNA coil. At s0 =7.2 mM, spermidine and
maximumvalueoff0,|ε|,isachievedats=s0 wherethefree its counter ion contribute to rs even more than monova-
DNA-spermidine complex is neutral. sc,d are the concentra- lentsalt(10mM).Treatingeveryspermidinemoleculeas
tions of spermidine at which the condensatedissolves. apoint-liketrivalention,wegetr =13˚Afromthestan-
s
′
dardDebye-Hu¨ckelexpression[10]. CalculatingC andC
If rs is comparable with R (as in experiment [6]), the according to Eqs. (5) and (10), we finally get |ε| = 0.11
approximationusedabovethatthecondensateisneutral k T/bp (1bp =3.4 ˚A) from Eq. (8).
B
hastoberevisedandtheCoulombenergyoftheconden- For these parameters, f0 and fp are shown together
′
sate should be taken into account. If we introduce C as with the experimental data in Fig. 3. We see that our
the effective capacitance per unit length of the complex result for f agrees pretty well with the experimental
p
in the condensate, similarly to Eq. (4), the free energy data. Noticethatcorrectionfp−f0 ismuchsmallerthan
per unit lengthis just −C′φ2/2. Accordingly,inEq.(6), f0 in the range of experimental data. This justifies the
′
C shouldbe replacedby C−C . Using Eqs.(2) and(3), use of the perturbative result Eq. (1).
we rewrite Eq. (6) as Finally, let us derive Eq. (1). For this purpose, we
consider the detached part of DNA as a wormlike chain
f0 =|ε|− (C−2ZC2′)ek2B2T2 ln2 ss0. (7) w(sietehFtihge. 1c)o,nwtohuilre tlehnegctohntLouarnldenegnthd-otof-tehnedwdhiostleanDceNAx
moleculeisL0. Wealsoassumethatthesystemisalways
The function f0(lns) is shown in Fig. 4 by an inverted inequilibriumduringthewholeprocess. The freeenergy
parabola. AccordingtoEq.(7),themaximumvaluef0 = of the system is
|ε|isachievedats=s0wherethefreecomplexisneutral.
Also, we get the two concentrations sc and sd at which F(x,L)=−(L0−L)f0+Fe
4
2
kBT x L x+L Solving Eq. (13) in the same limit, we get
=−(L0−L)f0+ + − ,
l 2L 4(1−x/L) 4
(cid:20) (cid:21)
(11) 1 k T
B
a=1− . (16)
2s lf0
where −(L0 −L)f0 is the free energy of the condensed
part of DNA, and F is related to the entropic elas-
e Substituting Eq. (16) in Eq. (15), we arrive at our fi-
ticity of the detached part of DNA. The expression
nal result Eq. (1). It gives the pulling force f in the
p
for F is obtained for a free wormlike chain with fixed
e first order perturbation theory in the small parameter
contour length L and end-to-end distance x: F (x) =
x ′ ′ e kBT/lf0. In experiment [6], we have kBT/lf0 = 0.06
f (x)dx +const.. Heref isthe forceneededtokeep
0 e e at s = s0. Therefore, Eq. (1) can be used in almost all
the end-to-end distance x for a free wormlike chain with p
R the range sc < s < sd. At s close to sc,d, where f0 → 0,
length L and is given by an interpolation formula [7]
the perturbation theory fails. In principle, we can calcu-
latef numericallyusingEqs.(13)and(14)inthe whole
k T x 1 1 p
f = B − + . (12) interval s <s<s .
e l L 4 4(1−x/L)2 c d
(cid:20) (cid:21) The authors are grateful to A. Yu. Grosberg, Y. Mu-
rayama, Y. Rabin, I. Rouzina, and J. Zhang for useful
The energy zeropointhas been chosenatx=0,L=L0,
discussions. This work is supported by NSF No. DMR-
i.e., at the free energy of a DNA-spermidine complex in
9985785and DMI-0210844.
its coil state.
For a given x, the contour length L of detached DNA
can be found from the condition of minimal free energy,
namely, ∂F/∂L| = 0. If we define a ≡ x/L < 1, this
x
minimum condition can be written as
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2 2 2 4lf0 2 and P. Walter, Molecular Biology of the Cell (Galland,
2a (1−a) +a − (1−a) =0. (13) New York,2002).
kBT [2] V. A.Bloomfield, Biopolymers 44, 269 (1997).
[3] E. Raspaud, M. Olvera de la Cruz, J.-L. Sikorav, and
Finding a from this equation and putting L =x/a back
F.Livolant, Biophys.J.74, 381(1998). M. Saminathan,
to Eq. (11), we calculate the magnitude of the pulling
T. Antony,A. Shirahata, L. Sigal, T. Thomas and T. J.
force Thomas, Biochemistry 38, 3821 (1999).
[4] M. OlveradelaCruz, L.Belloni, M.Delsanti, J.P.Dal-
f ≡ ∂F = f0 + kBT a − 1 + 1 . (14) biez, O. Spalla and M. Drifford, J. Chem. Phys. 103,
p
∂x a l 2 4 4(1−a) 5781 (1995).
(cid:12) (cid:12) (cid:20) (cid:21)
(cid:12) (cid:12) [5] T.T.Nguyen,I.Rouzina,andB.I.Shklovskii,J.Chem.
Notice(cid:12)(cid:12) tha(cid:12)(cid:12)t a does not depend on x and L (see Phys. 112, 2562 (2000).
Eq.(13)). Therefore,accordingtothe definitionx=aL, [6] Y. Murayama, Y. Sakamaki, and M. Sano, Phys. Rev.
the two lengths x and L are always proportionalto each Lett. 90, 018102 (2003).
[7] J. F. Marko and E. D. Siggia, Macromolecules 28, 8759
other. This result was also obtained in Ref. [11] using
(1995).
a different approach. From Eq. (14) it is easy to see
[8] A. Yu. Grosberg, T. T. Nguyen, and B. I. Shklovskii,
that fp > f0 since a < 1. There are two contributions Rev. Mod. Phys. 74, 329 (2002).
to fp − f0. The first is a geometrical effect (the term [9] I. Rouzina and V. A. Bloomfield, J. Phys. Chem. 100,
f0/ain Eq.(14)). Namely, to changethe end-to-enddis- 9977 (1996).
tanceby ∆x, alargerlength∆L=∆x/amustbe pulled [10] This calculation overestimates the screening effect be-
out of the condensate. The second contribution is the causeofthesignificantlengthofthespermidinemolecule.
free energy increment of the detached part of DNA, F . Wecangetaprettygood estimateofrs from abovecon-
e
sidering eachspermidinemoleculeasthreeseparatepos-
These two contributions are equal in the limiting case of
itive monovalent ions, which gives rs = 17 ˚A. Our final
kBT ≪ lf0 (see Eqs. (15), (16) and (1)). Since fp does result is not sensitive to this variation of rs: the corre-
notdependonx,wegetaforceplateauwithincreasingx, sponding variation of |ε| is less than 7%.
as observed in the experiment (see Fig. 2). This plateau [11] H.Wada,Y.Murayama,andM.Sano,Phys.Rev.E66,
endswhenxreachesx0 =aL0 andallDNA ispulledout 061912 (2002).
of the condensate. At x>x0, the pulling force starts to [12] Strictly speaking, at x → x0, our theory is not valid
increase with x according to Eq. (12) [12]. because the toroidal condensate becomes so small that
When kBT/lf0 ≪1, we have 1−a≪1, and Eq. (14) cfa0nstnaorttsbteoigdneocrreeadsein(tthheissluimrfaitc)e. Tenheerrgeyfooref tahtextocrlooside
becomes
to x0, the pulling force f first decreases a little bit with
f0 kBT increasing x, and then goes back to the wormlike chain
fp = + . (15) behaivor.
a 4l(1−a)