Eur. Phys. J. E 16, 199206 (2005)DOI: 10.1140/epje/e2005-00021-2 THE EUROPEAN

PHYSICAL JOURNAL E

Predicting the mechanical properties of spider silk as a modelnanostructured polymer

D. Porter1,a, F. Vollrath1, and Z. Shao2

1 Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK2 Department of Macromolecular Science and Key Laboratory of Molecular Engineering of Polymers of Ministry of Education,Fudan University, 200433 Shanghai, P.R. China

Received 4 August 2004 / Received in final form 6 October 2004Published online 22 February 2005 c EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2005

Abstract. Spider silk is attractive because it is strong and tough. Moreover, an enormous range of me-chanical properties can be achieved with only small changes in chemical structure. Our research shows thatthe full range of thermo-mechanical properties of silk fibres can be predicted from mean field theory forpolymers in terms of chemical composition and the degree of order in the polymer structure. Thus, we candemonstrate an inherent simplicity at a macromolecular level in the design principles of natural materials.This surprising observation allows in depth comparison of natural with man-made materials.

PACS. 87.15.La Mechanical properties 81.05.Lg Polymers and plastics; rubber; synthetic and naturalfibers; organometallic and organic materials

1 Introduction

Spider silk is a strong and tough polypeptide that is op-timised by nature to fulfil a wide range of functions usingsubtle changes in chemical composition and, importantly,morphological structure at a nanometer scale [1,2]. At-tempts to emulate the attractive properties of silk andother natural polymers are frustrated by a lack of quanti-tative structure-property relations, which are the subjectof this work.

In order to develop a quantitative model for silk, let ustake the radical step of looking at silk from the perspectiveof a user. Silk fibres are produced by a spider to managemechanical energy for different tasks without breaking: tostore elastic energy in supporting its own weight or for thestructural framework of a web, and to absorb kinetic en-ergy to capture flying insects. Here we identify the mech-anisms at a molecular level that dictate energy storageand dissipation in a polymer and derive straightforwardanalytical relations for the full range of mechanical proper-ties that are possible in silk. These relations are expressedin terms of a small number of energy-based parameterswith a direct fundamental link to chemical compositionand morphological order. In this way, we hope to eluci-date some of the key design principles in natural polymers.Moreover, this approach can be applied also to much sim-pler man-made polymers, thereby offering the potentialfor the design of improved synthetic polymers.

Contribution presented at the World Polymer CongressMACRO 2004, Paris, France, July 2004

a e-mail: DPORTER@qinetiq.com

Why look specifically at spider silk? Although silk-worm silk is strong, the oscillatory motion of the wormduring spinning leads to great fluctuations in mechani-cal properties [3,4]. Spider silks, on the other hand, areevenly spun, which provides a model natural fibre withsuperb combinations of strength and toughness. Such silksrange from drag-line with a modulus and strength of about10 GPa and 1 GPa respectively to capture thread with amore modest strength, but because of its high extensibilitya comparably high toughness.

Figure 1 shows a schematic diagram of the stress-strain profile of a silk fibre through a general loading cy-cle and also the deformation through to break. The ini-tial modulus, Ei, takes an upper value for dragline silk ofabout 15 GPa, down to about 4 GPa for capture thread.The yield strain, y, under ambient conditions is usuallyabout 2%. Other parameters of yield stress, y, post-yieldmodulus, Ey, and stress and strain to break, b and b re-spectively, are dependent upon the detailed compositionand morphology of the silk polymer. Generally, a high ini-tial modulus and failure stress has a lower strain to failure,and vice versa.

Recent experimental measurements of dynamic me-chanical and stress-strain properties of single fibres ofNephila Major Ampullate drag-line silk (over a wide rangeof temperatures from 100 to +350 C) suggest that themechanisms in silk deformation are no more complex thanfor any other semi-crystalline polymer [5]. The small insertin Figure 1 is a typical stress-strain plot at 15 C whileFigure 2 shows the temperature dependence of dynamictensile modulus and loss tangent for the same silk, with

200 The European Physical Journal E

Fig. 1. Schematic diagram of the stress-strain profile of a spider silk in a deformation cycle and during strain to break.The insert shows experimental curves for a dragline silk discussed in the text. The numbered regions relate to the followingdeformation processes: 1. Initial elastic zone through to yield: amorphous domains change from glass to rubberlike states atyield. 2. Post-yield elastic zone of a crystal-reinforced rubber. 3. Plastic zone as mechanical energy transforms rubber statesback to glass. 4. Recoverable elastic energy: approximately the sum of zones 1 and 2.

Fig. 2. Experimental DMTA data for tensile modulus and loss tangent and initial modulus from tensile tests for the mechanicalproperties of a dragline silk discussed in the text.

points also showing the initial modulus from stress-strainplots at different temperatures; loss tangent here is the ra-tio of energy dissipated to energy stored in a deformationcycle, and plays a key role in the toughness of a polymer.

The semicrystalline morphology of spider silk has beendiscussed extensively in literature, but the nanometerscale of the different domains are difficult to define interms of specific crystallographic forms with the sameclarity as synthetic polymers. Grubb proposes only 12%crystallinity in the dragline silk of Nephilia clavipes usingwide angle X-ray diffraction, which clearly represents the-sheet domains of poly(alanine) segments [6]. Van Beek ismore circumspect, again identifying the -sheet domainsusing NMR in silk from Nephilia edulis, but also a largefraction of helical structures in the glycine residues witha 3-fold symmetry alongside the truly disordered rigidamorphous phase [7]. The term non-periodic lattice isa useful way to describe order in silk, since neighbouring

polymer chain backbones can be aligned very regularlywith hydrogen bonding between the chains, but the irreg-ular distribution of side-chains along the backbone chainlength leads to a lack of precise order perpendicular to thechains [8].

Previous models have provided limited insights intothe mechanical properties of silk fibres [9,10], but aregenerally weak in their link to chemical composition andsemicrystalline morphology. For example, the networkmodel of Termonia simulates properties in remarkableagreement with observation. This model is based upona hypothesis of a hydrogen-bonded rubber matrix rein-forced by a fraction of rigid crystal domains, acting asphysical crosslink sites [11]; the stiff hydrogen bonds arefirst broken to allow the dynamic rubber phase to re-distribute the deformation field to predict the nonlinearlarge strain deformation. The NMR studies of van Beekshow that the non-crystalline phase in dry silk is not

D. Porter et al.: Mechanical properties of spider silk 201

rubber, but glassy, which means that care must be exer-cised in the use of the term rubber in any model [7]. Theconventional alternative to network models for polymersis detailed molecular dynamics simulations [12]. However,structures such as silk require enormous model calcula-tions to simulate dimensions and time scales far shorterthan those of experiment. Consequently understandingthe mechanisms that quantitatively control complex me-chanical properties are at present virtually impossiblewith such a black-box approach. The same comments onstructure-property relations can be applied equally well tosemicrystalline synthetic polymers and fibres, which aredominated by phenomenological models, based upon con-tinuum theory [13,14].

2 Model

Our work is based on two premises of energy storage andenergy dissipation. The first implies that the high modulusand great strength of silk are due to the high cohesiveenergy density of hydrogen bonding between the amidesegments of the backbone in the primary silk protein(Spidroin) molecules. The second premise implies that silktoughness is due to the high energy density absorbed inthe post-yield work-hardening phase through to break.

The premises are derived from the different molecularlevel mechanisms characterised in the stress-strain pro-file of Figure 1 i.e. 1) the change at yield of the amor-phous domains from glass to rubber 2), the post-yieldcomplex modulus 3), the plastic behaviour as the rubberystate converts back to glass and 4) the recoverable elasticenergy.

The analysis uses a mean-field modelling methodcalled Group Interaction Modelling (GIM) that has beenused before to predict the engineering properties of poly-mers like poly(styrene) and poly(carbonate) [15]. GIM ap-plies the ensemble average values of the thermodynamicenergy terms of a characteristic mer unit (at a molecularlevel) to a potential function that quantifies the relation-ship between intermolecular energy and separation dis-tance between polymer chains. Thus, GIM allows analyti-cal structure-property relations for mechanical propertiesto be derived by a combination of mathematical processingof the potential function and application of continuum-level physics of a materials energy-storage and energy-loss. Hence, the quantitative physical links between struc-ture and properties are transparent at all stages of themodelling process.

To avoid unnecessary controversy over crystalline frac-tions (see introduction), let us take the straightforwardposition that silk consists of ordered and disorderedphases, and that the more rigid ordered domains are dis-persed at a nanometer scale within the less rigid amor-phous phase. The nanoscale morphology allows the twophases to share energy at the molecular level and behavein bulk as a homogeneous material (e.g. [16]). Energy shar-ing at a nanoscale is a key aspect in natural materials, andhas been discussed quantitatively for bone, which consistsof organic and inorganic layers to form a rigid and toughnatural hybrid nanocomposite [17].

A detailed analysis of the component groups in theprimary structure of both Spidroin I and II suggests thatpoly(alanine) can be taken as a hypothetical model poly-mer with ensemble average parameter values that repre-sent the chemical composition of Spidroin [18]; the cal-culation of model parameters for Spidroin I is given inthe appendix to illustrate the use of ensemble averagedparameter values.

The alanine segment consists of a highly polar amidegroup and a non-polar hydrocarbon group, such that theinteractions at a molecular level are most likely to be be-tween like groups of atoms.

The model poly(alanine) exists in ordered or disor-dered states with fractions ford and fdis respectively. Themain difference here between the ordered and disorderedstructures is the number of hydrogen bonds per segment.The ordered segments have two hydrogen bonds per seg-ment due to favourable alignment of adjacent chain back-bones. Disorder reduces the inter-chain bonding to onehydrogen bond per segment due to unfavourable align-ment of backbone chains. However, both ordered and dis-ordered phases can be oriented in the fibre axis, with thedisordered phase simply having more non-minimum en-ergy conformers.

The polymer group structure is quantified by a limitednumber of model parameters that allow the main energyterms at a molecular level to be calculated for use in thepotential function. A set of parameters for poly(alanine)have been calculated using a combination of group addi-tivity tables [15,19], connectivity indices [20], and molecu-lar modelling [21,22]. The difference between ordered anddisordered phases is only in the cohesive energy, using10 kJ/mol per hydrogen bond, and the overall value ofcohesive energy for any silk is simply the sum of the or-dered and disordered fractions. The parameter N is thenumber of degrees of freedom per group and plays a keyrole in the energy dissipation model. The value of N isgiven for the glassy or crystal state of matter as skeletalvibrations perpendicular to the chain axis, butN increasesby 50% above the glass or crystal melt transition as thepolymer becomes mobile in the chain axis, which is re-flected in the change in heat capacity through the glasstransition [19,23]

Cohesive energy Co = 52 kJ/mol ordered or42 kJ/mol disordered

Van der Waals volume Vw = 40 cc/molMolecular weight M = 72Degrees of freedom N = 8Debye temperature = 400 K

The Lennard-Jones potential function for the bind-ing energy, C, per group for a strongly bonded one-dimensional polymer chain in a weak three-dimensionally

202 The European Physical Journal E

bonded lattice can be written in terms of the thermal en-ergy of skeletal mode vibrations, HT , and the configura-tional energy, Hc, by

C = Co +Hc +HT = Co +Hc + N RT D1(T )4= Co

((VoV

)6 2

(VoV

)3)(1)

where Vo = 1.26Vw is the volume at the minimum of theglobal potential well, R is the gas constant, and D1(T )is the one-dimensional Debye function consistent with thestructural model for the potential function [24,25]. Theconfigurational energy differentiates between ordered anddisordered states of matter and takes fractional values ofthe cohesive energy of 0.04 Co and 0.106 Co respectively;greater order has fewer excited conformer states, and al-lows the difference between crystal and amorphous statesof matter to be quantified.

The most direct predictions from the potential func-tion are volume, V , and the bulk elastic modulus, B, as afunction of temperature. Volume as a function of temper-ature can be obtained by solving the potential function,and gives density values in the range 1.2 to 1.3 g/cc. Bulkmodulus is particularly important here as a reference pa-rameter, and is given by

B = V d2 C

dV 2= 18

C

V(2)

which shows that elastic modulus is determined directlyby the cohesive energy density and supports the firstpremise of this model.

3 Elastic modulus and loss

The results of our Dynamic Mechanical Thermal Analy-sis of a single fibre of dragline silk are shown in Figure 2.They are consistent with the observation for polymers ingeneral that the steepest negative gradient in elastic mod-ulus is associated with peaks of loss tangent due to re-laxation processes at a molecular level [26]. Qualitatively,elastic modulus is a measure of the elastic energy storedin a material, so greater energy dissipation or loss mustbe reflected in a lower value of elastic modulus. Bondipointed out that the empirical proportionality betweenloss tangent and the temperature gradient of elastic mod-ulus could be used to predict elastic modulus if the mech-anism for mechanical energy transformation to heat couldbe identified [27]. Porter derived a general expression forthis proportionality in terms of the structural model pa-rameters and the chain length per group, L 0.3 nm,with A as the proportionality constant [15]. The deriva-tion relates the change in thermal energy in the polymer tothe mechanical energy input during mechanical deforma-tion via its effect on the theta temperature in the Tarasovform of the Debye theory for heat capacity; loss tangent

is then the fraction of mechanical energy converted irre-versibly to heat. The most appropriate form of the modelexpression is

tan ABE

dEdT

=(1.5 105 L

M

)B

E

dE

dT. (3)

The parameter A takes a general value of about unity inunits of K/GPa, and specifically for poly(alanine) A 1.5 K/GPa. This expression can be inverted to allow elas-tic modulus to be calculated from the loss profile of a poly-mer, which is easier to predict from the chemical structureusing models for relaxation mechanisms. Note that bulkmodulus is the reference value of elastic modulus beforeany relaxation mechanisms allow transient spatial redis-tribution of groups of atoms under strain to reduce thevalue of modulus

E = B exp( tan dT

A B). (4)

The loss tangent peaks of the silk in Figure 2 are due to re-laxation events in the different atomic groups of the poly-mer chain. In particular, the peaks below about 300 Care local phase transition events in the disordered fraction.The peak at about 70 C is conventionally attributed tohydrocarbon interactions [26], and the higher temperaturepeak at about 200 C is shown below to be attributed toamide segment interactions. The temperature of the peakscan be predicted as local glass transition temperature, Tg,events in terms of model parameters for the specific groupinteractions by the relation [15]

Tg = 0.224+ 0.0513CoN

. (5)

The model expression for the glass transition is derivedusing the Born criterion for a transition, from the energyat which the force between chains is a maximum using theLennard-Jones potential function, and has been validatedfor a dataset of about 250 polymers. The expression showsthat stiffer chains with a higher skeletal theta temperatureand higher cohesive binding energy give a higher glasstransition temperature, but that greater thermal energythrough more degrees of freedom reduces Tg.

The predicted transition temperatures are given in Ta-ble 1 and are in good general agreement with the peak po-sitions in Figure 2. Note that the extra degrees of freedomequal to fam, in the amide group are due to the extra de-grees of freedom developed in the alanine segment abovethe hydrocarbon transition temperature and the value ofTg = 479 K is for the silk used in this section as an exam-ple for predicting mechanical properties with ford = 0.66.

Each loss tangent peak is modelled as a Gaussian dis-tribution of transition events over a distribution of tem-peratures with a standard deviation, s, that is broader fora greater range of different structural group interactions;Figure 2 suggests a value of s 30 degrees. The total areaunder each Gaussian peak n is called the cumulative losstangent, tann, such that the area up to a temperature Tis expressed in terms of the activated fraction of groups up

D. Porter et al.: Mechanical properties of spider silk 203

Table 1. Segment interaction parameters and properties.

Segment Co (J/mol) N Tg (K/C) tan Fraction

Hydrocarbon 9000 4 205/68 19 0.5Amide: 1 H-bond 33000 4 + fdis 479/206 68 0.5

Fig. 3. Predicted dynamic mechanical properties for a model silk with ford = 0.66, to be compared with the experimental plotsin Figure 2.

to T , fn, in the disordered fraction, fdis, and the modelparameters in the GIM expression for tan

tan ndT = tannfnfam = 0.0085ConNn

fnfam. (6)

The values of tann are given in Table 1 for the hydro-carbon and amide interactions. The relation for tan =0.0085 Co/N is derived again from the irreversible changein thermal energy due to new degrees of freedom, N/2,that are invoked by forcing the structural groups througha transient glass to rubber phase transition by mechanicalenergy of deformation. The fractions of the hydrocarbonand amide interactions are their volume fractions, whichgives the correct value of tan = 45 for the model alaninesegment overall.

Figure 3 uses equations (2), (4), and (6) to reproducethe loss tangent and the bulk and tensile elastic moduliof a model silk with an ordered fraction ford = 0.66. Twocurves are shown for tensile modulus, which are the lim-iting upper and lower predicted values. The upper valueassumes that the activated hydrocarbon groups are pinnedby the amide groups and does not allow energy dissipa-tion until the model alanine segment as a whole becomesmobile. The lower value assumes spatial redistribution tooccur; the following discussion uses the upper curve. Notethat the dynamic mechanical property calculations stopat the point where observed modulus starts to rise at hightemperatures, since this marks the onset of thermally in-duced crystallisation in the sample, with a loss peak de-veloping with the increasing modulus in Figure 2.

The value of ordered fraction used in the predictionsfor Figure 3 was chosen ad-hoc to make the initially bestmatch with the measurements shown in Figure 2 for adragline silk. By changing the single variable of the frac-

tion of the ordered phase, we propose that the model canreproduce the full range of mechanical properties of spi-der silks. Qualitatively, a greater degree of order has fewergroups that undergo phase changes to dissipate energy,and is thereby stiffer and stronger. For example, a highproline content in a silk such as flagelliform is less stiffand more extensible due to the disorder induced by theproline. The next step in the model is to translate the dy-namic mechanical property predictions into engineeringstress-strain profiles over the range of possible values forordered fraction.

4 Stress-strain profileEnergy in the potential function used to predict thermo-mechanical properties in silk is dependent upon the di-mensions occupied by the interacting groups of atoms.This can be used to make a transformation of the temper-ature dependent dynamic mechanical properties to pre-dictions of stress-strain curves to break. To do this, tem-perature, T, is used as a dummy variable to calculateconsistent pairs of values of strain, , and tensile stress, ,using the linear coefficient of thermal expansion, , thatis predicted in GIM by the relation derived by differenti-ating the potential function of equation (1) with respectto temperature

0.45NCo

0.0001 K1 (7)

=

TT

dT =

TT

E dT . (8)

Five key parameters are needed to map the overall stress-strain profiles that are possible with spider silk: initial

204 The European Physical Journal E

(a) (b)

Fig. 4. (a) Predicted stress-strain profiles of silks with different degrees of order, ford. (b) Experimental stress-strain curvesfor silks within the predicted envelope of properties shown as a dashed line: silks curves identified by markers at break point: various dragline silks (4), Bombyx mori drawn evenly at different rates [3,4], Argiope trifasciata dragline silk [31], Laboratory spun recombinant silk based upon Spidroin II [32].

modulus, Ei, yield strain, y, yield stress, y, post-yieldmodulus, Ey , and strain to break, b. These parametersare shown in Figure 1.

Taking the ambient temperature to be 20 C, the yieldstrain can be estimated by associating the yield pointwith maximum rate of reduction in modulus through thepeak in loss tangent of the amide groups in the disorderedphase, at a temperature about 200 degrees above ambi-ent; yield is thereby considered to be a mechanical formof the glass transition condition, where a transient glassto rubber phase change occurs in the disordered fractionto dissipate elastic energy. This suggests a yield strain ofabout 2%. The yield stress is taken as a first approxima-tion to be the product i Ei.

The initial and post yield moduli are the values of E atambient temperature and at the upper end of the amideloss tangent peak at about 300 C, and are calculatedusing appropriate values of B and cumulative loss valuesin equation (4) after each of the two relaxation peaks;namely, tan values of 19 and 44 respectively. The upperand lower limiting values of Ei for totally ordered anddisordered silks are 14 and 2.6 GPa respectively, whichis in good general agreement with observation. The post-yield modulus has upper and lower values of 12 GPa andthe rubberlike plateau modulus of about 10 MPa. Notethat work hardening of the polymer above yield is notincluded in this first presentation of the model, but Eyincreases with strain as rubberlike states are transformedback to glassy states.

Finally, the strain and stress to break is calculatedfrom the mechanical energy absorbed in the post-yield de-formation, shown as regions 2 and 3 in Figure 1. In thisregion, mechanical energy transforms rubberlike states ofmatter back into glassy states as the polymer chains areforced closer together by the tensile strain, thus remov-ing the extra skeletal degrees of freedom above the yieldor glass transition condition, Ng = 0.5 N. The strainto break is calculated from the equivalence of mechanical

and thermal energy density using as a first approximation

y (b y) + Ey (b y)2

2= Ng R T D1(T )

V

1.2 108 J/m3. (9)The left hand side of equation (9) is the mechanical en-ergy density as the sum of plastic and elastic terms, whichare regions 2 and 3 in Figure 1. The right hand side ofequation (9) is the change in energy due to the change indegrees of freedom through the phase change, and its nu-merical value is a direct quantitative measure of the tough-ness of a polymer as the area under a stress-strain curveto break. For example, synthetic engineering polymerspoly(carbonate) and poly(ether ether ketone) (PEEK)have values 0.5108 and 0.4108 J/m3 respectively, andare thus far less tough than spider silk by a factor of morethan two. Polymers such as poly(ethylene) and Nylon mayhave comparable model measures of toughness, but theyrequire a high degree of molecular orientation to achievethe stiffness of dragline silk and do not have the highertemperature stability imparted by the hydrogen bondingin the high amide group concentration in dry silk.

For ease of calculation, the Debye function takes avalue of 0.7 for the model = 400 K and the volumeis taken to be V 1.5Vw = 60 cc/mol. Figure 4a showsthe full range of stress strain profiles predicted for themodel silk at different values of the single model vari-able ford, and all other parameters are calculated fromford using the structure-property relations derived above.Experimental plots of stress-strain for some typical silktypes are plotted in Figure 4b for comparison with pre-dictions, with each curve labelled or explained in the figurecaption. The overall limiting envelope of predicted stress-strain profiles has the correct general form, and futurework will explore the detailed bounds of the failure en-velope due to energy absorption mechanisms post-yield,such as the formation of new hydrogen bonds in the moreoriented strained structure.

D. Porter et al.: Mechanical properties of spider silk 205

Table A.1. Calculation of the average group parameters for Spidroin I.

Group Structure Number Co Vw M N

(kJ/mol) (cc/mol)

Peptide base - CCONH - 1 24.3 29 56 6

-R groups

Glycine (G) -H 16 0 0 0 0

Alanine (A) -CH3 10 4.5 10.2 15 2

Glutamine (Q) -CH2CH2CONH2 3 28.8 36.1 72 5

Tyrosine (Y) -CH2PhOH 1 35.8 58.5 107 4

Leucine (L) -CH2C(CH3)2 2 18 40.8 57 4

Arginine (R) -CH2CH2CH2NHC(NH2)2 1 45 52.8 101 7

Serine (S) -CH2OH 1 10.8 15.2 31 3

Average -R 7.7 12 12 2

Total average 32 41 68 8

5 Conclusions and discussion

Mean field theory for polymers has been used to pre-dict structure-property relations for the mechanical prop-erties of spider silk in terms of chemical compositionand morphological structure. The chemical compositionof Spidroin proteins is embodied in the model parametersof poly(alanine), whose morphology is quantified by thesingle variable parameter of the fraction of ordered phase.The dynamic mechanical properties of silk are derived interms of the ordered fraction, and are then transformedinto predictions for the whole range of stress-strain pro-files that are possible in silk. The predicted envelope ofsilk properties is in good quantitative agreement with ex-perimental observation. Hence the surprising outcome ofour approach was the demonstration that spider silk, ap-parently so complex in morphology, can be modelled accu-rately with relatively few and rather robust assumptions.Also, a key to the silks strength clearly lies in the peculiarmolecular and nano-scale structure of its morphology.

The initial premise of this work assumed that stiffnessand strength, on the one hand, are due to the high cohesiveenergy density of hydrogen bonding. Toughness, on theother hand, is due to the high energy absorption duringpost-yield deformation. The model shows both premises toform a useful basis for understanding the structural originsof the properties of spider silk: but not just spider silk. Ourmodel is fully portable not only to other bio-polymers butalso to the much less complex non-biological, man-madeengineering polymers such as poly(carbonate) and PEEK.Thus, with a curious twist, insights into silk, which is theoldest commercial high-strength fibre, will in the futureenable us to better understand its modern (although stillmuch less sophisticated) man-made derivatives.

Although the main variable parameter of degree oforder may appear vague and no assumptions have beenmade as to the morphology associated with order, it doesforce us to reconsider the rather narrow description ofcrystal and amorphous states in a protein. Ongoing workis looking at how processing the silk (rate, temperatureand chemistry [28]) and cyclical loading of silk fibres [29]changes the observed degree of order. Group Interaction

Modelling was originally formulated to develop structure-property relations for polymer processing rheology andto include such non-Newtonian effects as shear-thinningand strain-induced crystallisation due to macromolecularstrain [30]. Thus, we expect that the model presented herefor silk properties will provide a valuable direct link to helpunderstand the production of silk fibres with controlledmechanical properties.

For funding we thank the British EPSRC (grantGR/NO1538/01) and BBSRC (S12778) as well as theEuropean Commission (grant G5RD-CT-2002-00738), theNational Natural Science Foundation of China (NSFC,No. 20244005) and the AFSOR of the United States ofAmerica (grant F49620-03-1-0111).

Appendix A: Model parameters for spidroin

Spidroin I has the repeat sequence:

The structure of each peptide segment is different in theR group, such that highly irregular sequences of sidechains are attached to a very regular chain structure.

The detailed GIM model parameters for Spidroin I aretabulated above (Tab. A.1) and then averaged to showthat the ensemble average parameter set can be approxi-mated very well by those of poly(alanine), except the cohe-sive energy, which is take to be the average of all peptide

206 The European Physical Journal E

segments. In addition, the cohesive energy must also besupplemented by hydrogen bonding energy, at 10 kJ/molper hydrogen bond. Thus, the disordered and ordered frac-tions have +10 and +20 kJ/mol extra due to hydrogenbonds respectively. Hydrogen bonding between side chaingroups has been ignored at this stage, but would tend toincrease the average cohesive energy value by a maximumof 3 kJ/mol.

The Debye theta temperature is calculated from theaverage molecular weight per segment to be 1 = 400 K,and the average length per segment in the chain is takento be L 0.3 nm.

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