The extraordinary joint material of an articulated coralline alga. II. Modeling the structural basis of its mechanical properties

The seaweeds of wave-swept rocky shores survive, in large part, by being flexible, which provides them with the ability to bend and reconfigure when subjected to flow, thereby reducing the imposed hydrodynamic force (Koehl, 1984, 1986; Denny and Gaylord, 2002; Harder et al., 2004; Boller and Carrington, 2006; Martone et al., 2012). The need for flexibility would seemingly pose a problem for coralline red algae, which calcify their cell walls and are therefore inherently rigid. However, some coralline algae – the erect, articulated corallines – have avoided this limitation through the evolution of joints (genicula) that allow their fronds to be as flexible as those of other seaweeds. The flexibility provided by these joints depends on the unusual mechanical properties of the genicular tissue (Martone and Denny, 2008; Janot and Martone, 2016), and these properties are best understood for Calliarthron cheilosporioides Manza, a representative articulated coralline found on the West Coast of North America.

Calliarthron cheilosporioides' genicular tissue is 3–5 times more extensible and approximately 4 times stronger than typical algal tissue (B. Hale, Macroalgal materials: foiling fracture and fatigue from fluid forces, PhD thesis, Stanford University, 2001; Martone, 2007; Janot and Martone, 2016), and it is effectively immune to failure by fatigue (Denny et al., 2013). Furthermore, the tissue exhibits an unusual combination of elastic and viscous properties (Denny and King, 2016). As one would expect from an elastic solid, the stress (force per cross-sectional area) required to extend the material increases in proportion to strain (the change in length per unstressed length) (Fig. 1A), and, as is common in many biological solids, the tissue becomes less compliant (i.e. stiffer) the farther it is stretched (Fig. 1B). Neither breaking stress nor the material's compliance (strain per stress) is significantly correlated with the rate of strain, again typical of an elastic solid (Denny and King, 2016). By contrast, other aspects of the tissue's mechanical behavior reveal viscous tendencies. In particular, when subjected to repeated or continuous loads, genicular material accumulates deformation through time with no hint of a plateau (Fig. 2), a behavior characteristic of viscous fluids. In other loading regimes, the interplay of viscous and elastic properties is evident. When cyclically stretched, stress increases with strain during loading as expected of an elastic solid (Fig. 3A, red line), but the unloading curve (black line) falls below the loading curve, an indication that energy is dissipated to heat by viscous processes. An average of 42% of the strain energy required to extend a joint in tension is lost upon retraction, and at least part of this hysteresis can be attributed to unrecovered (that is, plastic) strain, which averages 13% of the maximum strain (arrow in Fig. 3A). Similarly, when a stress cycle is interrupted during extension, stress decreases through time (stress relaxation), a viscous process (Fig. 3B), but, when a cycle is interrupted during retraction, stress in the sample subsequently increases through time (stress recovery), the result of elastic recoil.

Together, these material properties allow genicula to provide the flexibility C. cheilosporioides' fronds need to survive – and even dominate space – in the surf zone, while resisting fracture from fatigue and lethal strain from creep (Denny and King, 2016). Indeed, C. cheilosporioides can act as an ecosystem engineer: its densely packed fronds serve as a shelter for invertebrate animals.

The unusual mechanical properties of C. cheilosporioides genicula stem from the chemical composition and morphology of the cells from which they are constructed, in particular from the chemical composition and morphology of genicular cell walls. We know some basic facts about these cell walls: they form 34% of the cross-sectional area in immature genicula and 54% of the area of mature cells (Martone, 2007), and they are composed of randomly oriented cellulose fibers in a matrix gel of highly methylated, highly sulfated galactan (Tsekos et al., 1993; Martone et al., 2010; P. T. Martone, unpublished data). However, it is currently unclear how the micro-scale architecture of C. cheilosporioides' cell walls is tied to the tissue-scale mechanical properties of this extraordinary material, and thereby to the performance of the fronds in nature.

To synthesize information from studies characterizing C. cheilosporioides' extensibility, compliance, strength, hysteresis and creep – and thereby to understand the mechanisms underlying the alga's ecological interactions – we propose a conceptual model of genicular cell wall material in which elastic elements (springs of a given stiffness) and viscous elements (dashpots of a given viscosity) are combined in the simplest possible configuration capable of mimicking the behavior of genicular material (Fig. 4).

Stress has units of newtons per square meter (i.e. pascals), viscosity has units of pascal-seconds and strain is dimensionless.

where dσ/dt is evaluated at the initiation of a stress relaxation experiment (described below). This leaves E₁ as a free parameter that we can estimate by choosing the value that allows the model to best match experimental results. Once values have been established for E₁, E₂, µ₁ and µ₂, we can calculate how the model will behave for any time course of applied stress or strain (see  Appendix 2).

In summary, we can use experimental data to populate a mathematical model of genicular material properties, by extension providing a means to tie cell wall structure to frond-scale performance.

From the high strain rate experiments conducted in the companion paper (Denny and King, 2016), we estimate that E₂ (at ε=0.2, a strain typical of our cyclical stress–strain experiments) is approximately 27 MPa. From the average creep of genicula in tension (Denny and King, 2016), we estimate (using Eqn 5) that μ₂ is approximately 5×10¹² Pa s, and from stress relaxation tests (Denny and King, 2016), we estimate that μ₁ is approximately 3.9×10⁸ Pa s. Through trial and error, we then estimate that E₁ is 40 MPa. The response times T₁ and T₂ are thus 14 s and 35 h, respectively (Eqns 3 and 4).

Using these values, the model mimics (qualitatively and, roughly, quantitatively) much of the tensile behavior of genicula as measured in our experiments (Fig. 5). A cycle to a stress of 3.63 MPa at a strain rate of 0.017 s⁻¹ has a hysteresis of 35.5% with a plastic strain of 0.038, 22% of maximum strain (Fig. 5A, compare with Fig. 3A). (The stiffness of the model is slightly lower than that of the specimen shown in Fig. 3A, but this is expected because this particular specimen's stiffness was above average.) The model exhibits stress relaxation after an increase in stress and stress recovery after a decrease in stress (Fig. 5B, compare with Fig. 3B). The model's modulus increases with increasing strain rate, but only for rates less than approximately 0.1 s⁻¹ (Fig. 5C); at the higher strain rates used in actual measurements (0.2–825), the model's compliance is independent of strain rate, as it was in the experiments. The model overestimates the creep compliance of genicular material by a factor of 4–7 (for reasons we address in the Discussion), but qualitatively reproduces the material's behavior: at long times (>1000 s), the model continues to creep in tension without a hint of a plateau, as it does in the real material (Fig. 5D, compare with Fig. 2A). For times less than approximately 20 s, the model creeps rapidly, as genicular material did for our (Denny and King, 2016) experiments in shear (Fig. 5D, compare with Fig. 3B). Note that, because the model predicts compliance in tension rather than shear, this last comparison can be made only qualitatively.

The ability of our model to mimic the behavior of genicular material is due to the interplay between its characteristic elastic moduli (E₁ and E₂) and its characteristic viscosities (μ₁ and μ₂). E₁ and E₂ are so similar (27 and 40 MPa) that we can think of them as a single characteristic stiffness, E. But μ₁ and μ₂ are vastly different (µ₁=3.9×10⁸ Pa s, µ₂=5×10¹² Pa s), which has two important consequences. On the one hand, a Voigt element's viscous nature is apparent only for stresses that are applied for periods comparable to T₁ (=μ₁/E). For loads applied more rapidly, the whole element is essentially frozen; for loads applied for longer periods, only E matters. Because μ₁ is relatively low, T₁ is of the order of 10 s, which explains genicular material's hysteresis and plastic deformation in short-term cyclic tests, its short-term stress relaxation and stress recovery behavior, and the initial rapid creep seen in the shear tests. On the other hand, dashpot 2's viscosity is so high (13,000 times that of µ₁) that the time constant for the Maxwell element (=μ₂/E) is of the order of 1.5 days. As a consequence, the Maxwell element's viscous nature is evident only for loads applied over many hours, accounting for the fact that genicular material can creep in the long term, yet avoid accruing lethal strain during a frond's lifetime. In short, our heuristic model suggests that the mechanical behavior of genicular cell walls is due to a single elastic structural element, and structural elements that involve two different viscous responses.

These results allow us to form a tentative understanding of the mechanics that link cell wall chemistry and structure to material properties, and thereby to the alga's function in nature. We suggest that the springs and dashpots of our model correspond (at least roughly) to the structural attributes of genicular micro-architecture – to the cellulose fibrils and galactan matrix, respectively. We propose that both E₁ and E₂ are provided by discontinuous cellulose fibrils. If these fibrils are long compared with their diameter (as is typical of cellulose), shear between fibrils and the matrix can effectively transfer stress to the fibrils, allowing them to act as the primary bearers of an imposed tensile load (Bunsell and Renard, 2005; Gosline, 2017). The slight difference between our estimates of E₁ and E₂ (27 MPa versus 40 MPa) may or may not be significant. Perhaps the stiffness of fibrils differs between the primary and secondary cell walls, accounting for the difference in moduli. However, it seems equally likely that the apparent difference between E₁ and E₂ is an artifact of our tweaking the model; the model is only marginally less realistic if E₂=E₁=27 MPa. Note that the model tacitly assumes that fibrils are discontinuous; if they were continuous from one end of a cell to the other, the material could not creep without limit as it apparently does.

While it is reasonably clear that cellulose fibrils act as springs, the correspondence of cell wall architecture to the model's dashpots is more nuanced. In the model, overall tensile compliance is accounted for by two dashpots that differ drastically in their viscosity. Thus, because stress is equal to the product of viscosity and strain rate (Eqn 2), when a given strain rate is applied to the model, the different viscosities produce different stresses. However, we can instead suppose that, in the cell wall, there is only one viscosity – that of the galactan gel – but that two strain rates act as the material is deformed. From this perspective, the apparent difference in dashpot viscosity (µ₁=3.9×10⁸ Pa s and µ₂=5×10¹² Pa s) is due to variation in how the matrix is strained locally. We propose that μ₁ is the viscous reaction of the matrix to bulk shear as a cell extends. Even in the absence of any interaction with cellulose fibrils, the matrix is sheared during extension as the cell wall changes shape – lengthening axially and contracting laterally; μ₁ is the viscous reaction to the rate of this shear. By contrast, we propose that μ₂ is due to the viscous reaction of the material as the discontinuous cellulose fibrils slide past each other, shearing the matrix between them. For a given overall tensile strain rate, local shear strain rate in the matrix confined between sliding fibrils is likely to be much greater than the bulk shear accounting for μ₁, resulting in our calculation of a relatively large value of μ₂. Any direct viscous interaction between cellulose fibrils (e.g. the slippage of entangled fibrils) would also contribute to μ₂.

where D is the material's overall tensile compliance, E is the material's overall tensile modulus, E_f is the tensile modulus of the fiber, E_m is the tensile modulus of the matrix and V_f is the fraction of the material's volume occupied by fibers. Coefficient k accounts for the orientation of fibers relative to the direction of applied force; k is 1 for fibers in line with force and 0 for fibers perpendicular to force. When fibers are randomly oriented, as they are in C. cheilosporioides' cell walls, k=0.375.

where C is the concentration of galactan in the hydrated matrix gel. For example, if the matrix is a 1% gel, the volume of gel is approximately 100 times that of dry galactan, and therefore 100 times that of cellulose; V_f is thus 0.0099. Wainwright et al. (1976) suggest that the modulus of wet cellulose in terrestrial plant cell walls is approximately 3×10¹⁰ Pa, and we can guess that the tensile modulus of galactan gel is similar to that of agar (a cell wall component of some other red algae), which ranges from 0.023 MPa at 0.5% concentration to 0.53 MPa at 5% concentration (Nayar et al., 2012). Inserting these values into Eqns 7–10 gives the results shown in Fig. 6. At a gel concentration of 1%, the observed compliance of genicular material (3.7×10⁻⁸ Pa⁻¹) is obtained if cellulose fibers have an aspect ratio of approximately 1200. The higher the galactan concentration, the shorter the fibers need to be to result in the observed compliance. At a 5% concentration, fibrils need have an aspect ratio of only ∼200. Gosline (2017) suggests that the aspect ratio of cellulose fibrils in terrestrial plants is at least 1000, so the aspect ratios required to match the observed compliance seem feasible. In summary, it seems likely that the measured tensile stiffness of genicula can be explained by treating the material as a two-part composite comprising cellulose fibrils and galactan matrix.

Note that Fig. 6 implies that genicula should become less compliant as they desiccate: the aspect ratio of fibrils stays the same, but the concentration of the gel increases as water is lost. Casual observation reveals that genicula do indeed become leathery and then stiffly brittle as they dry out.

Although this theoretical analysis (Eqn 7) is consistent with our proposed explanation of cell wall mechanics, it is important to note that it addresses only the static elastic nature of the material. Our conceptual model suggests that, at very low strain rates, the compliance of the material (1/ε) is strain-rate dependent (Fig. 5C). Thus, the compliance predicted by Eqn 7 in fact depends on how E_m varies with strain rate, and that information is not currently available. Further verification of our explanation of cell wall mechanics awaits more detailed examination of the properties of the matrix gel and its interaction with cellulose fibrils.

Our approach to the mechanics of C. cheilosporioides' genicula can potentially be applied to the joints of other articulated coralline algae. However, the morphology, chemistry and mechanical properties of joints vary substantially among taxa (e.g. Johansen, 1981; Janot and Martone, 2016). To determine whether our approach has general utility, we must wait until the necessary measurements (chemical content, ultrastructure, cyclic and high strain rate stress–strain characteristics, creep) have been made for a representative sampling of species.

There are several aspects of our conceptual model that must be taken with a grain of salt. For example, the model predicts that, across the range of strain rates used in our tests, tensile compliance is independent of strain and equal to 1/E₂. In reality, D_tan decreases with increasing strain (Fig. 1B), likely because cellulose fibrils reorient. (In fiber-reinforced composite theory, this would be accounted for by an increasing k.) A more accurate spring-and-dashpot model would need to specify that spring 2 has non-linear characteristics in which E₂ increases with increasing strain.

Our model overestimates the compliance measured for actual genicula. This discrepancy may be due in part to the fact that our estimates of E₁, E₂ and µ₁ are based on experiments conducted at 19–20°C, while the tensile creep experiments (from which µ₂ was estimated) were conducted at 12–13°C (Denny and King, 2016). In general, the compliance of viscoelastic materials is lower at lower temperatures (Ferry, 1980; Gosline, 2017). Until the tensile creep tests are repeated at higher temperatures, we will not know how much of the discrepancy can be explained by temperature alone.

Another limitation of the model is revealed when we use it to predict the behavior of genicular material in a dynamic test in which a geniculum is strained sinusoidally at a range of frequencies. In this experiment, there are two relevant metrics. The storage modulus, E′, is the portion of the overall stiffness of the material that is due to elasticity and is therefore in phase with the sinusoidally varying strain. By contrast, the loss modulus, E″, is the portion of the overall stiffness that is due to viscosity. Because the stiffness of a viscous material is proportional to the rate of strain, E″ is the portion of the overall stiffness that is 90 deg out of phase with the driving strain (Aklonis et al., 1972; Wainwright et al., 1976; Ferry, 1980). The model predicts the behavior shown by the solid red lines in Fig. 7: E′ increases with increasing frequency of loading, but only at the low frequencies corresponding to the Voigt element's time constant (f=1/T₁=0.07 Hz). At higher frequencies, E′ is virtually constant and approximately equal to E₂. Our model predicts that E″ is substantial only for a small range of frequencies around 0.07 Hz. These predictions are only a rough match to actual dynamic measurements on C. cheilosporioides' genicula (Denny and Gaylord, 2002), shown by the dashed black lines in Fig. 7. In those experiments, E′ increased gradually with frequency, while E″ was nearly constant across the entire range of frequencies.

These comparisons (creep compliance, dynamic stiffness) suggest that our heuristic model is too simple to be entirely accurate. Rather than a single dashpot in each of the Voigt and Maxwell elements, a more accurate model would employ a spectrum of dashpots with a range of viscosities. Such a spectrum could capture cell wall architecture and mechanics in greater detail. For example, at a given overall strain rate, local strain rate (and thereby apparent viscosity) may well vary with the local angle between cellulose fibrils and the applied force; the random arrangement of cellulose fibrils would result in a range of effective viscosities for both µ₁ and µ₂. A model incorporating a spectrum of dashpots could more closely mimic the dynamic properties of genicular tissue. Such a spectrum of dashpots could also produce a model whose creep characteristics are more similar to Fig. 2A than are those of our current model (Fig. 5D). However, it was not our intent to model genicular material as accurately as possible; rather, our intention was to model the material as simply as possible, and thereby gain insight into the connection between microscale structure and material properties. The model we propose here, rough as it is, suits this purpose.

We note again that our model provides predictions for genicular material only in tension. The behavior of genicula in shear is governed in large part by the properties of the middle lamina. Other than showing that compliance in shear is greater than that in tension, our tests have no ability to inform us about the properties of the middle lamina.

Lastly, in our companion paper (Denny and King, 2016), we report that, while breaking stress of genicular material is independent of strain rate, breaking strain increases with increasing strain rate. Our model offers no insight into this unusual behavior. The faster the material is stretched, the less time cellulose fibrils have to slip past each other before breaking stress is reached, so one might suppose that, in as much as breaking strain is proportional to how far fibrils slide, breaking strain would decrease with increasing strain rate. However, it is possible that tensile failure is initiated not by the average strain of the material, but rather by local strains. If, in the course of a slow stretch, the material is locally weakened as a few errant fibers have time to slide relative to each other, that local flaw could potentially trigger failure at relatively low overall strain. The faster the material is strained, the more resistant fibrils are to sliding past each other, making it less likely that local effects can proceed to the point where they trigger failure. The overall strain at breakage could thereby increase with increasing strain rate. However, this explanation is pure speculation.

The joint material of C. cheilosporioides is viscoelastic. On the one hand, its compliance is independent of strain rate across a wide range of strain rates – characteristic of an elastic solid – which allows joints to maintain their flexibility when loaded by the rapidly imposed hydrodynamic forces of breaking waves. On the other hand, because cellulose fibrils in cell walls are discontinuous, genicular material can creep – a characteristic of a viscous fluid. Viscosity causes the joints to dissipate much of the strain energy absorbed during cyclic wave loading – potentially reducing the chance of fatigue failure – and the viscosity of the cell wall matrix is sufficient so that creep may increase joint flexibility while preventing joints from gradually stretching to the point of failure. We tentatively assign structural elements of the cell wall to the elastic and viscous elements of a conceptual, spring-and-dashpot model that successfully mimics genicular performance, thereby allowing us to connect mechanism across scales, from the molecular architecture of cell walls to the properties of genicular tissue to the function of entire fronds in their ecological context.

where dσ/dt is evaluated at the initiation of stress relaxation.

The behavior of our model in response to a given time course of strain was solved numerically. The following pseudocode illustrates the algorithm. Here, Δε_tot(t) is the change in overall strain imposed in a particular increment in time, Δt; Δε_tot(t) can be positive for extension, or negative for retraction, of the apparatus imposing the strain. The magnitude and sign of Δε_tot(t) is specified by the type of experiment being conducted.

Initial conditions:

ε_tot=ε₁=ε₂=ε₃=σ=0.

Here, ε₁ is the strain of the Voigt element, ε₂ is the strain of spring 2 and ε₃ is the strain of dashpot 2.

For time t=Δt to the end of the experiment in increments Δt:

ε_tot(t)=ε_tot(t − Δt)+Δε_tot(t): increment the overall strain

ε₂(t)=ε_tot(t)−ε₁(t−Δt)−ε₃(t−Δt): calculate the strain in spring 2

σ(t)=E₂×ε₂(t): spring 2 acts as a force transducer allowing us to calculate stress

Δε₃=Δt×[σ(t) / μ₃]: calculate the increment in strain in dashpot 2

ε₃(t)=ε₃(t−Δt)+Δε₃: calculate the new strain in dashpot 2

Δε₁=Δt×{σ(t)−[E₁×ε₁(t−Δt)]}/μ₁: calculate the increment in Voigt element strain

ε₁(t)=ε₁(t−Δt)+Δε₁: calculate the new strain in the Voigt element

Iterate.

For the calculations to be accurate, the increment in time must be kept short; we used Δt=0.001 s. The efficiency of the numerical solution can be increased by implementing this basic procedure in a 4th order Runge–Kutta algorithm, but there is negligible increase in accuracy for the types of simple experiments documented in this study (linear strain ramps with slow stress relaxation/recovery, dynamic test with small-amplitude sinusoidal strain).

This approximation is only as accurate as its assumptions, which are unlikely to be precisely met in C. cheilosporioides' cell walls. We know, for instance, that there are other cell wall constituents besides cellulose and galactan. But for the present purposes, only a rough approximation is required.

We thank Sarah Tepler and Tad Finkler for assistance with the experiments that formed the basis for this synthesis, and Patrick Martone for advice and insight regarding all aspects of C. cheilosporioides' mechanics and natural history. This work is dedicated to John Gosline, who taught us how to think about materials.