A microscopically motivated model for the swelling-induced drastic softening of hydrogen-bond dominated biopolymer networks

Acta Biomaterialia 96 (2019) 303–309

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A microscopically motivated model for the swelling-induced drastic softening of hydrogen-bond dominated biopolymer networks Noy Cohen a, Claus D. Eisenbach b,c a

Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel Materials Research Laboratory, University of California, Santa Barbara, CA 93106, USA c Institute for Polymer Chemistry, University of Stuttgart, D-70569 Stuttgart, Germany b

a r t i c l e

i n f o

Article history: Received 9 May 2019 Received in revised form 26 June 2019 Accepted 3 July 2019 Available online 15 July 2019 Keywords: Resilin Reversible polymer networks Hydrogen cross-link bond Multi-scale analysis Swelling Softening of polymer networks

a b s t r a c t The penetration of water into rubber-like protein networks such as cross-linked resilin, which is found in insects, can lead to changes in stiffness that range over several orders of magnitude. This softening effect cannot be explained by the volumetric changes associated with pure swelling/deswelling used to describe networks with covalent bonds. Rather, this property stems from the reversible swellinginduced breaking of hydrogen cross-linking bonds that connect the chains in the network. This work presents a model for the swelling and the mechanical response of hydrogen-bond dominated biopolymer networks. It is shown that the penetration of water molecules into the network leads to the breaking of non-covalent cross-linking sites. In turn, the network experiences a reduction in the effective chaindensity, an increase in entropy, and a consequent decrease in free energy, thus explaining the dramatic softening. Additionally, the breaking of hydrogen bonds alters the micro-structure and changes the quantitative elastic behavior of the network. The proposed model is found to be in excellent agreement with several experimental findings. The merit of the work is twofold in that it (1) accounts for the number and the strength of non-covalent cross-linking bonds, thus explaining the drastic reduction in stiffness upon water uptake, and (2) provides a method to characterize the micro-structural evolution of hydrogen-bond dominated networks. Consequently, the model can be used as a micro-structural design-guide to program the response of synthetic polymers. Statement of Significance Hydrogen-bond dominated biopolymer networks are found in insects and have a unique structure that allows a dramatic reduction of several orders of magnitude in stiffness upon hydration. Understanding the micro-structure of such networks is key in the fabrication of new biomimetic polymers with tunable mechanical properties. This work introduces a microscopically motivated model that explains the dramatic reduction in stiffness and quantifies the influence of key micro-structural quantities on the overall response. The model is validated through several experimental findings. The insights from this work motivate further attempts at the fabrication of new biomimetic polymers and serve as a microstructural design guide that enables the programming of the elastic swelling-induced response. Ó 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction Resilin is an example of a rubber-like network dominated by hydrogen bonds that is found in the cuticle region of most insects [1–3]. Among other functions, resilin plays a major role in jumping [4], walking [5], and adhesion [6] mechanisms. Due to attractive properties such as high resilience, the capability of experiencing large deformations, and the ability to dramatically vary stiffness E-mail address: [email protected] (N. Cohen) https://doi.org/10.1016/j.actbio.2019.07.005 1742-7061/Ó 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

upon hydration and dehydration, resilin has been the subject of many investigations aiming to mimic its unique properties in synthetic materials [7,8] which can be used in various applications [9]. The unusual elastomeric properties of resilin are highly regulated by water content [1,3,6,8,10,11]. This stems from a unique molecular composition which differs from other structural proteins such as, e.g. elastin [12]. Resilin comprises a high proportion of hydrogen-bond forming and hydrophilic as well as polar amino acid residues (over 60 mol-%) with lysine and proline the dominating amino acids [9,12,13]. Both amino acid moieties provide high

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flexibility to resilin [13]. Furthermore, resilin contains about 6 mol% tyrosine residues which form dityrosine and trityrosine crosslinks, the ratio of which shows regional differences [14]. The distribution of hydrophilic and hydrophobic amino acids along the chain results in a segmented block copolymer type structure with predominantly hydrophilic segments, as shown in resilin from Drosophila [15]. The total sequence consists of three portions: exon 1 and exon 3, comprising 323 and 235 amino acids, respectively, functioning as the elastic constituents, and a comparatively short exon 2, with only 62 amino acids in between, which represents a typical cuticular chitin binding domain [15]. Short amino acid sequences within exon 1 and exon 2 are characteristic for their potential of secondary structure formation such as beta sheets and helices [16] and beta-turn conformation [13,15]. The difference in the hydrophilic/hydrophobic comonomer ratio in exon 1 and exon 3 is considered resulting in special assembly features, i.e. preferences for fibrillar and micellar structures in the range 10–50 nm, respectively [15]. Mechanical data obtained with swollen resilin specimen of different insects [1,3] and spectroscopic analysis of synthetic resilin [7] suggest that a network of ideally jointed chains serves as a good approximation of the elastic response. Additionally, the relatively broad range of elastic properties in resilin-based locomotion systems of insects [17,18] indicates that reversible cross-links, which are based on intermolecular interactions such as hydrogen bonding and related supramolecular features, also play an important role. Specifically, experimental studies demonstrate that upon water uptake, the moduli of resilin networks can vary by several orders of magnitude while experiencing small to moderate volumetric deformations [6,11,19]. These observations cannot be explained by the pure swelling effect associated with rubbers, suggesting that there are additional mechanisms that contribute to the elasticity. From a chemical viewpoint, the effects of hydration on a resilin network are well established. Water contained in resilin can be directly associated with the polypeptide by interacting with the peptide bond itself or with protein side groups (Fig. (1)); this has been referred to as ‘‘non-freezable” water [20]. Any water content that goes beyond this concentration is ‘‘free” water presumably within some sort of micro-pore structure [20], and finally resulting in the continuous aqueous phase of the swollen network. On the other hand, the removal of water content below a critical concentration required for complete solubilization of the polypeptide chains leads to the formation of intermolecular hydrogen bonds and may even result in beta-sheet structures [20]. A similar effect is found in other hydrogen bond dominated solids such as fibers or sheets made of cellulose and, to a certain extent, Nylon and wool [21,22]. From a mechanical viewpoint, the change in properties of resilin, ranging from a hard and brittle material in the dehydrated state

to a soft and highly elastic material in the swollen state can be understood as being related to the contribution of non-covalent, most likely hydrogen bond based cross-links between polypeptide chains. The amount of such temporary cross-links is correlated with the water content. Water molecules break intermolecular hydrogen bonds (Fig. 1), first acting similarly to a plasticizer [6,8,19,20]. As the water content in resilin increases, the number of non-covalent cross-linking sites decreases, ultimately leading to a swollen resilin comprising only tyrosine-based covalent cross-links. Generally speaking, water molecules interact with polar groups of the resilin polypeptide [15,16] and thus reduce the attraction between polymer chains, which in turn results in additional chain mobility and network flexibility. The aim of this work is to derive a methodical microscopically motivated and statistical-mechanics based model that (1) captures the mechanisms of dissociation in non-covalent cross-linking bonds, (2) accounts for the micro-structure of hydrogen bond dominated polymer networks, (3) allows to determine the changes in entropy and free energy resulting from cross-link dissociation, (4) quantitatively explains the solubilization-induced reduction in stiffness, and (5) captures the overall elastic performance. We point out that several analyses of networks with transient crosslinks have been carried out in the past [21–28]. However, these works did not explicitly account for the micro-structure of the network, quantified the entropic gain and the free-energy as bonds break, or explained the origins of the drastic stiffness changes observed in biopolymer networks [6,19,29]. The energetically based microscopically motivated model introduced in this work addresses all of these phenomena and qualitatively and quantitatively agrees with various experimental findings. 2. A microscopically motivated model In the following, we introduce a model that aims to capture the response of polymer networks with breakable cross-link bonds that is immersed in a liquid bath. The model we propose assumes that: 1. The network is idealized as a network of freely jointed chains [29], where a chain is defined as a coiled segment connecting two cross-linking sites. It is emphasized that by making use of the freely-jointed chain model we neglect the valence angle and the rotational barriers between neighboring monomers. It is also important to note that a chain is not referred to as a full length resilin molecule. This idealized description of the network topology derives from the fact that the resilin building blocks, with exon 1 and exon 3 in particular, are similar with regard to their capability of forming reversible cross-links via intermolecular hydrogen bonds between peptide groups.

Fig. 1. Schematic of hydrogen bond breaking in polypeptides such as resilin by water. DH and AX are hydrogen bond donor and acceptor groups, respectively, typically from the amide and carbonyl groups of the peptide backbone and polar functional groups (such as hydroxyls in the side-chain) (cf. [34]). The hydrogen bond DH


AX is disrupted by water attacking the energetically most favored entity, resulting in solvated chains with water being hydrogen-bonded to the donor and acceptor groups of the polypeptide.

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2. The network contains a sufficiently large number of cross-links consisting of covalent bonds and non-covalent, e.g. hydrogen bonds [30]. 3. The interactions between chains are negligible such that the total entropy of the network is the sum of the entropies of the individual chains [31,30]. 4. The chains experience affine deformations [30,32]. 5. The resilin polymer molecules and the water molecules are incompressible [30,32,33]. 6. The network is in equilibrium. Consider an elastic network cube with an edge length L, as seen in Fig. 2. The network comprises uc and up chemical and non-covalent cross-linking bonds per unit referential volume, respectively. Due to the high dissociation energy associated with breaking covalent bonds, we treat the covalent cross-links as permanent. The non-covalent cross-links formed by intermolecular associations can be broken by an influx of water molecules and the forces stemming from the swelling-induced osmotic pressure. Fig. 1 illustrates the breaking process in polypeptides such as resilin by water.

305

The referential chain-density in the network is
 N t ¼ uc þ up f =2, where f is the average number of chains meeting at a cross-link junction. Recall that a chain is defined as a coiled segment connecting two cross-links. Typical values for polypeptide networks are f ¼ 4 and f ¼ 6 [3,14]. A representation of a network comprising f ¼ 4 is illustrated in Fig. 3 and schematics of the chemical structure of a hydrogen cross-linking bond with f ¼ 4 and f ¼ 6 are depicted in Fig. 2 of the Supplemental Material. In the reference configuration, the chains are randomly oriented and uniformly distributed [29] and comprise n amino acid repeating molecular units, which are henceforth referred to as monomers, of length l. Following assumption (1), the referential end-to-end length of all the chains is set to the root mean square pffiffiffi R ¼ l n [30]. The elastic network is placed in an incompressible liquid solvent at a pressure p and temperature T and allowed to swell. In the equilibrium swelling configuration, the chemical potential ls of the m solvent molecules that penetrated the network equals to the chemical potential l ¼ p v s of the surrounding liquid solvent, where v s is the volume of a liquid molecule and we set the

Fig. 2. A schematic of the reference, the swollen, and the stretched states.

Fig. 3. A schematic of the effects of swelling and bond-breaking in a polymer network.

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chemical potential at the reference pressure and temperature to zero. The network swells isotropically and homogeneously and we denote the ratio between the current and the referential volumes by J (see Fig. 2). Accordingly, the volume fractions of the polymer cp and the solvent molecules cl can be written as cp ¼ 1=J and cl ¼ ðJ  1Þ=J, respectively. Microscopically, the swollen network is characterized by chains that are randomly oriented and uniformly distributed with an average end-to-end distance J 1=3 R (assumption (4)). Next, the swollen network is subjected to an isochoric uniaxial tension such that the ratio between the vertical edge of the cube before and after the stretch is k (see Fig. 2). Due to the incompressibility constraint (assumption (5)), the network shortens along the pffiffiffi transverse plane by 1= k. The uniaxial tensile force distorts the network and leads to a non-uniform distribution of the end-toend vectors of the chains [32]. The Helmholtz free energy-density of the deformed network comprises four contributions (Supplemental Material, Section 1): (1) the entropic energy stemming from the deformation of the network wn , (2) the free energy-density resulting from the mixing process of the network and the solvent molecules wm , (3) a constraint enforcing the incompressibility of the network and the solvent molecules, and (4) the work of the external solvent on the swollen network p J [32]. Accordingly, the Helmholtz free energy-density is



 1 w ¼ wn þ wm þ P  J  p J þ w0 ; cp

ð1Þ

where P is a Lagrange multiplier that describes the total pressure of the solvent-polymer mixture and w0 is the Helmholtz free energydensity of the dry undeformed polymer. We emphasize that p is the prescribed external pressure of the surrounding solvent while P is an unknown quantity that is determined from the boundary conditions. As shown in Section 1 of the Supplemental Material, the stress can be derived from the function w. As previously described, the solvent molecules and the osmotic pressure in the network are capable of breaking the non-covalent cross-link sites. We conjecture that below a critical volume fraction of solvent ccrit l , any non-covalent bond that breaks has a high probability of quickly reforming [21,22]. Accordingly, for any cl < ccrit l the response of the network is governed by the swelling effect associated with volumetric changes. However, since the formation and dissociation of hydrogen-bonds is a predominantly cooperative phenomenon, we postulate that for cl > ccrit there is a sufficient l number of solvent molecules in the network to trigger the breaking of many non-covalent cross-link bonds [21,35]. The reduction in the number of cross-links leads to a significant decrease in the effective chain-density and an increase in entropy. In turn, the network softens and undertakes a lower free energy configuration. We emphasize that once all of the non-covalent bonds are broken, the addition of water molecules simply swells the network. Three additional points are worth mentioning before proceeding: (1) we expect a distribution of hydrogen cross-linking bond strengths in the network, (2) the weakest hydrogen cross-linking bonds will be the first to break upon exposure to water, and (3) the energetic cost of breaking non-covalent bonds in a network is significantly higher than the energy associated with bond rotation. In the following we conjecture that bond breaking in response to a mechanical loading is negligible when water is present in the network. To model cross-link dissociation, we define the probability of breaking a single cross-link pb ¼ exp ðU b =kb T Þ, where U b is the dissociation energy required to break the cross-link by water. Following our conjecture, below a volume fraction cl < ccrit the l probability of breaking a cross-link bond and preventing it from forming is pb ¼ 0. It is once again emphasized that in this range

of volume fractions the non-covalent cross-link bonds quickly break and reform and do not influence the macroscopic behavior. Beyond the critical solvent volume fraction ccrit < cl , the probability l of dissociation increases as additional water molecules penetrate the network. Accordingly, we propose the form   U b ¼ a kb T= cl  ccrit . Here, a is a measurement of the average l cross-link strength that depends on the dissociation energy of a hydrogen bond in an aqueous environment and the cooperative index (or the number of cross-links that break as a small amount of water molecules are added to the network) [21,35]. Since it is assumed that chemical (i.e. covalent) bonds are permanent, the constant a  1 for covalent bonds, leading to pb ! 0. We underscore that the dependence of the dissociation energy U b on the amount of water in the system serves to compute the probability that the cross-link bond will break, and in this context the probability increases with water content. For a given volume fraction cl , the density of broken cross-links in the network is up pb . It is convenient to define the fraction of broken cross-links

8 > <0

0 6 cl < ccrit l   /¼ : a crit uc þ up > cl 6 cl  1 : exp  c ccrit

up

l

ð2Þ

l

 In networks with junctions connecting f ¼ 4 chains with n monomers, the breaking of a hydrogen cross-link bond results in  monomers, as illustrated the formation of f =2 new chains with 2 n in Fig. 3 where the red mark denotes a hydrogen bond. The situation is slightly more complicated in networks characterized by f ¼ 6. Specifically, the dissociation of a cross-link can result in the formation of three new chains or the formation of a new chain and a junction characterized by f ¼ 4. This process is described in Section 2 of the Supplemental Material. In this work we assume that on average, the breaking of cross-links results in the formation of f =2 new chains such that f does not change. To elucidate the consequences of non-covalent cross-link bond dissociation, consider two polymer chains in a network comprising nb monomers of length lb . One end of each chain is fixed in a common non-covalent cross-link bond, marked with a red dot in Fig. 3, while the other ends are attached to separate cross-linking sites, marked by green dots. In the dry reference configuration, the pffiffiffiffiffi end-to-end distances of the two chains are Rb ¼ lb nb . The network is subjected to a mechanical deformation such that the end-to-end distances of the two chains in the dry deformed state are r b ¼ kb Rb and the angle between their end-to-end vectors is g, as illustrated in Fig. 3. Next, the deformed network is placed in a liquid bath and swells. Upon sufficient water uptake, the hydrogen cross-link bond dissociates. We assume that the cross-links marked by the green dots experience the same translation after the central physical cross-link breaks. As a result, the two chains under consideration unite to form one long chain with 2 nb monomers and an end-to-end distance 2 kb Rb sin g2 (Fig. 3, right). Note that since this relation holds for any g, the new chain does not necessarily experience the macroscopic deformation. However, any additional macroscopic deformation will cause this chain to deform affinely. Therefore, we emphasize that this calculation is purely based on trigonometrical considerations. As cl increases beyond ccrit l , the cross-links dissociate to form / N t new chains per unit referential volume with 2 n monomers and, accordingly, we compute that there are ð1  2/Þ N t chains per unit referential volume with n monomers. Note that the breaking of bonds leads to a decreased chain-density per unit referential volume ð1  /Þ N t . Once / ¼ 1=2, enough cross-links broke such that the network is only made of chains with 2 n monomers. The penetration of additional solvent molecules, i.e. a further increase

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in cl , triggers the dissociation of the remaining cross-links and the formation of longer chains with 4 n monomers. This process repeats itself and is governed by the water content cl , ultimately resulting in a highly swollen gel comprising only covalent crosslinks. The number of monomers comprising a chain at any equilibrium configuration can be written as 2b n, where b ¼ 0; 1; 2; ::. At a swollen configuration, chains with various contour lengths (or different number of monomers) can co-exist. For simplicity, we conjecture that any equilibrium configuration can be characterized by two average chains with the distinct contour lengths 2b n l and 2bþ1 n l. To compute the composition of the network for a given cl , we note that the chain-density is given by ð1  /ÞN t ¼ N 2b n þ N 2bþ1 n , where N 2b n and N 2bþ1 n are the chain-densities associated with a number of monomers denoted by the subscript in the current configuration. Additionally, since the number of monomers in the system is constant, n N t ¼ 2b n N 2b n þ 2bþ1 n N 2bþ1 n . Accordingly, we find that


 N2b n ¼ 2 Nt ð1  /Þ  2ðbþ1Þ
 ; N2bþ1 n ¼ Nt 2b  ð1  /Þ b

ð3Þ

 1 N2b n hrc i2b n þ N 2bþ1 n hrc i2bþ1 n ; J

ð5Þ

    where hrc in ¼ rcu n  rct n and rcu n and rct n denote the average stress components along the uniaxial and the transverse directions,  monomers (derivation respectively, for a group of chains with n shown in Supplemental Material, Sections 1 and 3). The average stress components are computed via the micro-sphere technique (Supplemental Material, Section 4) [32,37]. In the limit of small to moderate deformations, Eq. (5) reduces to



ru ¼ G k  where

G¼

Nt kb T J 1=3

 ; k2 1

ð6Þ

! 2bþ2 ð1  /Þ  1 3bþ1

;

ð7Þ

is the shear modulus of the swollen network. The volumetric deformation is related to the external solvent pressure via


 kb T ln cl þ cp þ vc2p þ Pv s ¼ p v s ;

ð8Þ

ðbþ1Þ

where 1  2 6 /  1  2 and the value of b is determined from the bounds of the fraction of broken cross-links /, which is computed via Eq. (2). In the special case / ¼ 0, corresponding to a network in which none of the cross-links have dissociated, we find that b ¼ 0 to satisfy 0 6 /  1=2. Next, we compute the entropy of a newly formed chain. Following assumption (4), the chains deform according to the macroscopic deformation gradient such that rn ¼ kc R, where kc is the ratio between the current and the referential end-to-end distances. Note that kc depends on the orientation of the chain and the external loading (Supplemental Material, Section 1). Let us consider two chains with an end-to-end distance r 2b n that meet at a noncovalent cross-link bond, where the subscript 2b n denotes the number of monomers in the chain in the current configuration. The angle between the end-to-end vectors of the two chains is g, as shown in Fig. 3. Once the cross-link dissociates, the two chains unite to form a single chain with 2bþ1 n monomers and an end-toend distance 2 r 2b n sin g2. Following assumption (2), it is reasonable to assume that the distribution xðgÞ of the angle 0 6 g 6 p is proportional to the solid angle subtended by a differential surface area of a unit sphere, i.e. xðgÞ  sin g dg. Accordingly, we propose the Rp probability density function xðgÞ ¼ sin2 g such that 0 xðgÞdg ¼ 1 [30]. Consequently, the average end-to-end distance of the newly formed chains is

r 2bþ1 n ¼ 2 r 2b n

ru ¼

Z p sin 0

g 2

xðgÞ dg ¼

4 4 r b ¼ 3bþ1 r n : 3 2n

ð4Þ

 monomers and a ratio The entropy of a polymer chain with n

q ¼ r=n l between its end-to-end distance r and its contour length is given by [30,36]



Þ sinh sðq ; q  q  Þ  ln  Þ ¼ c  kb n  sðq Sð n ; sðq Þ  is  Þ ¼ L1 ðq  Þ, and LðsÞ ¼ coth s  1=s ¼ q where c is a constant, sðq b  the Langevin function. Thus, by setting n ¼ 2 n and employing Eq.  r  ¼ 2bb n ¼ 46 b pkcffiffin, we obtain the entropy (4) to compute q 2 nl
  b S2b n ¼ S 2b n; 46 pkcffiffin of a chain. Note that if all of the cross-links are made of covalent bonds, / ¼ 0; b ¼ 0 and the classical entropy is recovered [36]. By employing assumption (3), the stress per unit swollen area along the direction of stretch is determined,

where P is the total pressure, determined from the traction along the transverse plane rt ¼ p (Supplemental Material, Section 3). 3. The stiffness of hydrogen-bond dominated biopolymer networks To validate the model, we compare its predictions to several experimental findings. The Young’s modulus at the tip of the seta of a ladybird beetle (Coccinella septempunctata) was measured to be E ¼ 7:2 GPa and E ¼ 1:2 MPa in the dehydrated and the hydrated states, respectively [6]. By assuming incompressibility (assumption (5)), the shear modulus G ¼ E=3 can be obtained by taking uc ¼ 0; a ¼ 0:64
103 , and ccrit ¼ 0:1, corresponding to the l dissociation of almost all cross-links (/ ¼ 0:995). We point out that there is no evidence that there are zero chemical cross-links in the network, and the choice of parameters should be interpreted as uc  up . This suggests that the resilin network in the tip of the seta of the beetle contains a large number of cross-links made of weak hydrogen bonds. The reversibility of this process, i.e. the ability to reform these cross-linking sites, suggests that the beetle can tune the stiffness of the seta by regulating water content. A more thorough study into the influence of water content on stiffness was carried out on the cuticle of an untanned maggot (Calliphora), which contains resilin [19,38]. This work reported a modest decrease in stiffness for cl < 0:8, corresponding to a pure swelling-related effect. However, the penetration of water beyond ccrit ¼ 0:8 led to a sharp decrease in modulus, suggesting the disl ruption and the dissociation of hydrogen bonds and an increase in the effective chain-density. Fig. 4(a) plots the shear modulus as a function of the volume fraction cl according to the proposed model (Eq. (7)) for various values of a, where we set N t kb T ¼ 0:62 GPa. The circle marks denote the experimental measurements from [19]. We recall that the shear in the regime 1 cl < ccrit l , the shear modulus G depends on cl through J ¼ ð1  c l Þ corresponding to a pure swelling effect. A fit to the experimental measurements of the untanned maggot yields a ¼ 0:018. To illustrate the significance of this parameter, the plot shows that networks characterized by a  1 comprise extremely weak crosslinks that are easily broken in the presence of water, leading to an immediate drastic reduction in stiffness. At intermediate values of a, the cross-links are strong and the reduction in stiffness as a function of cl is moderate. For comparison, the pure swelling

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Fig. 4. (a) The shear modulus (Eq. (7)) as a function of the volume fraction cl of water in the swollen network. The circle marks correspond to the experimental findings in Ref. [19]. (b) The fraction of broken non-covalent cross-links / as a function of the solvent fraction cl .

response pertaining to a network made solely of covalent crosslink bonds is illustrated by the dashed curve marked a ! 1. Fig. 4(b) depicts the fraction of broken cross-links / as a function of the solvent volume fraction cl . As the parameter a increases, the network has stronger hydrogen cross-link bonds. Consequently, more water molecules are required to break these cross-links. We point out that the penetration of additional water molecules also increases the osmotic pressure, a factor which further encourages cross-link dissociation. The maximum number of cross-links which can be broken is

/¼

up

uc þ up

exp 

a 1  ccrit l

! :

ð9Þ

A comparison of the model parameters between the cuticle of the untanned maggot and the seta of the beetle reveals that the latter structure is made of weaker bonds. Specifically, the smaller a value and critical solvent volume fraction ccrit suggests a higher l susceptibility to water and lower dissociation energy and cooperative index of the hydrogen cross-linking bonds in the seta.

4. The non-linear response of hydrogen-bond dominated biopolymer networks

predictions of the proposed model to available experimental data on the stress-strain response under uniaxial tension of resilin from the tendons of Aeshna Grandis(cp ¼ 0:48) and Aeshna Cyanea (cp ¼ 0:45) [29]. This work infers that there are twice as many non-covalent cross-link bonds than chemical cross-links in the resilin network, thus posing a limitation on the maximum number of cross-links that can be broken. Specifically, the fraction of broken bonds in the network /  0:5. It is emphasized that since the water content on the tested tendons does not change during the uniaxial tensile test, we do not specify the values of a and ccrit l . Figs. (5a) and (b) plot the experimental findings of [29] and Eqs. (5) and (6) that were derived in the proposed model. The tensile tests were carried out at room temperature and the model parameters were fitted to the experimental measurements (these are listed in Table 1). We find that Eq. (5) captures the macroscopic response in the entire range up to failure and the approximation in Eq. (6) is useful up to a stretch  1:5 and  1:9 in A. Grandis and A. Cyanea, respectively. Interestingly, we find that in the swollen resilin network most or all of the hydrogen bond based crosslinks dissociate. Additionally, the swollen resilin in the tendon of the A. Grandis comprises shorter chains than the A. Cyanea, thus explaining the difference in stiffness. For completeness, we substitute atmospheric pressure and

v s ¼ 3
1029 m3 , corresponding to the volume of a water molecule,

The proposed model provides a systematic method to compute the macroscopic non-linear response of networks comprising non-covalent cross-link bonds. In this section, we compare the

in Eq. (8) to estimate the values of the interaction parameters v (see Table 1). These are within the range of typical values 0 6 v  1 [30,31] and are comparable to v  0:817 measured for

Fig. 5. The nominal stress versus the stretch according to Eq. (5) (continuous curve) and Eq. (6) (dashed curve) for (a) A. grandis and (b) A. cyanea. The circle marks denote the experimental findings of Ref. [29].

Table 1 The model parameters for A. grandis and A. cyanea.

A. grandis A. cyanea

G ½MPa

cp

/ (% of broken CL)

n

Chain groups

v

0:72 0:62

0:48 0:45

0:5 0:4

6 12

N 6 =N t ¼ 0 and N 12 =N t ¼ 0:5 N 12 =N t ¼ 0:2 and N 24 =N t ¼ 0:4

0:78 0:74

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the elastomeric polypeptide elastin [39]. To the best of the authors knowledge, experimental data regarding the interaction parameter of resilin/water have not yet been reported in literature. Elastin resembles resilin in the type of amino acid composition but contains a higher fraction of non-polar amino acid units [12]. Therefore, a lower chi-parameter (v  0:74; 0:78) is to be expected for the more hydrophilic resilin. This is in accordance with our calculations.

[5]

[6]

[7]

5. Conclusions

[8]

In conclusion, we derived a robust thermodynamics-based framework that describes the influence of micro-structure on polymer networks that swell and deform under external loadings. The proposed model suggests that the breaking of non-covalent crosslinks leads to a reduction in the effective chain-density and, consequently, an increase in entropy and a decrease in free energy. The model captures the softening response observed in arthropods, quantifies the number of broken cross-links, and describes the structure of the chains at any swollen equilibrium configuration. Note that while the derived model is based on non-Gaussian rubber elasticity theory, the consideration of several families of chains with a different number of monomers, formed by the breaking of cross-link bonds, is unique to this model and provides a deeper understanding of the micro-structure of a swollen network. The model is also capable of capturing the stress-strain response under tensile tests. To the best of the authors knowledge, the response of resilin to other deformation modes has not been experimentally characterized. We underscore that the proposed model can be used to examine and better understand the behavior of hydrogen-based networks under such loadings and we hope that this work will motivate such experiments. The merit of the model is twofold. First, the model provides insight into the influence of the number and the strength of noncovalent cross-linking bonds, thus explaining the significant reduction in stiffness that is experimentally observed. Second, the framework introduces a quantifiable method to characterize the micro-structural evolution of the polymer chains in hydrogenbond dominated networks. The insights from this work can be used as a design guide in the fabrication of new bio-inspired synthetic polymers. Specifically, by setting the initial chain-density, the initial number of monomers in a chain, and the number and the type of non-covalent cross-linking bonds, we can program the dependence of the stiffness on water content and the response of a network to an applied load.

[9] [10]

[11] [12] [13]

[14]

[15] [16] [17]

[18] [19] [20] [21] [22] [23]

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Acknowledgements This work was supported in part by the National Science Foundation Materials Research Science and Engineering Center (IRG-3) DMR 1720256. Appendix A. Supplementary data

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Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.actbio.2019.07.005.

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