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Semiflexible Polymers Confined in Soft Tubes F. Brochard-Wyart,* T. Tanaka, N. Borghi, and P.-G. de Gennes Laboratoire PCC Institut Curie/CNRS UMR 168, 11, rue P. & M. Curie, 75005 Paris, France Received October 20, 2004. In Final Form: February 16, 2005 We discuss various conformations for a polymer (of persistent length lp) confined into a deformable tube (the wall being a lipid bilayer with a certain surface tension σ and curvature energy K). Our study assumes that there is no adsorption of the chain on the wall. Three states are compared: (a) an unperturbed tube, plus a confined chain, (b) a tube swollen in all the region surrounding the chain (similar to a snake eating a sausage), (c) a globule, a roughly spherical coil surrounded by a strongly deformed tube. We construct a (qualitative) phase diagram for these systems with two variables: the surface tension σ and the degree of polymerization N. Our main conclusion is that “globules” usually win over “snakes”.
1. Introduction Certain natural processes, such as the (rare) conjugation of bacteria or the infection of plant cells, require the transfer of DNA through a thin tube between the two partners. These days, it is also possible to transfer DNA between two giant vesicles (GUVs), which are connected by a lipid tube1 (Figure 1). The natural size of these tubes is fixed by a compromise between curvature energies (rigidity K) and surface tension (σ). In most practical systems the tension σ of a GUV is extremely small but finite. Then the diameter D0 corresponds to the minimum of energy per unit length
Ftube(D) )
[
πD K +σ 2 (D/2)2
]
(1)
giving
D0 )
(2Kσ )
1/2
(2)
This size is of order 100 nm for floppy vesicles (σ ∼ 10-5 N m-1, K ∼ 10kT ∼ 4 × 10-20 J) and 10 nm for tense vesicles (σ ∼ 10-3 N m-1). It is thus much smaller than the standard size Rf (where f stands for free) of a DNA coil in water. The scaling structure of Rf was obtained by Nakanishi,2 using an extension of Flory’s approach, and minimizing a free energy functional of the scaling form 2
Fchain(R) R2 N 2lp a ) 2+ kT g R3 R0
()
(3)
Figure 1. Semiflexible polymer confined in soft tubes: (a) unperturbed tube, short chains are ideal (UI), while long chains are swollen (US); (b) perturbed tube, snake eating an ideal sausage (PI) or a swollen sausage (PS); (c) globules, the size of the globule is controlled by a balance of capillary energy against confinement entropy (GI) for short chains, and excluded volume (GS) for long chains.
Onsager3) is
v = alp2
A minimization of Fchain(R) (eq 3) then leads to a swollen chain radius
Here kT is the thermal energy, R0 is the ideal chain size 2
R0 ) Nalp
(4)
N is the number of base pairs, a = 0.5 nm is the size of one base pair. The second term in eq 3 describes a chain as a sequence of N/g rods, each of g ) lp/a base pairs, with lp = 50 nm the persistence length. The excluded volume between these rods (following a classical argument by * E-mail:
[email protected]. (1) Karlsson, A.; Karlsson, R.; Karlsson, M.; Sott, K.; Lundqvist, A.; Tokarz, M.; Orwar, O. Nanofluidic networks based on lipid membrane technology. Anal. Chem. 2003, 75, 2529-2537. (2) Nakanishi, H. J. Phys. (Paris) 1987, 48, 979-984.
(5)
() lp a
Rf = N3/5
1/5
a
(6)
Equation 6 is valid for very large chains. The criterion for this is that the last term of eq 3, when computed in the unperturbed state (R ) R0), be larger than unity. This leads to
N > NC3 ≡
() lp a
3
(∼106)
(3) Odijk, T.; Houwaart, A. C. J. Polym. Sci. 1978, 16, 627.
10.1021/la0474114 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/29/2005
(7)
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On the other hand, when N < NC3, we return to an ideal coil Rf ) R0. This is realized for most cases of interest. In the following we shall assume N < NC3. We must compare Rf with the tube diameter, D0. If Rf < D0, the chain is moving freely in an unperturbed tube. This will occur for very short chains. The threshold length for this is N ) NC1 , NC3, and is given by
R0(NC1) ) D0
(8)
corresponding to
NC1 )
D02 K = alp σalp
(9)
NC1 ∼ 102 for σ ∼ 10-5 N m-1. We are interested here in chains longer than NC1 or membrane tension higher than σC1 ) (K/lp2)(g/N). What sort of distortion will they impose on the tube? What conformation will the chains have? In section 2 we consider the case of a semiflexible chain confined in a rigid tube of diameter D0. In section 3 we consider the case of an extended chain surrounded by a swollen tube, with a certain diameter D (Figure 1b). In section 4 we consider another extreme, where the chain is roughly a spherical coil, and the tube adjusted to this (Figure 1c). All our discussion is limited to thermal equilibrium. It may happen that certain nonequilibrium conformations be prepared and be metastable; we do not discuss this. We also constantly rely on a Flory approach, extending eq 3 to confined objects and adding surface energies from the tube. It is known that the Flory method gives a good estimate of sizes (for coils, brushes, etc.) but a slightly incorrect estimate of polymer energies. But we cannot go beyond without very heavy complications. 2. Semiflexible Polymers in Rigid Tubes This situation represents an important starting point for our later discussions. It is also of interest in connection with recent experiments using a piece of genome trapped into a thin glass capillary.4 Our main assumptions are (a) DNA does not adsorb on the tube wall and (b) the persistence length lp is smaller than (or at most comparable to) the tube diameter D. This is not unreasonable when the tension σ is small (σ ∼ 10-6 N m-1). The opposite case (lp > D) is discussed in the Appendix. We may then write the free energy of a swollen chain spanning a length L of rigid tube (“rt”) in the scaling form 2 Frt L2 N 2 alp ) 2+ kT R g LD2 0
()
(10)
(since the volume involved is now LD2). The optimal L is given by
L ) Na
()() a D
2/3
lp a
N > N* ≡
()() lp a
1/3
D a
4/3
(12)
∼103 for lp ∼ D ∼ 50 nm. This regime is easier to achieve in a tube than in bulk solution (compare eqs 7 and 12). The chain can be pictured as a train of N/N* ideal blobs of size L*) (N*alp)1/2: L ) (N/N*)L*. It is of some interest to estimate the free energies ∆F required to squeeze the DNA inside the tube, because this will be related to a threshold (osmotic pressure or electric field) for entry into the tube. (i) N < N*. The length of the chain is R0, and the energy is dominated by the elastic term
∆F ) kT
Nalp
(13)
D2
The corresponding force for entry is
f)
R0 ∆F ) kT 2 R0 D
(14)
and the threshold osmotic pressure is
Π=
R0 f ) kT 4 2 D D
(15)
If q is the charge per monomer, the threshold electric field E1 for entry of a single chain is given by NqE1 ) f or
qE1 ) kT
R0
(16)
ND2
(ii) N > N*. Here the free energy is due to confinement of N/N* ideal subunits
∆F ≡ kT
Nalp N ∆F* = kT N* D2
(17)
The threshold force is
(N*alp) ∆F ∆F* ) ) kT f) L L* D2
1/2
(18)
and is independent of chain length. It is equal to the force f* required to push in a single subunit (of length N*). From eq 18 we can obtain scaling laws for the thresholds in osmotic pressure (Π*) or in field (E*)
Π* ) kT
(N*alp)1/2
E* ) E1*
D4 N* N
(19) (20)
where E1* is the field required to push in a single subunit.
1/3
(11)
This swollen chain regime inside an unperturbed tube holds if L > R0, or (4) Tegenfeldt, J. O.; Prinz, C.; Cao, H.; Chou, S.; Reisner, W. W.; Riehn, R.; Wang, Y. M.; Cox, E. C.; Sturm, J. C.; Silberzan, P.; Austin, R. H. The dynamics of genomic-length DNA molecules in 100-nm channels. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 10979-10983.
3. Semiflexible Polymer in a Soft Tube: The Snake Model An empty tube has an equilibrium diameter D0 ) (2K/ σ)1/2. What happens to the tube diameter D when a polymer has entered? In this section we discuss a model reminiscent of a snake which has eaten a sausage (Figure 1b). The overall energy is the sum of tube energies (eq 1) and chain confinement energy (eq 13 or eq 17)
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∆F ) πL
(2KD + σD2 ) + kT
Brochard-Wyart et al.
Nalp D2
(21)
Here the length L is given by our discussion of section 2
(Nalp)1/2 (N < N*) L = L* (N ) N*) (N/N*)L* (N > N*)
(22)
Coexistence between the swollen region and the empty tube imposes that the osmotic pressure Π (discussed in section 2) be balanced by the pressures due to curvature and to the Laplace capillary term
-
2K σ + )Π D3 D
(23)
where
∏=
R0 kT 4 D Π* ) kT
(N < N*) (24)
(N*alp)1/2
(N > N*)
D4
Notice that Π(D0) ∼ σ2 while the Laplace pressure is σ/D0 ∼ σ3/2. This will explain why a tube swells at high σ, which is not an intuitive result. This leads to different regions in the (σ, N) plane as shown in Figure 2. (i) The tube is not deformed if Π(D0) , 2σ/D0. (a) For ideal short chains (N < N*), this condition becomes
(Nalp)
kT σc)
(28)
σ1 ) σc defines a critical value of N ) Nc ) (kT/K)4(lp/a)3, which satisfies σ*(Nc) ) σc.
(29) (30)
The cross over between short (ideal) and long (swollen) regime is given by N ) N*(D), or 5/4 kT lp σ2 ) 7/4 13/4 N a
or
σ < σc )
(kTσ )
( )
(b) For long swollen chains, it leads to
(N*alp)1/2
(ii) Expanded tube. We expect a swelling of the tube if the osmotic pressure overcomes the Laplace pressure, i.e., for σ > σ1 (short chains) or σ > σc (long chains). Equation 21 leads to
D)
or
σ < σ1 )
Figure 2. Snake regimes for semiflexible polymers (degree of polymerization N) into a tube of surface tension σ: U ) unperturbed tube; P ) perturbed tube; I ) ideal chain; S ) swollen chain. We take here (and further on) K = 10kT, a = 5 nm, and lp = 50 nm for the calculation of phase boundaries.
(31)
The phase diagram shows that swollen tubes should arise only for σ > σc (Figure 2). For DOPC vesicles, we measured K = 7kT.5 With a/lp ) 10-2, σc ∼ 10-4 N m-1. For K = 10kT, σc ∼ 10-3 N m-1 is higher than the standard surface tensions of large vesicles. 4. Globules When the confinement energy of the chain in the snake model becomes too high, the chain can adopt a completely different globular configuration (Figure 1c). The energy of the chain encapsulated in the membrane (tension σ) is the sum of three contributions: excluded volume, surface energy, and reduction of entropy by confinement
Fch N2a3 σR2 R02 ) + 2 + kT kT R3 R
(32)
(5) Borghi, N.; Rossier, O.; Brochard-Wyart, F. Hydrodynamic extrusion of tubes from giant vesicles. Europhys. Lett. 2003, 64, 837.
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The radius of the globule is given by the minimization of Fch (δFch/δR ) 0). Depending upon N*, we expect two regimes: (i) If N < Ng*, the excluded volume is negligible
(
R ) N1/4 alp
kT σ
)
1/4
) N1/4(alpb2)1/4
(33)
(ii) If N > Ng*, the entropic contribution is negligible, and
R ) N2/5
(kTσ a ) 3
1/5
) N2/5b2/5a3/5
(34)
where b is defined by σ ) kT/b2. The cross over between the two regimes occurs for Ng* ) (lp/a)5/3(b/a)2/3 or
σg* )
lp5 kT
(35)
a 7 N3
For typical values of lp (DNA) and b (soft vesicles), Ng* ∼ 104-105. The scaling form of energy of the globule is 2
Fg = σR )
(NσkTalp)1/2
(N < N*)
(N4σ3(kT)2a6)1/5
(N > N*)
(36)
Figure 3. State diagram (σ, N) for semiflexible polymers (degree of polymerization N) in a tube of surface tension σ with a diameter D larger than the persistence length lp: U ) unperturbed tube; G ) globular tube; I ) ideal chain; S ) swollen chain.
(37)
We have to compare the energy of the DNA extended in the tube (eq 21) to the energy of the DNA collapsed inside a globule (eq 32). In the unperturbed regime, where D ) D0, the energy of the confined DNA is identical to the energy in a rigid tube, always written as
Comparing eqs 25 and 26 with eqs 39 and 42, we notice that σgi = (kT/K)σ1 and σcg = (kT/K)4σc. The curve σgi is below σ1 in the limit kT/K , 1 and the snake regime is masked by the globular phase (Figure 3). For the practical case (kT/K ∼ 0.1), without exact numerical coefficients we cannot have a definitive answer until experimental observations. If σgi > σ1, we can predict a snake regime above σc.
R02
6. Conclusions
5. Globule versus Snake
Fconf ) kT
) kT 2
D0
R02 σ K
(38)
The equality Fg ) Fconf given by eqs 33 and 34 defines two boundaries in the (σ, N) diagram: (i) When N < Ng*, we expect from eq 33 a characteristic tension σgi (gi refers to ideal confined chains), for the onset of globules, given by
σgi )
K2 kTNalp
(39)
(ii) When N > Ng*, we expect from eq 34 a characteristic tension σgs (gs refers to swollen confined chains) given by
σgs )
( ) () K kT
5/2
kT a a2 lp
5/2
1 N1/2
(40)
The plots of σgi and σgs(N) cross the plot of σg*(N) at a special chain length
()
(41)
( )( )
(42)
N ) Ncg )
kT lp K a
3
corresponding to a tension
σcg )
K K lp2 kT
2
a lp
2
Let us focus our attention on the usual situations, where the bending modulus K of the membrane is much larger than the thermal energy kT. Then we expect to see only two major conformations: (a) an unperturbed tube with an extended chain inside; (b) a “globule” varicose deformation of the tube, containing a roughly spherical coil. The coil radius R ∼ N1/4 for small chains (N < Ng*) and R ∼ N2/5 for large chains, where excluded volume overcomes confinement. This prediction appears to agree with certain observations by the Swedish group.6 They study DNA electrophoresis in lipid tubes. The fluorescent DNA is followed by confocal microscopy, with a resolution ∆x ∼ 0.8 µm ∼ 2 to 3L*. For a long snake (L > ∆x) we would expect the fluorescence intensity inside the detection volume to be independent of chain length. But what is found experimentally for the longer DNAs is different. The intensity increases linearly with chain length, as expected for one globule (smaller than ∆x). What is the speed of a globule under an electric field E0? We can propose a qualitative answer for the simplest case where (a) the membrane is not charged (no elelctroosmosis) and (b) the dissipation due to membrane deformations is weak. Then the main effect is a dilution of field lines inside the globule. The local field E is proportional to the local electric current density j. This current is spread out over the whole cross section of the globule; thus we (6) Orwar, O. Private communication.
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Brochard-Wyart et al.
κ(D2/λ3) ∼ kT and the persistence length becomes λ ≡ ˜lp = (lpD2)1/3: this is the “Odijk” regime.7 (i) Perturbation of the Tube. For simplicity we assume no hairpins. The energy of confinement is kT times the number of Odijk segments
∆F ) kT
L (lpD2)1/3
(44)
For small swellings of the tube ((D - D0)/D0 , 1) the free energy of tube plus chain has the scaling form (derived from eq 21)
(
F ) D0Lσ
)
D - D0 D0
2
+ kT
L (lpD2)1/3
(45)
and with D2 ) K/σ, the optimum corresponds to
()
D - D0 kT D ) D0 K lp Figure 4. Final state diagram (σ, N) for semiflexible polymers (degree of polymerization N) in a tube of surface tension σ, including small and large tube diameters: U ) unperturbed tube; G ) globular tube; I ) ideal chain; S ) swollen chain; R ) ultrathin tubes (lp > D); B ) unconfined chain (Rf < D0).
expect the internal field to be reduced by a factor ∼D2/R2, and the global electrophoretic velocity Vg to be
Vg ∼
D2 Vfree R2
1/3
(46)
Each of these factors is small: the tube is unperturbed. (ii) Globule versus Ultrathin Snake. When lp . D, the ultrathin snake has the energy of eq 44. We must compare this to the globule free energy (eq 32)
Fg - ∆F N2a3 σR2 R02 L = + 2 + kT kT R3 R (lpD2)1/3
(47)
(43) The first or the third term is comparable to the second (capillary) term. Thus the overall free energy is
where Vfree is the velocity of DNA in bulk solution (and is independent of N). We are currently starting more detailed experiments on these problems. Acknowledgment. We thank B. Akerman, P. Silberzan, J.-F. Joanny, and O. Owar for illuminating discussions on DNA. We have received generous support (grants and a fellowship to Tomoki Tanaka) from the Human Frontier Science Program (Research Grant 52/2003) and from the Curie Institute. Appendix: Very Narrow Tubes When the surface tension σ of the tube is relatively high (σ ∼ 10-3 N m-1) the tube is narrow (D0 ∼ 10 nm), and we have D0 < lp for semirigid rods such as double strand DNA in usual conditions (lp ∼ 50 nm). The correlation length λ is given by the bending energy of the polymer chain under thermal fluctuations: κ(θ2/λ) ∼ kT (where θ is the curvature angle and κ the bending elasticity of the chain). For an unconfined chain, the angle required to loose correlation is θ ≡ π and λ ≡ lp. A chain confined in a tube of diameter D < lp undergoes a series of deflections with the tube walls such that θ = D/λ, then
σR2 Na F ∼ kT kT (lpD2)1/3
(48)
The globule is more stable when Fg is negative, which corresponds to
σ < σrgi )
K kT 3 N lp2 K g
( )( )
3
(49)
(where r refers to “rodlike” chain) for the regime GI (entropy driven), and to
σ < σrgs )
( )() ()
K kT lp2 K
9/4
N g
3/4
lp a
3/2
(50)
for the regime GS (excluded volume dominant). The phase diagram including lp < D0 is shown in Figure 4. LA0474114 (7) Odijk, T. On the statistics and dynamics of confined or entangled stiff polymers. Macromolecules 1983, 16, 1340-1344.
