Investigating the Relationship between Network Mechanics and

Article pubs.acs.org/Macromolecules

Investigating the Relationship between Network Mechanics and Single-Chain Extension Using Biomimetic Polysaccharide Gels Erich Schuster,† Leif Lundin,‡ and Martin A. K. Williams*,†,‡,§,⊥ †

Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand Food Future Flagship and Division of Food and Nutritional Sciences, CSIRO, Werribee, Australia § MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington, New Zealand ⊥ The Riddet Institute, Palmerston North, New Zealand ‡

ABSTRACT: The anionic polysaccharide pectin can form biomimetic networks which predominantly consist of single biopolymer chains connected by calcium-mediated junction zones of limited extent. As these biopolymeric networks have stress-bearing filaments whose force−extension relations are independently accessible in single-molecule experiments, they thus provide a rather unique test of our current understanding of the relationship between the mechanical properties of a gel and those of its constituent chains. Here we show that remarkable agreement is found between the predictions of a numerical simulation, based solely on the concentration, connectivity, and individual mechanical responses of the biopolymers, with the experimentally measured bulk network properties.

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in this regard.11,12 Here we show that, by exploiting recent findings in the polysaccharide field, biopolymeric networks can be assembled which not only yield generic mechanical properties comparable to those reported for protein networks but also have, in contrast, predominantly single biopolymer chains as stress-bearing filaments.13−15 As the force−extension behavior of these polysaccharide sections has been measured in independent single-molecule experiments,16 these systems can thus provide a rather unique test of our current understanding. Pectin17 is an anionic polysaccharide present in land plants and is a crucial component in controlling the mechanical properties of the cell wall. While complex in detail, the crucial aspects of the polymer fine structure relevant for the work described herein are the backbone of galacturonic acid and the degree and pattern with which these sugar residues are methylesterified. Pectin has a long history of use as a gelling agent, both in acidic conditions and in those exploiting the chelation of divalent metal ions, particularly calcium. Commercially, extracted pectin is typically pretreated in order to control the degree of methylesterification (DM) and thus affect its calcium sensitivity before being sold. It should be noted that as a contiguous sequence of around 10 un-methyl-esterified groups is required in order to form a stable calcium-binding crosslink,18 that the sequence of these groups is also extremely important, and this has motivated the use of different demethyl-esterification methodologies that can produce random (chemical saponification) or blocky (enzymatic) structures.

INTRODUCTION Biopolymeric Networks. In attempting to understand how Nature assembles functional three-dimensional scaffolds that inter alia generate, transmit, and transduce stresses, modeling the macroscopic mechanical properties that arise from collections of semiflexible hierarchical biopolymer assemblies such as rods, tubes, and fibrils, has become the prevailing paradigm.1−7 Indeed, only recently a plethora of diverse protein-based networks generated by the in-vitro cross-linking of such semiflexible entities have been shown experimentally to exhibit generic rheological properties, akin to those of intact cells, including strain stiffening.1 Furthermore, it has been demonstrated that such behavior emerges whenever the underlying force response of the constituent filaments of the network can be described as that of a worm-like chain. The worm-like chain is an extensively tested model which describes, among other things, strain stiffening and the subdiffusive highfrequency microrheological response found in many biopolymer systems in a generic manner.1 Three-dimensional numerical simulations of actin8,9 and collagen10 networks have been found to successfully regenerate reported mechanical properties by treating the network elements as generic semiflexible filaments. However, owing to the complexity of the underlying multimeric filaments, direct independent measurements of their mechanical properties have proved difficult to obtain. As alluded to, previous work on semiflexible networks has focused almost exclusively on proteins, motivated largely by the desire to understand the mechanics of the cytoskeleton.1−7 Polysaccharides, on the other hand, while also important in fulfilling structural and mechanical roles in biological systems, have received comparatively little attention © 2012 American Chemical Society

Received: April 8, 2012 Revised: May 14, 2012 Published: May 24, 2012 4863

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Table 1. Biomimetic Pectin Networks Investigateda

Thus, prior manipulation of the DM and the subsequent release of ions into pectin containing solutions has been, until recently, the prevailing method for ionotropic gel formation. In plant cells, pectin is synthesized in the Golgi apparatus with a high DM. This high DM prevents the formation of gels that otherwise might be induced by omnipresent Ca2+ ions while pectin is being transported to the plant cell wall. The pectin chains are subsequently modified in muro by pectin methylesterase (PME) enzymes, which de-esterifiy the polymer and subsequently allow calcium binding. This process of gelation can therefore be seen as the controlled release of ionbinding groups in contrast to the more traditional release of ions into pre-de-esterified low DM substrates. This realization has recently prompted studies in which biomimetic networks have been formed in vitro.13,19,20 Interestingly, recent SAXS studies on such pectin systems15 support the hypotheses, initially developed from high-frequency microrheological data,13 that when using plant PME (pPME) to release ionbinding groups from high DM pectins in the presence of calcium in a biomimetic approach gels form in which single pectin chains play an important stress bearing role. For such pPME-induced gelation it has been hypothesized that once the processive enzyme activity has freed a sufficient number of carboxyl groups to form a stable egg-box junction zone, these newly induced steric constraints hinder the enzyme activity on this strand, so that the pPME moves to another chain and creates another blocky calcium binding site of minimal length, before being again encouraged to move on by the induced association. In this way minimal-length connections are formed and bundling into thicker multipolymer aggregates, as found with ion release into pre-de-esterified substrates, is discouraged. In this work, enzymatically induced biomimetic gels of pectin were formed in a rheometer. As described, the enzyme in question de-methyl-esterifies the galacturonic acid sugar rings of the pectin backbone, revealing charged groups that subsequently chelate calcium ions into intermolecular junction zones.17 Following gel formation, the dependence of the differential shear modulus of these gels on applied prestress was investigated. Additionally, a simulation of the bulk network rheology was undertaken using the experimentally measured mechanical properties of single chains.

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sample (i) (ii) (iii) (iv)

[PME] 10 69 25 25

(fresh) (frozen) (fresh) (frozen)

[pectin] (%)

[CaCl2]

G0 (Pa)

1 1 1 1

3.3 1.1 1.1 1.1

1300 10 35 50

a

The volume of enzyme stock solution added to each mL of pectin stock solution (μL/mL); the final pectin (% w/w) and calcium concentrations (mM); and the zero shear moduli exhibited by the systems investigated.

volume of 50 mM HEPES buffer, in order to reach a final polymer concentration of 1% w/w. As the starting pectin was of a high DM, it did not gel in the presence of the added Ca2+. Before loading 20 mL of the solution into the rheometer, the desired quantity of enzyme solution as given in Table 1 was added to the pectin solution. Rheological Measurements. The viscoelastic moduli (G′, G″) of the pectin gels were measured using a stress-controlled Anton Paar Physica 301 rheometer with CC27 double-gap sandblasted geometry. The real part of the storage modulus reduces to the shear modulus G0 at zero frequency. The sample was loaded as quickly as possible onto the rheometer (preset to 298 K) after the introduction of the enzyme. A thin layer of Whiterex Oil was placed on top of the sample to prevent evaporation. First, the gelation profile of the sample was monitored over time using small deformations with a strain of 0.5% and a frequency of 1 Hz until a gel of desired strength was formed. Subsequently, a differential measurement23 was utilized to quantify its nonlinear behavior. The system was held at a constant average prestress σ, while the differential response dγ to a small additional oscillatory stress dσ was measured to yield the nonlinear differential modulus K = dσ/dγ. The first rotational stress was 1 Pa in magnitude. Subsequent prestresses were, in order, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 868, 1024, and 1536 Pa. At each rotational stress interval, small deformation oscillations were conducted at 1 Hz for 120 s. The magnitude of each oscillation was set at 10% of the respective rotational prestress during each interval. Plastic Embedding of Pectin Gels. Samples for transmission electron microscopy (TEM) were prepared by being embedded in a plastic resin and then thinly sliced with an ultramicrotome. The protocol for the sample preparation was as follows: Small bodies (about 2−3 mm wide) of gels were placed in a fixating solution (2 vol % glutaraldehyde in HEPES buffer (50 mM, pH 7.5) including 1 g/ dm3 ruthenium red) for 1 h to fix and stain the network structures. HEPES buffer (50 mM, pH 7.5) was used to remove any unbound glutaraldehyde and ruthenium red in two washing steps. The samples were then further fixed and stained in 2% osmium tetroxide solution for 1 h. Ultrapure water was used to remove any unbound osmium tetroxide in three washing steps. The samples were dehydrated in multiple baths of ethanol50%, 70%, 90%, and absolute alcoholand then embedding media (LR White) was added to infiltrate the samples. Finally the polymerization of the resin was allowed to occur at 333 K overnight. Electron Microscopy. 150 nm sections of epoxy-embedded gel, as described in the sample preparation section, were places on 400 mesh hexagonal copper grids and were loaded into a high-tilt cryotomography sample holder. A JEM-1400 transmission electron microscope (JEOL, Japan) with a tungsten filament and operating at an accelerating voltage of 120 kV was used for data acquisition. The sample was cooled to the lowest stable temperature possible with this holder system (about 99 K) and then preirradiated at low electron flux for 10 min to permit occurrence of gross shrinkage and mass loss. Modeling Methods. To computationally simulate the shear deformation of a three-dimensional pectin network, in which single pectin chains predominantly form the connections, a previously described 2-D method24 was simply extended to 3-D. It has previously been shown in single-molecule experiments that the force−extension curve of pectin can be modeled as a clickable-extensible-worm-like

EXPERIMENTAL DETAILS

Materials. Pectin. The material used had a degree of methylesterification (DM) of 78 and has been previously described in detail elsewhere.16 It has a molecular weight in the order of 100 kDa corresponding to chains of around 560 monomers in length. The sugar monomers typically reside is the 4C1 chair conformation, in which a monomer spacing of 0.4592 nm is found,21 giving an estimate for the contour length of a free pectin chain as lC = 260 nm. Viscometry studies and SANS experiments have estimated the persistence length lP to be between 7 and 10 nm.22 Pectin methylesterase (PME) [EC 3.1.1.11] was purchased from Sigma-Aldrich (P5400). Either (a) stock solutions of the enzyme were prepared by dissolving 0.01 g of dried PME in 20 mL of Milli-Q water with 0.58 g of NaCl, stored at 253 K in Eppendorf tubes, and thawed immediately prior to conducting experiments, or (b) 0.01 g of dried PME was dissolved in 40 mL of Milli-Q water with 1.2 g of NaCl and aliquots were used immediately. Methods. Enzymatically Induced Pectin Gels. Stock pectin solutions of 1.25% w/w were prepared by dissolving the pectin powder in a 50 mM HEPES buffer made with Milli-Q water, adjusting to pH 7.5 with 1 M NaOH, and stirring for 1 h. Subsequently, desired volumes of a CaCl2 salt solution were added, giving concentrations as reported in Table 1. Finally the sample was mixed with the desired 4864

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chain (CEWLC), and thus this functional form is utilized (as depicted in Figure 1).

Figure 2. Gel evolution as a function of time during in situ deesterification of the samples reacting with different activities of each enzyme and at different concentrations of CaCl2, as in Table 1. G′, G″: gel (i) filled squares; gel (ii) open squares; gel (iii) open circles; gel (iv) filled circles. G′ is the higher modulus in all cases.

Figure 1. Force−extension curve of a CEWLC that fits the experimental force−extension curve of a single pectin chain, as used in the simulations herein. The CEWLC parameters have been chosen in order to produce a curve consistent with experimentally measured data on the polysaccharide pectin.16,27 Specifically, lC = 128 nm, T = 298 K, chair length = 0.4592 nm, boat length = 0.5176 nm, inverted chair length = 0.5576 nm, specific stiffness = 20 nN, and the energy differences for the two conformational transitions ΔG = 12 and 17 kJ mol−1 respectively; this CEWLC reproduces the experimental data very closely with r2 = 0.9996.

Table 1. In a recent study the enzyme activity was found to be the major determinant of the speed of the gelation in these systems,20 and indeed this is supported by our study (compare the time evolution of gel (ii) and gel (iv)). However, it is also conspicuous that the activity of the enzyme after being thawed from a previously frozen aliquot that had been stored in a freezer was significantly lower than that found using a sample freshly made from the powdered enzyme, with the former exhibiting an additional delay of around 6 h before a significant gelling behavior could be measured. The reason for this was not pursued in detail but may be indicative of a renaturing of the enzyme. It is also clear that, as expected, the elasticity of the fully cured gel has a plateau modulus, G0, that is sensitive to calcium concentration, the source of cross-link forming ions. The calcium concentration was found to be a limiting factor in the gelation of gel (iii), which plateaued after only 2 h, whereas an increase in the initial CaCl2 concentration resulted in moduli 1 order of magnitude stronger than observed before (see gel (iv)). After the gel formation had been monitored for some 10 h, the nonlinear elasticity of the networks formed was tested by performing measurements of the differential modulus K as a function of prestress. These measurements took around 2 min per prestress value, so that even in the cases where the modulus was still slowly evolving the experimental results were not influenced by the ongoing kinetics, and they were performed up to prestresses at which the additional oscillatory stress caused a breakdown or a dramatic rearrangement of the network. The resultant plots from all four samples exhibit scaling behavior and collapse onto a master plot (Figure 3c), indicating that the same single-chain physics is governing the observed responses irrespective of the different initial conditions. Comparison with Previous Work on Protein Networks. Results from the two samples gel (i) and gel (iv) (the extreme cases studied of a strong and a weak gel) were compared with previous measurements made on a variety of protinaceous biomaterials in which the single filaments are not themselves single chains but assembled intermediate structures. The general agreement between the behavior of the previous ecclectic selection of protein gel samples with the biomimetic polysaccharide network investigated here can be seen to be remarkable. To illuminate the comparison and permit further a facile comparison with theory, the strain and stress dependence of the differential moduli was evaluated (Figure 3). The mapping of

As affine deformations were previously found to describe the bulk mechanical response of biopolymer gels successfully,1,25 especially at intermediate to high polymer concentrations,3,14 we focused simulations on this affine case. The estimates in the Materials section confirm that the bending of chains can be neglected (lP ≪ lC), and as such the mechanical response of the CEWLC chains captured by the experimentally measured force−extension curves (Figure 1) should be valid for use in networks. The networks were formed by randomly placing nodes into a three-dimensional cell, so that the position and orientation of connections was equidistributed. The connection of these nodes formed the network architecture. This step allowed the assignment of the interaction neighbors at each node, whose interaction is determined in the due course by the experimentally determined CEWLC potential describing pectin single-chain extension. During all further computations all node-connecting chains are assumed to be freely rotating so that the system Hamiltonian depends only on the node-to-node distances. The force−extension curves of the CEWLC were used to numerically derive the potentials describing how connected nodes interact. The CEWLC parameters were selected so that the model represents actual single molecule experimental data for pectin, and consequently two clear transitions assigned to two conformational changes in the sugar rings are evidenced at higher forces. With the potential and the node positions in hand the equilibrium state of the network is found by minimizing the energy using a conjugate gradient routine. Once the relaxed start configuration is found the network can subsequently be strained and the process repeated after each strain step. Affine shear deformations were applied to the bounding X−Y planes in the Y-direction: nodes in the bounding X−Z planes were moved affinely, while Lees−Edwards boundary conditions26 were applied to the bounding Y−Z planes. The simulations were carried out on 1 μm3 systems, with ensemble averages taken over five runs.

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RESULTS AND DISCUSSION Several enzymatically induced pectin gels were formed in the rheometer and the evolution of their viscoelastic properties measured, via dynamic shear modulus measurements as described, at a frequency of 1 Hz (Figure 2). The investigations centered around using a constant polymer concentration and differing concentrations of enzyme and calcium, as detailed in 4865

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Figure 3. Differential shear moduli, K, as a function of strain, γ, and prestress, σ. (a) The differential shear moduli, K, of two representative biomimetic pectin gels (i and iv) plotted versus strain, compared with measurements of protein networks.1 (b, c) The data from all four pectin samples investigated collapse onto master plots (samples as described in Table 1; gel (i) filled squares; gel (ii) open squares; gel (iii) open circles; gel (iv) filled circles). (b) The moduli have been scaled by the corresponding modulus value at zero prestress, G0. (c) The data have been scaled by the corresponding values of the prestress at which the gels stiffened 50%, K = 1.5 × G0, and plotted against prestress (solid line). Equation 1 is utilized to calculate the differential modulus for networks of filaments characterized by an entropic and enthalpic spring constant,28 with kentropic/kenthaplic = 0.009 for the case of single pectin chains.16

Comparison with an Averaging Approach. The results can further be compared with the predictions of a 3dimensional averaging approach to calculate the shear modulus of affine deformations, as introduced by Storm et al.1 The model considers a regime where the effects of bending modes and torques are minor, and stretching and compression of the constituent monodisperse filaments dominate the stress response of the network, with the molecular force−extension relation depending only on the node-to-node distance. Assuming affine deformations one can express the node positions r′ under strain as a linear transformation of the unstrained positions r by just one isotropic Cauchy deformation tensor Λ: r′(γ) = Λr, which results in the following expression for the shear strain:

prestress measurements onto a strain dependent differential modulus K is valid for permanently cross-linked systems29,30 where the results obtained from strain sweeps and those obtained by the differential prestress protocol agree well. Such “pinned” network structures have been suggested for pectin networks,14 and this contention has been supported by a previous SAXS study. The mapping is performed via a numerical integration, with the strain measured at the lowest prestress of 1 Pa used as the initial value. Power Law Behavior. For semiflexible filaments whose force−extension relation diverges as they extend toward their contour length, a power law dependence of the differential shear modulus upon stress would be expected with an exponent of 3/2.3,5,29,30 However, it has also been shown that the introduction of a more realistic extensible chain potential at higher extensions can produce deviations from the simple prediction and indeed that a weaker scaling can be expected if the nonlinear-entropic regime and such an enthalpic regime are not substantially separated.28,31 The application of such a model by Blundell and Terentjev28 is also shown in Figure 3c and provides a reasonable description of the data K (σ ) = G0(kentropic/kenthalpic + (1 + σ /σc)−3/2 )−1

σ=

ρ det Λ

f (|Λr |)

(Λr )x (Λr )y |Λr |

|r |= LR

(2)

where ρ is the number of links per unit volume and f(r) is the force−extension relation that governs the node-to-node potential. The average is taken over all spatial configurations with one end-to-end extension, LR, being assumed for all the single chains comprising the network. The following 3dimensional area-conserving affine shear deformation tensor was used:

(1)

with σc being the critical stress at which nonlinear scaling begins and the modulus at zero prestress G0. The ratio of entropic and enthalpic spring constant, kentropic/kenthaplic = 0.009, was evaluated from the experimentally measured single-chain force−extension curves for pectin, as the slope at high and low extensions, respectively.

⎛1 γ 0⎞ ⎜ ⎟ Λ = ⎜0 1 0⎟ ⎜ ⎟ ⎝0 0 1⎠ 4866

(3)

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Figure 4. Network properties compared with the results of simulation. (a) The differential shear moduli, K, of a biomimetic pectin gel, (i) in Table 1, plotted versus strain, compared with the results of a numerical simulation (solid line). (b) Visualization of a typical simulation slice (the node linkages are shown and not the trajectories of actual polymer walks), parallel to the shear plane, at 0 strain, with the number of nodes reduced in order to capture an experimentally measured zero-strain modulus; compared with a TEM image of the corresponding experimental network, (iv) in Table 1. (c) A comparison of the experimental and simulated strain dependence of the differential shear moduli of the network gel (iv) depicted in (b). The error bars show the standard deviation of the simulation results estimated from ensemble averaging over five runs.

Comparison with Simulations. A simulation was first devised in which all the pectin chains in the simulation box were connected to the network in order to try and capture the properties of the strongest gel, which had had its calcium and enzyme concentration controlled with this aim in mind. First, the effective length of the pectin chain sections between nodes (cross-links) in the fully cured network was estimated using basic statistical considerations. Assuming the basic single polymers are pinned together by a random enzyme activity, this allows estimates to be made of (a) the average chain length between two randomly chosen cross-link points along the polymer backbone and (b) the functionality of the nodes generated by the attachment, i.e., how many of connections are to nodes with additional connections and how many chains are just elongated by a connection which only pins together two chains. By assuming process (a) can be modeled by an equidistributed random process (a classical Bernoullian process where each event occurs independently with the probability of 1/L), then the probability density ρ of two selected points along the polymer backbone being separated by the distance d is proportional to

and eq 2 used to calculate a shear stress σ for each strain step γ. The predicted differential modulus K was subsequently obtained by a numerical differentiation of this stress−strain curve. The experimental results reported here for all the biomimetic pectin gels, in the form of the differential moduli vs strain plot collapsed onto a master plot as described previously and a comparison with the calculations of the averaging approach reveals that using the experimentally measured force−extension relation is sufficient to fit this master plot (solid line in Figure 3b). The averaging approach evaluates the pectin-describing CEWLC force−extension relation (as in Figure 1) at one extension and averages the result over all orientations. The calculation of this normalized stress−strain behavior depends crucially then on the value of this end-to-end extension, LR, the extension at which the force−extension relation is evaluated at 0 strain. While this approach is efficient the value of LR is extracted by fitting and difficult to estimate otherwise. The result, that LR equals 53% of the chain’s apparent contour length between nodes is qualitatively consistent with the experimental data within this theoretical framework and indicates that the chains are extended toward the end of the linear entropic regime. Further applied strain extends the chains into the nonlinear regime and causes the stiffening response. The data recorded for the two samples gel (i) and gel (iv), the extreme cases of strong and soft gels, studied here are further compared with quantitative predictions obtained from a numerical simulation that does not require any fitting parameters.

ρ ∝ (1/L)2 (1 − 1/L)d

(4)

After normalization of the probability distribution the average distance ⟨x⟩ between two randomly chosen points can be calculated as ⟨x⟩ = 4867

p L (L ln(p) − 1) + 1 (p L − 1) ln(p)

(5)

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with p = (1 − 1/L), and this leads directly to an estimate of an average distance between nodes of around 110 nm. This can be seen by examination of the EM data (Figure 4) to be eminently reasonable. The functionality of nodes can be captured by considering the likelihood for the occurrence of more than two connections per node. Assuming again that the node formation follows a simple Bernoullian process, the probability for a node with c connections to be formed follows a power law pc1, with the probability p1 for the existence of a single connection at the point of interest. Considering this power-law decay of the probability for nodes of high connectivities (and spurred on by attempting to limit the computational expense of the modeling approach), it is argued that the node forming physics is captured within functionalities of up to 4, with the occurrence of nodes with higher connectivities assumed negligible. Considering the dense reasonably homogeneous semidilute starting state, every functionality is assumed to have the same weighting factor, and therefore (with the cutoff described above) it can be assumed that p12 + p13 + p14 = 1

investigations found that enzymatic induced pectin networks are dominantly composed of single chains, such that an identification of the thinnest strands in the micrograph as single chains gets strongly encouraged. To capture the behavior of this weaker network and motivated by the gaps observed in the TEM micrograph, the numerical network forming process mentioned above was adapted, so that the initially connected network had several nodes removed. This was implemented by removing “chunks” of 5000 neighboring nodes. Unsurprisingly, a reduction of nodes was found to reduce the number of the chain segments contributing to the mechanical modulus and yields a lower modulus value in the spirit of classical rubber elasticity or cascade theory. The number of nodes was reduced in this random fashion until the experimentally measured modulus G0 (in the order of 100 Pa) was reached in the simulation. This procedure revealed that only around one-third of the available chains effectively contribute to the mechanical response in this weak gel. Interestingly, assuming that chains are not retained for the EM visualization process unless they are connected to the percolated network suggests that only around one-third of the chains should be visualized, which appears very reasonable comparing with Figure 4b. The resulting simulated strain dependence of the differential modulus is depicted in Figure 4c; using 44 000 chains in an identical simulation to that performed above, simply using one-third of the total available number of chains (consistent with the modulus and the imaging). Once again it is seen that there is remarkably good quantitative agreement between the predicted and the measured mechanical behavior. A snapshot of such a sparse network at zero strain, via modeling gel (iv), and a comparison with the TEM micrograph is depicted in Figure 4b. It should be remembered that such a visualization of the simulation essentially shows the node linkages and not the trajectories of actual polymer walks. Conformational Transitions. Lastly, the numerical simulation allows the monitoring of the node-to-node distances and the investigation of whether force-induced conformational transitions of the sugar ring found in single-molecule studies might be triggered in pectin during network straining. This question has been addressed previously in two dimensions,24 but in this case the 3D model is a more realistic model for considering the likelihood of such events in gels or even in vivo. To illustrate the investigation of this issue Figure 5 shows the distribution of the node-to-node distances of the fully percolated network, mimicking gel (i), and its evolution during

(6)

This results in p1 = 0.68 and therefore in a probability of 47% for the creation of a “2-node”, following p12. These statistical estimates were subsequently used to setup of a 3-dimensional CEWLC numerical simulation of the gel structure, with the aim of predicting the mechanical response of the 1% w/w pectin gel (i). The realization of the system simulated used a network of 130 000 chains (corresponding to the 1% w/w pectin gel in a 1 μm3 simulation box). In this framework the above-mentioned 47% of “2-nodes” are taken into account via doubling the average effective contour length between nodes, whereas the other effective chain sections between junctions are represented by an average chain length of 110 nm, as described above. Gel (i) resembles a reasonably stiff gel so that although it does not “break down” or dramatically reorganize until prestresses of kPa are reached this does limit the amount of data available to compare with the prediction. Nevertheless, Figure 4a shows that the modeling is remarkably successful in predicting the experimentally observed network properties from those of the constituent polymers. Similarly, the stress−strain response for the gel (iv) system has been investigated within the CEWLC model and the statistics elaborated above. It is clear that the plateau modulus G0 at zero strain differs by almost 2 orders of magnitude from the gel investigated above that was hypothesized to have all the chains connected to the network. A further insight into the network structure of the weaker gel can be found in an electron micrograph, obtained as described in the Experimental Details section (Figure 4b). It seems clear that while the network is a percolated structure of mainly thin-stranded sections, there do exist obvious gaps in the network, suggesting that not all the chains were attached to the network and that these were perhaps lost in sample preparation for the microscopy. The fixation and staining with metal compounds during the sample preparation is known to impact the observed strand thickness and therefore omits an accurate image analysis of the micrographespecially once the size limit of dimers and single chains gets approached. In an attempt to gauge the strand sizes the findings of a previous SAXS study,15 a complementary technique, with the advantage of not distorting the internal structure by any sample preparation, was utilized. These SAXS

Figure 5. Distribution of the of the node-to-node distance of a fully percolated network, mimicking gel (i); unstrained equilibrated (gray line) and strained γ = 0.5 (black line). The extensions are normalized with respect to the length of the fully extended skew-boat conformation of the CEWLC. 4868

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the shear deformation. It is observed that it is very unlikely for chains to be extended into an extended ”clicked” (boat or extended chair) state of the pyranose ring. For gel (i), out of the 130 000 available chains, 60 chains are in a “clicked” state at strain γ = 0 and 430 chains are found to be clicked at strain γ = 0.5. The same conclusions can be drawn for the case of gel (iv). Similar to the previous results in 2-D, the overall node-to-node distribution is shifted toward higher extensions, and the number of clicked states does increase under applied shear stress. This redistribution causes the observed strain stiffening, and while a small proportion of the chains are found to be extended sufficiently to yield “clicks” in the 3-D enzymatically induced pectin gels investigated here, there is the potential of this being a signaling mechanism in muro.

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CONCLUSIONS Our experiments and simulations highlight not only that physical models developed for protein gels and the cytoskeleton are applicable to the study of biologically important pectin networks but also that such polysaccharide networks can, in their turn, provide rewarding experimental systems for the continuing exploration, development, and testing of models of the underlying physics of biopolymer networks.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge Stephen Homer for the help with the prestress experiments and Sofia Øiseth, Andrew Leis, Sandra Crameri, and Alex Hyatt for helpful discussions about high resolution imaging and conducting the electron microscopy work. This research was supported by CSIRO, Australia. Romaric Vincent, Aurelie Cucheval, Niklas Loren, and Yacine Hemar are also acknowledged for helpful discussions.

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dx.doi.org/10.1021/ma300724n | Macromolecules 2012, 45, 4863−4869