Hierarchical Chain Model of Spider Capture Silk Elasticity

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by  Haijun Zhou

Hierarchical Chain Model of Spider Capture Silk Elasticity
Haijun Zhou1 and Yang Zhang2
1Max-Planck-Institute of Colloids and Interfaces, 14424 Potsdam, Germany
2Center of Excellence in Bioinformatics, University at Buffalo, 901 Washington St., Buffalo, NY 14203
(Dated: 21 January,2005)
Spidercapturesilkisabiomaterialwithbothhighstrengthandhighelasticity,butthestructural
5 design principle underlying these remarkable properties is still unknown. It was revealed recently
0 byatomic force microscopy that an exponentialforce-extension relationship holds both for capture
0 silk mesostructures and for intact capture silk fibers [N. Becker et al., Nat. Mater. 2, 278 (2003)].
2 In this Letter a simple hierarchical chain model was proposed to understand and reproduce this
striking observation. In the hierarchical chain model, a polymer is composed of many structural
n
motifs which organize into structural modules and supramodules in a hierarchical manner. Each
a
J modulein thishierarchyhasits own characteristic force. Therepetitivepatternsin theamino acid
sequenceof the major flagelliform protein of spider capture silk is in support of this model.
4
[Phys. Rev. Lett. 94, 028104 (2005)]
2

PACSnumbers: 87.15.-v,78.40.Me,81.05.Lg,87.10.+e
]
t
f
o
s Thecapturesilkisanaturalmaterialproducedbyorb- about 105 times that of a molecule [3]. It appears that ℓ
t. web weaving spiders. Spiders rely on it to entrap flying holds an approximately linear relationship with the con-
a
preys [1]. Like the spider dragline silk and many other tour length.
m
naturally occurring silks, the capture silk has a tensile
The exponential force-extension curve is significantly
- strength that is comparable to steel; but, unlike steel, it
d different from the data observedduring stretching single
n is also extremely elastic, with the ability to be stretched
double-stranded (ds) DNA molecules [6] or single titin
o to almost 10 times its relaxed contour length without
proteins [7, 8]. The behavior of dsDNA can be under-
c breaking[2,3]. Thisperfectcombinationofstrengthand
[ stood by the wormlike chain model of entropic elasticity
extensibility conveys a high degree of toughness to the
[9, 10], and that of titin by a two-level system coupled
2 capture silk: its breakageenergy per unit weightis more
with entropic elasticity [11]. Similar exponential force-
v than 20 times that of a high-tensile steel [2]. With the
8 extension data were also observed by Dessinges et al.
aim to produce synthetic silks with similar mechanical
3 whentheystretchedasingle-stranded(ss)DNAmolecule
properties,materialsscientistshavedevotedmanyexper-
5 at low salt conditions [12]. The data was explained as a
2 imental and computational efforts to the understanding
result of the interplay of electrostatic repulsion and en-
1 of spider silk’s structural organization [4]. Despite these
tropic elasticity [12, 13]. However, the success of this
4 painstaking efforts, the mechanism behind spider silk’s
0 model depends on the specific ionic concentration (i.e.,
remarkable strength and elasticity is still largely miss-
/ low salt and high electrostatic interaction); and it could
t ing,partlybecauseofthedifficultytoobtainhigh-quality
a not naturally reproduce the exponential stretching data
m crystallized structures of silk proteins.
atextremelyhighforce(i.e.,whentheextensionislarger
Single-molecule manipulation methods have been re-
- than 1.1 times the ssDNA backbone length) if higher or-
d cently applied on spider silks to obtain very detailed in-
der deformation energy terms are not included [12]. In
n formation on their force-extension response [3, 5]. In a
o the spider capture silk experiment, the exponential be-
recent experiment, Hansma and co-workers [3] attached
c havior was observed at both fluid and air within a force
v: capturesilkmesostructures(probablycomposedofasin- range from about 100 piconewton (pN) to about 106 pN
gle protein molecule) or intact capture silk fibers to an
i [3].
X atomicforcemicroscopytipandrecordedtheresponseof
r thesamplestoexternalstretchingforce. Theyfoundare- Equation (1) indicates the following: (i) Because the
a markableexponentialrelationshipbetweenthe extension capture silk is highly extensible, a great amount of ex-
x and the external force f, tralengthmusthavebeenstoredinitsrelaxedform. (ii)
Sinceextensionincreaseswithforcelogarithmically,some
f ∝exp(x/ℓ), (1) fraction of the stored length must be easy to be pulled
out, some fraction be harder to be pulled out, and till
whereℓ,the lengthconstantreferredto in[3],is afitting some other fraction be even harder to be pulled out. To
parameter whose physical meaning needs to be decided modelthis kindofheuristiccascadingresponses,herewe
(see below). ℓ=110±30 nm for a capture silk molecule proposeahierarchical chainmodelforspidercapturesilk.
and ℓ = 11±3 mm for an intact capture silk fiber [3]. Inthehierarchicalchainmodel,thepolymeriscomposed
The length constant of a silk fiber is about 105 times ofmanybasicstructuralmotifs;thesemotifsarethenor-
that of a silk molecule; its relaxed contour length is also ganized into a hierarchy, forming structural modules on
2
moreandmorelongerlengthscales. Atthe deepesthier- h
archy level h , the structural motifs could be β-sheets,
m
β-spirals, helices [4] or microcrystal structures [14]. The
interactions among some of these motifs are much more
strongerthantheir interactionswith othermotifs, there- h+1 h + 1
fore they forma structuralmodule at the hierarchylevel
(hm−1). These level-(hm−1) modules are then merged h+1
intolevel-(h −2)modulesthroughtheirmutualinterac-
m
tions. This merging process is continued; and finally at
the global scale, the whole spider silk string is regarded
as a single module of the hierarchy level h=0. 2
In nature, the structures of many biomaterials are in- h + 2 h+
2
+
deed hierarchically organized. As a composite material, h
chromosomeisamixtureofDNA,histone,andothernon- h+2
histone structural proteins [15]. The DNA molecule first
wrapsontohistoneproteinstoformnucleosomeparticles, FIG.1: Thehierarchicalchainmodel. Ateachhierarchylevel
the basic units of chromosome. With the help of H1 his- hastructuralmoduleM is composed of atandemsequence
h
tone, this linear sequence of nucleosomes are then coiled ofmhsubmodulesMh+1ofhierarchylevelh+1. Underexter-
and folded to form the 30-nm chromatin fiber. With nalstretching,Mhrespondsby(i)adjustingthearrangements
of its m subunits and making them more aligned along the
the help of other scaffoldproteins,the chromatinfiber is h
force direction, and (ii) extending these m subunits. The
thenfurthercoiledandfoldedatseverallevelstoformthe h
thickbrokenlinesbetweensubmodulesofeachhierarchylevel
compactchromosomestructure. Asanotherexample,the
indicate theexistence of sacrificial bonds.
amino acid sequence of a protein first forms basic struc-
tural modules of α-helix and β-sheet, called secondary
structures. Bydifferentwaysofconnectionsofthese sec- displacement and position-rearrangement of its m1 sub-
ondary modules, the protein constructs various tertiary units; and x2(f) denotes the extension because of the
topologiesthatarecriticalforits specificbiologicalfunc- inherent deformation of a level-2 subunit M2. As this
tions. Inahigherlevel,thesetertiarydomainsthenform structuralhierarchyiscontinued,wearriveatthefollow-
quaternary structures of protein complexes consisting of ing expression concerning the total extension x(f):
multiple chains [16]. Recent experimental and theoreti-
cal studies have shown that the force–induced unfolding x(f)=∆x0(f)+ m0m1...mh∆xh+1(f). (3)
of proteinmolecules is indeed processedin a hierarchical hX=0
way, i.e. the tertiary structure precedes the secondary
It is reasonable that the average extension ∆x (f) of
structures to be pulled over at increasing external forces h
a M contributed by the reorientation or rearrangement
[7, 17]. Our hierarchical chain model may also serve as h
ofitslevel-(h+1)subunits isproportionalto therelaxed
a framework to understand the mechanical property of
contour length L of this module. To facilitate the fol-
these biomaterial systems. h
lowing analytical calculation, let us assume ∆x (f) has
ConsiderapolymerofcontourlengthL0. Itisregarded h
thefollowingnon-linearform(wewillshowthatthefinal
as the module of hierarchylevel h=0 (the module M0).
force-extensionrelationshipisnotsensitivetothespecific
M0 is composed of a tandem sequence of m0 subunits
assumption made here):
M1 of contour length L1 = L0/m0 (Fig. 1). Under the
action of an external force field f, the extension of M0 αL f/f , f <f
is denoted as x0(f). It was observed that water-induced ∆xh(f)=(cid:26)αLhh, h f ≥fhh (4)
protein mobility is a significant contribution to capture
silk elasticity [18]. Therefore, we decompose x0 into two where α is a dimensionless proportional constant; fh is
parts: (i) the extension∆x0 causedby the displacement the characteristicforce needed to displace and rearrange
and position-rearrangement of these m0 subunits and, the positions of the mh submodules contained in a Mh
(ii) the total extension m0x1 caused by the inherent de- (during this process some sacrificial bonds are broken).
formations of these m0 subunits: Equation (4) is in agreement with the experimental ob-
servation [3] that, between adjacent rapture events, a
x0(f)=∆x0(f)+m0x1(f). (2)
capture silk responds to external stretching in a linear
Similarly, at the hierarchy level h = 1, each unit is way. Denote ∆E as the energy cost of breaking all the
h
itself composed of m1 level-2 subunits M2 of length sacrificialbondsbetweenalevel-hmodule’smh subunits.
L2 = L0/(m0m1). Therefore, x1(f) can be written as From Eq. (4) we know that fh+1/fh =mh∆Eh+1/∆Eh.
x1(f)=∆x1(f)+m1x2(f), where ∆x1(f) is the contri- Consideralevel-(h+2)moduleMha+2,itisinMha+1which
bution to the extension of a level-1 module due to the in turn is in Ma. Ma feels an internal energy ǫ due to
h h+2
3
its interaction with other subunits in Ma , and it feels to 10−3N, it appears that 4–5 levels of hierarchy were
h+1
anexternalenergyǫ′duetoitsinteractionwithothersub- probed.
units in Ma but not in Ma . Based onFig. 1, we know The exponential relationshipshownin the figure is in-
h h+1
′
that ∆Eh+1 = mh+1ǫ/2 and ∆Eh = mhmh+1ǫ/2. The sensitive to our particular assumption Eq. (4), as long
hierarchical organization of the polymer requires that as the elastic response at each hierarchy level is non-
′
ǫ>ǫ, so as to ensure that structuralmodules of shorter linear and bounded. As an example, the solid curves
length scales will be formed earlier. Based on these con- in Fig. 3 show the resulting force-extension relationship
siderations,we arriveatthe following self-similarscaling when Eq. (4) is replaced by
form:
∆x (f)=αL 1−exp(−f/f ) . (8)
′ ′ h h h
fh+1 =(ǫ/ǫ)fh =βfh (β ≡ǫ/ǫ >1). (5) (cid:0) (cid:1)
The same exponential behavior as in Fig. 2 is obtained.
The parameter β characterizes the degree of coherence
However, the hierarchical scaling form Eq. (5) is needed
in the modular organization of the polymer: a large β
for the exponential force-extension correlations. For ex-
valuemeansthatasubmodulehasmuchstrongerinternal
ample, when Eq. (5) is replaced by a power-law, f ∝
h
interactions compared with its external interactions. f0hγ, the response is not exponential (the dotted line in
FromEqs.(3)and(4)wefindthatwhenfh−1 <f ≤fh Fig.3). Wealsonoticedthat,wheninEq.(5)theparam-
dx(f) = αL0 = αβL0f−1. (6) reatenrgeβoifsβn>ot1a,tchoensrteasnutltibnugtfoflruccet-ueaxtteesnsoiovnercsuormveeisfinstiitlel
df hX′=h fh′ β−1 exponential (Fig. 3, dashed lines).
Equation (6) therefore recovers the experimental expo- In summary, we have developed a hierarchical chain
nential force-extension relationship of Eq. (1) with modeltounderstandthestrengthandelasticityofspider
silks. Remarkably, this simple model was able to repro-
αβ ducethepeculiarexponentialforce-extensionresponseof
ℓ= β−1L0. (7) spider capture silk reported by Becker et al. [3]. The
model can also be used as a framework to understand
The lengthconstantℓis proportionaltothe relaxedcon-
the elasticity of other spider silks and other biopolymers
tour lengthL0 of the whole polymer, inconsistence with
with hierarchically organized structures.
Ref. [3].
Figure 2 demonstrates the numerically calculated
force-extensioncurve based onEqs.(3), (4), and (5). As
a comparison, the experimental data [3] on intact spider
capture silk is also shown. As β ≃ 2 and the exprimen- 100
tal exponential force range is roughly from 6×10−5N
10-1
N)10-2
10-2 orce (
f10-3
10-3 10-4
N)
force ( 10-50 1 2extension (x)3 4 5
10-4
FIG. 3: The force-extension relationship of a hierarchical
chain is insensitive to the assumption made to the response
10-50 0.5 1 1.5 2 ∆xh(f) of Eq. (4), but is sensitive to the hierarchical scal-
extension (x) ing form of the characteristic force f . The solid lines are
h
obtained by assuming ∆x (f) has the form of Eq. (8), while
h
other parameters are the same as those in Fig. 2. The dot-
FIG. 2: Exponential force-extension relationship for the hi- ted line shows the change in the force-extension curve when
erarchicalchainmodel. Equation(4)isusedinthenumerical additionally a power-law form of fh = f0hγ with γ = 3.0 is
calculation. The parameters are f0 = 10−4 N, α = 0.3, and assumed for fh. The dashed lines are obtained by assum-
β = 2 (the upper curve) or β = 1.75 (the lower curve). Ex- ing Eq. (8) and Eq. (5), with β fluctuating uniformly within
tensionisinunitsofL0. Symbolsareexperimentaldatafrom [1.25,2.25] (the lower curve) and within [1.5,2.5] (the upper
Fig. 4 of [3]. curve).
4
Beckeret al. [3] haveproposedan alternateand inter- forcespectroscopy. The characteristicforcesf may cor-
h
estingideatomodelthespidersilkasabranchednetwork respondtotheforcesatthesaw-toothraptureeventsob-
of interconnected springs. In their model, the system re- served by Hansma and co-workers [3]. Although the ex-
sponds to external stretching by first unfolding the sin- perimentalcurvesofRef.[3]isinagreementwithEq.(5),
gle spring at the root level, then the m springs at the more systematic investigations are necessary to draw a
first branching level, then the m2 springs at the second solid conclusion.
branchinglevel,andsoon. Thehierarchicalchainmodel
developed here is different from Becker et al.’s model.
First, the molecule in the hierarchical chain model con-
We are grateful to H. Hansma for communicating
sistsofatandemsequenceofstructuralmodulesofdiffer-
her experimental results with us prior to publication
entlengthscales. Therefore,noassumptionofbranching
and for the experimental data used in Fig. 2, and to
structure is made in our model. Second, the response
D.E.Makarovforcommentsonthemanuscript. Thanks
of the chain to external perturbations is in a hierarchi-
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cal manner. If the external force is small, only those
nick for their support.
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tural units at short and moderate length scales will re-
main intact. As the external perturbation is elevated,
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