Stepwise Unwinding of Polyelectrolytes under Stretching
M. N. Tamashiro*,† and H. Schiessel‡
Materials Research Laboratory, University of California, Santa Barbara, California 93106-5130, andDepartment of Physics and Department of Chemistry and Biochemistry, University of California,Los Angeles, California 90095
Received December 3, 1999; Revised Manuscript Received April 17, 2000
ABSTRACT: We consider the unwinding of a tethered, flexible, and weakly charged polyelectrolyte in asalt-free poor solvent under an externally imposed strain. Using scaling arguments, we predict that therestoring force of a stretched polyelectrolyte shows a sawtooth pattern as a function of the externallyimposed end-to-end distance. This nonmonotonic behavior arises from a cascade of conformationaltransitions between necklace-like structures with different number of beads. The transitions result fromthe interplay between the unscreened Coulomb repulsion of charged monomers and the short-rangeattraction of the backbone.
1. Introduction
Various water-soluble polymers, including essentialbiopolymers such as DNA and proteins, carry ionizablegroups, which dissociate upon contact with a polarsolvent. The solubility of these charged polymersscalledpolyelectrolytes (PE)sis significantly enhanced by thepresence of the charged groups. The electrostatic repul-sion between the charged monomers improves the PEsolubility in solvents that would be poor for the un-charged backbone, providing a stabilizing mechanismagainst collapse and eventual precipitation. From thetheoretical point of view, the challenge represented bythe long-range Coulomb interaction makes the analyti-cal description of PE more difficult than that of theirneutral counterparts. (For a review on scaling ap-proaches to PE, cf. ref 1 and references therein.)
Especially, the widespread case of a PE in a poorsolvent shows an intricate behavior resulting from thecompetition between electrostatic repulsion and surfacetension. The shape that minimizes the free energy ofsuch a chain is usually the necklace configuration, whichconsists of compact charged globules connected via thinstrings (see Figure 1). Such a conformation was firstproposed by Kantor and Kardar2 for polyampholytes(polymers with positively and negatively charged mono-mers) and was then extended to uniformly charged PEin poor solvent by Dobrynin, Rubinstein, and Obukhov.3Besides the first analytical study by Dobrynin et al.,3based on scaling arguments, this scenario is also sup-ported by a variational approach4 as well as MonteCarlo3 and molecular dynamics simulations5 of dilutesolutions of weakly charged PE. Small-angle neutronand small-angle X-ray scattering experiments per-formed with polystyrenesulfonate in water6 are at leastconsistent with the picture of necklace structures insemidilute solutions,7 although they do not represent adirect proof. We note that for a finite concentration ofcounterions (always present for a finite concentrationof chains) a part of the counterions condense on suf-ficiently highly charged chains. This may lead to a chaincollapse as demonstrated in a molecular dynamics
simulation for multivalent counterions,8 consistent withscaling theories.7,9-11 Necklace configurations are, how-ever, observed in molecular dynamics simulations ofhighly charged PE with partially condensed monovalentcounterions.12,13 It still remains an open question whetherand under which circumstances thermodynamicallystable necklaces with condensed counterions reallyexist.
Besides recent theoretical progress in the understand-ing of charged polymers, there are also new experimen-tal methods on-hand that allow the manipulation ofindividual macromolecules.14 Force measurement ap-paratus, including the atomic force microscope (AFM)and optical tweezers, allow to observe the stretching ofsingle polymer chains. Experiments were performed ona variety of systems, including DNA,15-18 single chro-matin fibers,19 the polysaccharide dextran,20 and theprotein titin.21-25 In particular, in the case of the giantprotein titin, responsible for the elasticity of muscles,the force-extension profile obtained under stretchingexhibits a prominent sawtooth pattern,24,25 which isattributed to the successive unfolding of immunoglobu-lin domains. This effect may be well explained asstepwise increases in the contour length of a polymerwhose elastic properties are described by the wormlikechain model (WLC). In fact, Monte Carlo simulationscombining WLC elasticity with a thermodynamical two-state model26 reproduce the experimental data verywell. A similar two-state model has also been proposed
† University of California, Santa Barbara.‡ University of California, Los Angeles.* Corresponding author. e-mail: [email protected].
Figure 1. Necklace configuration composed of Nbead sphericalbeads with diameter dbead joined by (Nbead - 1) cylindricalstrings (Nbead ) 6 in the figure) with length lstr and diameterdstr. Each bead (string) contains mbead (mstr) monomers, re-spectively. The monomer density of the beads and the stringsis the same and follows from the densely packed thermal blobsof size êT, i.e., F ) 6τ/πb3.
to explain the stress-induced transformation of DNAbetween the B and S forms.27 Both treatments focus onthe elastic properties of the chain and do not discussthe role of the Coulomb interactions in the stress-driventransitions.
Motivated by these experiments, we perform a scalinganalysis of the stretching of a weakly charged PE underpoor solvent conditions, i.e., of a necklace. The processof unwinding such a structure may be of some impor-tance for the denaturation of biopolymers under defor-mation or mechanical stress as well as for the stretchingof synthetic PE. It is known that globular unchargedpolymers subject to external forces undergo an abrupt(first-order) coil-stretched transition.28,29 Similar effectsmight be expected in the case of a necklace structure.Because most biopolymers, particularly DNA and pro-teins, are in fact electrically charged, it is important toinvestigate how the Coulomb repulsion affects theirconformation and response to mechanical stress andstrain (cf., e.g., ref 30). In fact, we will demonstrate inthis paper that the necklace undergoes a stepwiseunfolding under an externally imposed strainssimilarto the behavior observed for titin mentioned above.Thus, a PE in a poor solvent represents a simplenonmicroscopic system to study this phenomenon.
The paper is organized as follows: In section 2 wereview and refine the scaling arguments that lead tothe necklace configuration for weakly charged PE underpoor solvent conditions. In section 3 we obtain thesteady-state configuration and the static restoring forceof a stretched PE when the end-to-end distance isexternally imposed. In section 4 we consider the effectof friction forces when a tethered PE is stretched fromthe rest at constant drift velocity. We conclude in section5 with a discussion of the relevant mechanisms that leadto stepwise unfolding and give further examples ofpolymeric systems that may exhibit similar features. InAppendix A we obtain the contributions to the freeenergy of the necklace structure. In Appendix B we givethe free energy of the tadpole configuration. In AppendixC we show the validity of the quasi-static steady-stateapproximation.
2. Scaling Theory for Weakly Charged PEWe consider a dilute solution of flexible PE under poor
solvent conditions at a temperature ∆T below the Θtemperature. This is the temperature at which thesecond virial coefficient between polymer segmentsvanishes so that the conformation of each chain corre-sponds to that of an ideal random walk. The chainscontain N . 1 monomers of size b, and a fraction φ ofthe monomers are charged. To avoid the phenomenonof counterion condensation, which decreases the effec-tive charge of the PE and may change its conformationalproperties,10,11 we restrict our analysis to the case ofweakly charged PE (u2φ , 1), where u ) lB/b is the ratioof the Bjerrum length lB ) e2/εkBT to the monomer sizeb (e is the electronic charge, ε is the dielectric constantof the solvent, kB is the Boltzmann constant, and T isthe temperature). Highly charged PE and the effect ofcounterion condensation, which may lead to an instabil-ity of the necklace structure, will be discussed in aforthcoming paper.31 Furthermore, we will assume asalt-free environment, so that there is no screening ofthe Coulomb repulsion between the charged monomersdue to excess ions.
For uncharged polymers (u2φ ) 0) and close to the Θtemperature (τN1/2 < 1), where τ ) ∆T/Θ denotes the
relative deviation from the Θ temperature (a measurefor the solvent quality), the chain behaves like an idealGaussian coil32 of size
where the symbol ∼ is used to state a scaling relation,which ignores numerical prefactors. On the other hand,under poor solvent conditions occurring for tempera-tures sufficiently below the Θ point or sufficiently largechains (τN1/2 > 1), the short-range attraction betweenmonomers of the backbone results in a collapse of thechain into a spherical globule32,33 of radius
and density
On length scales smaller than the size of the thermaldensity fluctuations, êT ) b/τ, the attractive interactionscan be neglected, and the chain obeys Gaussian statis-tics. On length scales larger than êT the collapsed chainmay be pictured as composed of close-packed thermalblobs of size êT containing gT ) (êT/b)2 ) 1/τ2 monomers.
For weakly charged PE (u2φ , 1) and close to the Θtemperature (τN1/2 < 1), the length scale below whichthe chain preserves its ideal shape, remaining unper-turbed by the Coulomb repulsion, is the electrostaticblob size,34 êel ) b(uφ2)-1/3. On length scales larger thanêel, the electrostatic repulsion dominates and extendsthe chain into a linear array of electrostatic blobs of sizeêel containing gel ) (êel/b)2 ) (uφ2)-2/3 monomers. In closeanalogy to the Pincus blob picture for a stretched chainunder external forces,35 the chain assumes a rodlikeshape of length
For weakly charged PE (u2φ , 1) under poor solventconditions (τN1/2 > 1), the conformation of the chain isgoverned by the competition between the long-rangeCoulomb repulsion of charged monomers and the sur-face tension originated from the short-range attractionof the backbone (poor solvent condition). Khokhlov9
suggested that, to optimize its free energy, the collapsedchain deforms into an elongated cylindrical globule oflength
and thickness
However, the cylindrical geometry is locally unstable,analogous to the classical problem of a charged liquiddroplet, considered by Lord Rayleigh36 more than onecentury ago. He predicted that a charged liquid droplet,above a certain critical charge, spontaneously deformsand splits into smaller droplets separated by an infinitedistance, each droplet carrying a lower charge than thecritical one. Because of the covalent bonds of thebackbone, it is not allowed for the monomers to dissoci-ate, and this equilibrium state is unreachable for a PEchain. However, the system can reduce its energy by
R ∼ N1/2b (1)
R ∼ (N/τ)1/3b (2)
F ∼ τ/b3 (3)
L ∼ Ngel
êel ) Nb(uφ2)1/3 (4)
Lcyl ∼ Nb(uφ2)2/3/τ (5)
D ∼ b(uφ2)-1/3 ) êel (6)
5264 Tamashiro and Schiessel Macromolecules, Vol. 33, No. 14, 2000
rearranging into a set of smaller compact chargedglobules (beads) connected by narrow stringssthe neck-lace configuration.3
Consider now the necklace configuration with Nbeadspherical beads of size dbead joined by (Nbead - 1)cylindrical strings of length lstr and width dstr, depictedin Figure 1. The total length of the necklace is given by
The number of monomers per bead is denoted by mbeadand the number of monomers per string by mstr. Sincethe electrostatic repulsion between charged monomersdoes not affect significantly the volume occupied by thePE, we assume that the beads as well as the stringsare still composed of close-packed thermal blobs, i.e.
resulting in a monomer density F ) 6τ/πb3. (One has gT) 1/τ2 monomers per spherical thermal blob of radiusb/2τ.) Introducing the total number of monomers insideall the strings,
we are led to
The total free energy of the necklace configurationmay be split into five terms,
where the different terms account for intra-string, intra-bead, inter-bead, inter-bead-string, and inter-stringinteractions, respectively. In contrast to previous simplescaling arguments,3 we perform here a more detailedcalculation taking all numerical and geometrical pref-actors into account. These refinements are required herebecause the energy barriers between the differentnecklace structures are smaller in the stretched statesunder investigation than in the unstretched case. Wealso mention that the optimum lowest energy necklacestructure for a given number of beads is not comprisedby identical subunits, but rather contains slightlypolydisperse nonequidistant beads. This effect followsfrom the long-ranged nature of the Coulomb interactionand finite-size effects. A necklace composed of mono-disperse beads would correspond to the optimal config-uration only if the beads would not interact. Neverthe-less, since the determination of the exact lowest energypolydisperse-bead structure is hardly feasible and leadsonly to minor deviations from the monodisperse-beadnecklace, we assume in the following that all beads havethe same size and are equally spaced.
The different contributions to the free energy of thenecklace, eq 13, are rather complicated and are rel-
egated to Appendix A. The minimization of the total freeenergy, eq 13, with respect to Nbead, Mstr, and dstr leadsto the optimal values of the necklace configuration. Ina previous calculation, Dobrynin et al.3 neglected allnumerical prefactors, treated Nbead as a continuousvariable, and made additionally two approximations.The total length of the necklace structure was obtainedin the limit of a string much longer than a bead, lstr .dbead,
and the sum of the three inter-energy repulsion terms,given by eqs A.11-A.13, was approximated by
With these approximations, Dobrynin et al. were ableto show that, for a certain range of parameters,3 theequilibrium configuration of a weakly charged PE in theabsence of external forces is a necklace structure withnumber of beads
and length (at rest)
As we will consider the transitions driven by thecompetition between the different terms by stretchingthe PE, it is important to keep all numerical prefactors.Especially, as mentioned earlier, the distinct necklaceconfigurations with different number of beads havesmall free energy differences under external forces,because all structures converge to a stretched cylinderupon strong strain. Therefore, we will use the morerefined approximations given by eqs A.11-A.13, whichyield a repulsion energy that depends on the numberof beads, instead of eq 15. Therefore, in our formulationthe unstretched equilibrium configuration is not givenby eqs 16 and 17, but rather follows from the numericalminimization of the total free energy (13).
Up to now we considered just necklace structures withNbead g 2. However, by stretching a PE, we may alsofind structures with just one bead (Nbead ) 1). Figure 2sketches two of these configurations: a half-dumbbell(tadpole) and a symmetrical double-tadpole. In thetadpole configuration, we have just one single rodattached to a spherical globule. In the symmetricaldouble-tadpole configuration, we have two identical rodspointing in opposite directions, radially aligned with thecenter of the spherical globule. It turns out that the free
L ) Nbeaddbead + (Nbead - 1)lstr, Nbead g 2 (7)
dbead ) b(mbead
τ )1/3
(8)
lstr )2b3mstr
3τdstr2
(9)
Mstr ) N - Nbeadmbead (10)
mbead )N - Mstr
Nbead(11)
mstr )Mstr
Nbead - 1(12)
F ) Fs + Fb + Fbb + Fbs + Fss (13)
Figure 2. Two stretched conformations with Nbead ) 1: thetadpole and the symmetrical double-tadpole configurations.
energy associated with all possible double-tadpole con-figurations is always higher than that associated withthe tadpole configuration. This is due to the bead-stringrepulsion, which is smaller for the tadpole configuration.The free energy of the tadpole configuration is discussedin Appendix B. Although the tadpole configuration isenergetically not favorable in the absence of appliedforces, we will show in the next section that it occurswhen a necklace chain becomes highly stretched.
Further stretching of the PE (after the unraveling ofthe last bead of the tadpole configuration) yields a highlystretched cylindrical globule, similar to that found underΘ conditions. In the same spirit of the approximationsperformed for the strings of the necklace structure (seeAppendix A), we compute the free energy of the cylin-drical configuration of length L ) êTL,
where N ) Nτ2, ω(x) ) arccosh(x)/x1 - x-2, and recall-ing that D ) êTD is the thickness of the Khokhlovunstretched cylindrical globule, eq 6. The crossoverlength Lcross ) êTLcross with
is the length at which the thickness of the stretchedcylinder becomes of the thermal blob size êT. At PElengths L < Lcross the free energy of the cylinder has asurface tension contribution, whereas for L > Lcross themain contribution of the free energy comes from theentropic stretching of a Gaussian coil. At this length thefree energy of the cylindrical and the tadpole configura-tions should be the same, although we observe a verysmall difference due to the approximations performed.In fact, the free energies of all necklace structuresshould also reduce to the cylinder value at the crossoverlength. Despite the approximations, we are able toobtain this convergence within fractions of kBT. Weobserve that, for L < Lcross, the cylindrical configurationhas always a higher free energy than that of thenecklace/tadpole. Only after the last bead is unfolded,which occurs around the crossover length Lcross, weexpect that the lowest energy configuration of the PEis a Θ solventlike cylinder with thickness smaller thanthe thermal blob size êT.
3. Steady-State Configuration of a Stretched PEwith Externally Imposed End-to-End Distance
A typical experimental setup in which a singlemacromolecule is stretched by an AFM tip20,24 is de-picted in Figure 3. In our PE model picture, we startfrom a necklace configuration at rest with an initialnumber of beads Nbead
0 and natural length L0, bothdetermined by the minimization of the total free energy(13). The polymer chain is adsorbed at one end to a flat(gold or mica) surface, and the other end is bounded toan AFM tip. The flat surface is moved away from theAFM tip (see Figure 3), imposing an external strain L> L0 on the chain. We assumed that the PE chain is
chemically adsorbed onto a neutrally charged flatsurface and that image-charge effects due to the solvent/surface dielectric contrast are negligible. In the case ofphysical adsorption, when the chain is electrostaticallybound to the oppositely charged surface, the detachmentof single PE chains leads to a plateau in the force-extension profile.37,38
Instead of using the approximated linear relation 14for Mstr as a function of L, we evaluate it by taking thereal positive solution of the cubic equation
where Mstr ) Mstrτ2. For a fixed set of parameters (τ, u,φ, N) and an externally imposed end-to-end distance L,we minimize the total free energy with respect to dstrfor all possible structures (necklaces with Nbead g 2, thetadpole, and the cylindrical configurations), choosing theconformation with the lowest total free energy. For therange of parameters that we have explored, it turns outthat the total free energy is minimized when the stringthickness dstr assumes the thermal blob size êT, whichcorresponds to dstr ) 1. The free energy minimizationprocedure allows us to construct the phase diagramsshown in Figures 4 and 5. The phase diagrams showthe different states (characterized by the number ofbeads) in two cuts through the parameter space, one inthe φ × L plane and the other in the τ × L plane (seefigure captions for details). It can be seen that, byimposing a strain on the necklace, one induces transi-tions between states with different number of beads.Usually, the number of beads decreases with increasingend-to-end distance. Note, however, that in many casesthere is an initial increase in the number of beads uponstretching, which occurs due to the reentrant behaviorof the transition boundaries near the unstretched equi-librium length. Although at first sight unexpected, thisreentrant behavior is required to match the Dobryninet al. unstretched equilibrium results3 and the highlystretched cylinder at strong strain. Therefore, theconformational transitions of the PE under stretchingare closely related to Dobrynin’s force-free abrupttransitions upon variation of φ and/or τ.
In Figures 6 and 7 we present some geometricalproperties of the necklace structure upon unraveling.In particular, Figure 7 shows that the total number of
Figure 3. Our PE model system corresponding to theexperimental setup in which a single PE chain is stretchedby an AFM tip. The adsorbed chain, immersed in a salt-freepoor solvent, takes at rest a necklace-like conformation. Thechain is attached to an AFM tip and stretched by moving theflat surface away from the tip. In most cases the number ofbeads decreases as the PE is stretched. The force exerted onthe polymer as a function of the end-to-end distance L ismeasured by a small deflection δ of the cantilever spring.
F cyl
kBT) 6N2
5D3Lω[(3L3
2N)1/2] + (23NL)1/2
, for L < Lcross
) 6N2
5D3Lω(3L2
2N) + 3L2
2N, for L > Lcross (18)
Lcross ) 23N (19)
L ) LêT
) Nbead2/3(N - Mstr)
1/3 + 23dstr
2Mstr (20)
5266 Tamashiro and Schiessel Macromolecules, Vol. 33, No. 14, 2000
monomers in all strings Mstr has only small jumpsduring the transitions. This means that the monomersof the unfolded bead essentially redistribute betweenthe remaining beads. The transitions take place when-eversafter sufficient stretchingsthe beads are so smallthat one of the beads can redistribute its monomersbetween the remaining beads without violating theRayleigh criterion (corresponding to an upper bound forthe bead size dbead). For the first transition, however,due to the steep increase of the restoring force in thevicinity of the unstretched length, the critical upperbound for the bead size decreases faster than the(shrinking) size of the beads. Therefore, even thoughthe size of the beads decreases, there is a Rayleighinstability that gives origin to the reentrant behavior
described in the last paragraph and the initial increasein the number of the beads.
To keep the PE chain at the externally imposed end-to-end distance L > L0, one needs to apply an externalforce F(L). In the experimental setup, this force may bemeasured by a small deflection δ of the AFM tip (seeFigure 3). In the case of a static, time-independent,steady-state configuration, the external force F(L) coun-terbalances simply the restoring force -f(L),
Therefore, once we have obtained the lowest energyconformation, the restoring force may be evaluated bydifferentiation of the total free energy with respect tothe end-to-end imposed length L. Typical static exten-sion-force profiles are presented in Figures 8 and 9.They show a prominent sawtooth pattern, associatedwith the abrupt transitions between necklace configura-tions with a different number of beads. Despite thedifferences in the mechanism of unfolding, the stepwisebehavior of the force-extension profile resembles thesame effect observed in titin.24,25 For the set of param-eters used in Figures 8 and 9, the necklace structures
Figure 4. Diagram of states in the φ × L space for N ) 5000,u ) 2, and τ ) 0.4. The number of beads labels distinctnecklace configurations. “1” corresponds to the tadpole con-figuration, and “0” is a stretched cylinder with thicknesssmaller than the thermal blob size êT. The gray regioncorresponds to compressed states, which were not treated inthis work. The rightmost boundary of the gray region corre-sponds to the equilibrium lengths L0 in the absence of externalforces. In this case, we can clearly see the discontinuous natureof the transitions related to the jumps of the unstretched end-to-end distance L0, which become weaker as we increase thefraction φ of charged monomers. Note that for many values ofφ one has first an initial increase in the number of beads uponstretching due to the reentrant behavior of the transitionboundaries near the unstretched equilibrium line.
Figure 5. Diagram of states in the τ × L space for N ) 5000,u ) 2, and φ ) 1.5%. The interpretation of the numbers andthe gray region are the same as in Figure 4. The inset presentsa magnification of the region near the unstretched equilibriumline, showing the discontinuous nature of the transitions inthe absence of external forces. Contrary to the previousexample of constant τ and increasing φ, in this case thediscontinuous jump of the unstretched end-to-end length L0becomes weaker as we decrease τ. For many values of τ thereis also an initial increase in the number of beads uponstretching due to the reentrant behavior of the transitionboundaries near the unstretched equilibrium line.
Figure 6. Number of monomers inside each bead mbead andsize of each bead dbead, for the same set of parameters shownin Figure 9: N ) 5000, u ) 2, τ ) 0.4, and φ ) 1.5%.
Figure 7. Total number of monomers in all strings Mstr andlength of each string lstr normalized to their final values, forthe same set of parameters shown in Figure 9: N ) 5000, u )2, τ ) 0.4, and φ ) 1.5%. For the two last stages of theunraveling, the dumbbell (Nbead ) 2) and the tadpole (Nbead )1) configurations, the two normalized curves are given by thesame expression. The dashed curve represents the linearapproximation 14 for Mstr as a function of L (taking thenumerical prefactor 2/3 into account). Although this approxi-mation describes well the behavior for high stretching, thedeviation increases for smaller strains.
at their equilibrium lengths in the absence of externalforces contain initially three (Nbead
0 ) 3) and eight (Nbead
0 ) 8) beads, respectively. After an initial increasein the number of beads, they are successively unfoldedas we increase the length of the PE.
We also give here the restoring force associated withthe highly stretched state that follows from eq 18 withL > Lcross,
which corresponds to the Pincus blob picture35 (if weneglect contributions from the electrostatic repulsion).The PE forms a linear array of blobs of size êP ) kBT/|f|containing gP ) (kBT/|f|b)2 monomers, leading to a linearrelation between externally imposed end-to-end distanceand force,
We close this section by noting that there is afundamental difference between an experiment wherethe end-to-end distance L is externally imposed (strainensemble) and a situation with a given external force F(stress ensemble).39 In the latter case one will not finda sequence of stepwise unfolding processes as responseto an increasing applied tension. Instead, the structurewill unfold at once when a critical value of the tensionis reached. This value can be estimated by comparingthe energy gain by stretching the necklace with the pricethat one has to pay by increasing its surface. If thenecklace is stretched by a length êT (the thermal blobsize), one gains -FêT. At the same time, one thermalblob is transferred from a globule to a string, whichleads to an increase σêT
2 ) kBT in the surface energy,where σ is the surface tension. The critical tension isFc ) kBT/êT, i.e., the tension at which the size of thePincus blobs êP ) kBT/F equals the size of a thermalblob êT. For F > Fc the chain can be described by asequence of Pincus blobs of total length L ∼ Nb2F/kBT.
The static restoring force -f presented in Figures 8and 9 corresponds to an equilibrium steady-state con-figuration. In other words, in the previous section weassumed that the flat surface is kept fixed after onereaches the externally imposed end-to-end length L >L0. Most measurements, however, use a dynamicalmethod to measure the force, being performed atconstant pulling velocity V,24-26
Besides the static restoring force -f, we need to add afriction contribution, ffriction(t), proportional to the prod-uct of the diameter of the bead, dbead, by its drift velocityVi, i ) 1, ..., Nbead(t),
where η is the shear viscosity of the solvent. To obtaineq 25, we have assumed a quasi-static equilibrium forthe PE chain. In this case, the velocities of the beadsare given simply by
Figure 8. Static restoring force -f as a function of theexternally imposed end-to-end distance L for the set ofparameters: N ) 5000, u ) 2, τ ) 0.4, and φ ) 1%. For theseparameters, the natural length L0 of the PE is L0 ) 48.9êT,and the necklace structure contains initially three beads (Nbead
0 ) 3). Using the Bjerrum length of the water at roomtemperature, lB ) 7 Å, the normalizing factors are kBT/êT )4.73 pN and êT ) 8.75 Å. The number of beads labels thedistinct necklace structures. Note the initial increase in thenumber of beads, which leads to a narrow negative-forcewindow. The unwinding of the last bead occurs through atadpole configuration (a spherical globule terminated by acylindrical rod), labeled by “1”. After the bead size dbead reachesthe string size dstr, which turns out to be of the thermal blobsize êT, the conformation of the PE is a highly stretchedcylinder with thickness smaller than êT, labeled by “0”. Asshown in the inset, for this highly stretched regime we expectthat the force-extension relation is governed by the Pincusblob picture, |f| ∝ L.
Figure 9. Static restoring force -f as a function of theexternally imposed end-to-end distance L for the set ofparameters: N ) 5000, u ) 2, τ ) 0.4, and φ ) 1.5%. For theseparameters, the natural length L0 of the PE is L0 ) 116.2êT,and the necklace structure contains initially eight beads (Nbead
0 ) 8). Using the Bjerrum length of the water at roomtemperature, lB ) 7 Å, the normalizing factors are kBT/êT )4.73 pN and êT ) 8.75 Å. Here we also have an initial increasein the number of beads. Analogous as shown in Figure 8, thestrong-stress regime corresponding to a highly stretchedcylinder with thickness smaller than êT is governed by thePincus linear relation, not presented here. To allow a betterappreciation of the force-extension sawtooth pattern, the insetshows a magnification of the end portion of the curve.
-fêT
kBT) 1
kBT∂F cyl
∂L) - 6N2
5D3L2ω(3L2
2N) +
18N5D3
ω′(3L2
2N) + 3LN
(22)
L ∼ NgP
êP ) Nb2
kBT|f| (23)
L(t) ) L0 + Vt (24)
ffriction(t) ) -3πη ∑i)1
Nbead(t)
dbead(t) Vi(t) )
-3
2πηVNbead(t) dbead(t) (25)
Vi(t) )(i - 1)V
Nbead(t) - 1(26)
5268 Tamashiro and Schiessel Macromolecules, Vol. 33, No. 14, 2000
Since the polymer has a constant drift velocity V, thesum of all forces on the PE must vanish, F(t) + f +ffriction(t) ) 0. The measured force F(t), given by
is plotted in Figure 10 for some pulling velocities V. Ourtreatment of the unraveling dynamics of a PE is basedon the assumption of a quasi-static equilibrium of thenecklace (see Appendix C), where the beads preservetheir spherical geometry at each instant of time. Fur-thermore, we assume that there is a vanishing transienttime for the beads to rearrange between the abruptconformational transitions. Finite transient times yielda slower relaxation to the new conformation, broadeningthe troughs of the force-extension profile after thetransitions. Despite the approximations, we see thatsome features resemble those experimentally observedin the unfolding of the titin:24 (a) roughly equidistantpeaks, (b) positive slope of the peaks (for small pullingvelocities), and (c) increasing peak amplitude.
5. DiscussionIn this paper we have shown that a PE in a poor
solvent unfolds in a stepwise fashion under an exter-nally imposed strain. The underlying mechanism is thesuccessive unwinding of the beads of the necklacestructure. As a result, the force-extension profile showsa characteristic sawtooth pattern. A similar profile isfound in unfolding experiments on titin.24 However,while in titin the sawtooth pattern is due to theunfolding of protein domains associated with specificinteractions, here they arise from conformational tran-sitions of an initially structureless PE. This suggeststhat, despite the diversity in the mechanisms of unfold-ing, the stepwise behavior of the force-extension profilemay represent a rather general phenomenon.
What are the minimal requirements that lead tostepwise unfolding? And are there other systems thatexhibit similar features? In fact, one should expect awide variety of systems that show this behavior. Espe-cially, polymer chains that self-assemble into a chain
of subunits connected by strings are promising candi-dates. Polyampholytes show a necklace structure withmultidisperse beads.2 Polysoaps (polymers where afraction of the monomers is amphiphilic) self-assembleinto a chain of micelles.40 Polyelectrolytes that complexwith oppositely charged macroions may form necklaces,where each “pearl” consists out of a macroion with apart of the PE chain wrapped around.41 The nucleosomefilament of chromatin, a complex of DNA with oppositelycharged histone octomers, is a famous example for thistype of structure.42 We expect that all these systemsunfold in a stepwise fashion. Important is here that theself-assembled subunits do not condense into one bighomogeneous aggregate, like is the case for a neutralpolymer in a poor solvent. In this particular exampleone expects just a plateau in the force-extension curve,which corresponds to the unraveling of the globule, butno sawtooth pattern.28 Thus, one needs a mechanismthat ensures an upper bound for the size of the subunits.The other important ingredient is a mechanism thatleads to an increase in the free energy when the size ofthe subunit is decreased upon stretching. The interplaybetween these two competing mechanisms, which arenot necessarily of the same origin, ensures an optimalsize for the subunit. To make this more transparent,we give in Table 1 an overview over different polymericsystems and list the corresponding self-assembled sub-units, the mechanisms that limit the size of thesesubunits, and the penalty one has to pay when onelowers their size as a result of chain stretching. Forproteins, the unfolding depends on their detailed struc-ture and associated specific interactions (cf., e.g., ref 43).
The discrete stepwise character of the unfolding is notappreciated by most of the treatments on these systems.For polysoaps, for instance, one focuses on the coexist-ence of free amphiphiles and amphiphiles in the micel-les, which leads to a plateau in the force-extension pro-file.40 Indeed, also our analysis shows roughly a plateau,but a closer inspection reveals the sawtooth pattern ontop of it (cf. for instance Figure 8). Other investigationsare devoted to the case of an externally applied constantforce. This situation is much simpler: when the forcereaches a certain threshold value, all subunits unfoldsimultaneouslysat least, for identical subunitssasso-ciated with a jumpwise increase of the end-to-end dis-tance. The resulting monotonic stress-strain character-istics was studied, for instance, for the case of a poly-ampholytic necklace in an external electric field (see ref44 and references therein). We hope that micromechani-cal experiments will be able to test these predictionsdirectly.
Note Added: After the submission of this work webecame aware of the preprint by Vilgis et al.,45 in whichcomplementary topics of the same subject are consid-ered.
Acknowledgment. We are grateful to P. Pincus forsuggesting and discussing this problem and to A. W. C.Lau for bringing our attention to some relevant relatedwork. M.N.T. acknowledges the financial support of theBrazilian agency CNPqsConselho Nacional de Desen-volvimento Cientıfico e Tecnologico. H.S. was supportedby the National Science Foundation Grant DMR-97-08646. This research was also partially supported bythe MRL Program of the National Science Foundationunder Awards DMR-96-32716 and DMR-96-24091.
Figure 10. Measured force F(t) as a function of the externallyimposed end-to-end distance, L ) L0 + Vt, for the same set ofparameters shown in Figure 9: N ) 5000, u ) 2, τ ) 0.4, andφ ) 1.5%. The different curves show the effect of progressivelyfaster pulling velocities, varying from V ) 1 µm/s to V ) 1mm/s. We used a shear viscosity η ) 10-2 J s/m3. For theslowest drift velocity, the friction forces are too small, and themeasured force is indistinguishable from the static restoringforce presented in Figure 9. As we increase the pulling velocity,however, the friction contribution becomes more important.
Appendix A. Free Energy of the NecklaceConfiguration
We give here a detailed discussion of the contributionsto the total free energy of the necklace configuration,eq 13.
The intra-string free energy Fs accounts for theelectrostatic repulsion of charged monomers inside thestrings plus their surface tension energy. If dstr de-creases to a value smaller than êT, we should replacethe surface tension energy by the entropic energy of astretched Gaussian coil. Hence
with
The geometrical prefactor for the string electrostaticenergy was obtained by approximating it by a prolateellipsoid of major semiaxis a> ) lstr/2 and two identicalminor semiaxes a< ) dstr/2 carrying a charge Q ) eφmstr,whose electrostatic self-energy is given by46
The intra-bead free energy Fb accounts for the elec-trostatic repulsion of charged monomers inside thebeads plus their surface tension energy,
The geometrical prefactor for the charged-sphere elec-trostatic self-energy was obtained using the limitlimxf1ω(x) ) 1.
The inter-bead free energy Fbb takes the electrostaticrepulsion between different beads into account,
where γ ) 0.5772... is the Euler constant and ψ is thedigamma function.47 The above contribution correspondsto the energy of a linear arrangement of Nbead pointcharges eφmbead equidistant lstr + dbead apart.
The inter-bead-string free energy Fbs follows fromthe electrostatic repulsion between beads and strings,
The strings are assumed to be cylinders of length lstrand width dstr carrying a charge eφmstr.
Finally, the inter-string electrostatic repulsion Fssleads to
The leading term corresponds to the interaction energyof charged lines with linear density eφmstr/lstr alignedwith the center of the beads and equally spaced dbeadapart along the longitudinal direction. Since the exactintegral is awkward and this term leads only to minorcontributions to the total free energy, we consider justthe leading correction in dstr to the charged-line expres-sion.
It is convenient to introduce scaled dimensionlessvariables,
Table 1. Examples of Polymeric Systems That Self-Assemble into a String of Subunits; the Subunits Are Listed Togetherwith the Mechanisms That Control Their Optimal Size
polymeric system subunit that unfolds limiting-size mechanism of subunits penalty for lowering size of subunits
polyelectrolyte in poor solvent spherical globule Rayleigh instability surface tensionpolyampholyte spherical globule (polydisperse) Rayleigh instability (excess charge) surface tensionpolyelectrolyte-macroion complex macroion plus wrapped chain overcharging underchargingpolysoap micelle headgroup interaction, coronal loops surface tensionprotein folded domain specific specific
5270 Tamashiro and Schiessel Macromolecules, Vol. 33, No. 14, 2000
in terms of which, after some algebraic manipulations,we may rewrite the contributions to the free energy,
with
Appendix B. Free Energy of the TadpoleConfiguration
The total free energy of the tadpole configuration hasthree contributions,
accounting for intra-string, intra-bead, and bead-stringinteractions, respectively.
The intra-string contribution F st for the tadpole
configuration is given by eq A.9, while the intra-beadfree energy F b
t may be obtained by taking Nbead ) 1 ineq A.10. Finally, the bead-string interaction reads
with
Appendix C. Validity of the Steady-StateApproximation
In section 4 we assumed a quasi-static steady-stateequilibrium for the beads of the necklace configuration.This approximation is valid if the passage time
is larger than the bead relaxation time trel.29 To estimatetrel, we consider an infinitesimal deformation of anisolated spherical bead of radius R ) dbead/2 into aprolate ellipsoid of major semiaxis R> ) R + δR andtwo identical minor semiaxes R< ) R - δR/2. Thiselongated shape ensures that the volume remainsconstant, R3 ) R>R<
2 + O(δR2). The energy (in units ofkBT) of a prolate ellipsoid of eccentricity
The first term corresponds to the electrostatic self-energy,2,46 while the second term is the surface tensioncontribution taking the constraint of fixed volume intoaccount.2 Performing an expansion about the sphericalshape (ε ) 0) yields
Therefore, keeping only quadratic terms in δR, therelaxation equation reads
and the ellipsoid relaxes exponentially to the sphericalshape with an inverse relaxation time
For the range of parameters corresponding to Figure10, the ratio between the passage and the relaxationtimes,
is always larger than one, and therefore a quasi-staticapproximation is allowed.
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MA992025S
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5272 Tamashiro and Schiessel Macromolecules, Vol. 33, No. 14, 2000
