Theory

 

I. Background

 

Among one of the most studied and marveled at biological materials is spider silk, which is known for being extremely tough and extensible, even surpassing steel and other industrial materials. These properties can be seen by sheer observation, judging from the fact that spiders use their silk to support their own weight, and even absorb kinetic energy to capture prey.  

 

Scientists are currently exploring these mechanical properties and how they can be channeled in various ways, including tissue regeneration, medical sutures, parachutes, and other biomedical applications.  Spider silk is an especially important biomaterial because of its degradability. The exploration of this property has caused silk biomaterials to be successfully employed in wound healing as well as tissue engineering.  

 

The hierarchical structure of the silk is crucial in order for the silk to retain its mechanical properties.  This structure consists of extremely organized, hydrogen-bonded beta-sheet nanocrystals arranged within a semi-amorphous protein matrix consisting of less organized protein structures.  

 

To download this application, press on the link above.

 

Once launched, the application is dynamic and constantly running.

 

 

 

This simulation is meant to demonstrate a simple nonlinear spring representation of

spider silk.  By changing the behavior of each of the characteristic regions (see

descriptions below) we can learn a lot about the forces applied on spider silk and

how they can cause the silk to deform.  Four phases of deformation can be seen in

this demonstration.

 

Program Author(s):

 

 

Webpage Author(s):

 

 

Advisor:

 

 

 

[1] Andrea Nova, Sinan Keten, Nicola M. Pugno, Alberto Redaelli, Markus J. Buehler, 2010.   Molecular and Nanostructural Mechanisms of Deformation, Strength and Toughness of Spider Silk Fibrils. Nano Letters 10 (7), 2626-2634.

 

[2] Keten, S. and M.J. Buehler, Nanostructure and molecular mechanics of dragline spider silk protein assemblies. Journal of the Royal Society Interface, 2010., in press, published online June 2, 2010, doi: 10.1098/rsif.2010.0149

 

[3] Sinan Keten, Markus Buehler, 2010. Atomistic Model of the Spider Silk Nanostructure. Applied Physics Letters 96.

 

[4] S. Keten, Z. Xu, B. Ihle, M. J. Buehler, "Nanoconfinement controls stiffness, strength and mechanical toughness of β-sheet crystals in silk", Nature Materials, 2010, 9, p. 359-367.

 

III. Coarse-Grained Mesoscale Model - Nonlinear Springs in Series

 

The lab has developed ways to study spider silk and demonstrate how it actually deforms on a nanoscale level.  Through various tests using differing beta-sheet nanocrystal sizes, the lab has determined that a “coarse-grained” model can be implemented to represent the deformation of the spider silk system, as opposed to a full atomistic representation (this is due to the fact that the results from each test still showed the characteristic stress-strain graphs).  

 

Though this course-grained model does not take into account many variables in spider silk deformation, it is quite useful and relevant in this context, as it does explore the role of beta-sheet nanocrystal size (the key factor in deformation rate) on deformation.  What we have

adapted is known as a “bottom-up” approach, as we are exploring silk on the fundamental level.

 

The downloadable application we created explains the theory behind our coarse-grained model.  Our model represents the semi-amorphous and crystal regions of the spider silk as beads connected by nonlinear springs in a serial arrangement.  This model reduces the tensile loading of the spider silk to nothing more than a force-displacement model between two springs, which can be translated into stress-strain values.  

 

The displacement of the “springs” or “amorphous” regions depends on the stiffness of those regions.  In nonlinear spring models however, the total stiffness changes as a function of strain; it is necessary to know the strain to know if that particular spring has yielded or not.  It is reasonable to use a serial spring model in this instance because at the nanoscale the crystals are connected by repeating protein sequences.

 

The characteristics of the silk are due to the order of the amino acid residues in the protein sequence of silk: the poly-(Glycine-Alanine) and poly-Alanine domain makeup of the beta-sheet nanocrystals.  

 

One of the current ways to analyze the properties of spider silk is by using a one-dimensional coarse-grained model, which essentially models the beta-sheet nanocrystal and semi-amorphous regions using beads connected by nonlinear springs.  

 

Through a number of molecular dynamics simulations, it has been determined that the physical size of the beta-sheet nanocrystals affects the tensile strength of the silk as a whole. When the beta-sheet nanocrystals are smaller in size is when the spider silk is the toughest mechanically, highest in strength, and stiffest.  The importance of the beta-sheet nanocrystals is highly evident when stretching of the silk occurs; they reinforce the partially extended macromolecular chains by forming interlocking regions that transfer the force load between the chains, which allows for greater extensibility of the amorphous region.  The one-dimensional model that will be explained in greater detail reveals that the semi-amorphous region dominates deformation at small deformation levels (the region starts to unravel when the silk is stretched), but the beta-sheet nanocrystals dominate after the semi-amorphous region has essentially stretched to capacity. When an axial force is applied to spider silk, the silk experiences four distinct regimes that include a high tangent modulus, a softening effect, a stiffening effect, and then finally complete failure.

 

II. Four Deformation Stages

 

After extensive experiments of applying loads to spider silk researchers have observed that deformation in the silk experiences four distinct regimes, each with its own characteristic.  

 

    a high tangent modulus.  The occurring stretching of the silk causes the hydrogen

    bonds in the semi-amorphous regions to rupture. As the helices and beta-turns in

    the semi-amorphous domain of spider silk are quite rich in hydrogen bonds, the

    initial regime culminates in an early yield point for the silk (at strain values of about

    15%).  This yielding causes a significant decrease in the tangent stiffness of the silk,

    which then leads into the second regime of deformation.  

 

    the physical nanostructure of the spider silk; this physical behavior is a direct result

    of the early yielding point inherent to the first regime.  If we drill-down to a

    nanoscale level, we can see that the protein chains in the semi-amorphous region

    actually start to align with the direction the load was applied.  This process is

    gradual, due to the significant yet hidden length of the polypeptide chains; the

    polypeptide length allows the silk to still experience low stress values despite

    increases in strain values, hence why this regime is also known as the plateau

    regime.  Up to this point the behavior of the silk has almost been completely

    dependent upon the behavior of the semi-amorphous regions.

 

    strain curve follows one more final high-stiffness (and covalent) regime; this

    phenomenon occurs at strain values of around 50%.  It is in this regime that the

    deformation switches from semi-amorphous region influence to influence from the

    beta-sheet nanocrystals.  Due to the high-stiffness of the system as a whole, the beta-

    sheet nanocrystals start to sustain larger amounts of strain.  As shown in figure 1b,

    altering the size of the beta-sheet nanocrystals can reduce the amount of stress the

    system incurs.  Through experimentation performed at our lab, we can now see that

    silk fibrils break at lower stress values when the size of the crystals is larger.  

    Consequently, the third regime lasts significantly longer when the crystal size is

    smaller.

 

    maximum tensile strength of the silk (the maximum tensile strength varies for

    different systems and parameters).  One visible sign of failure includes some

    individual beta-strands being completely pulled out.  To reinforce how strong spider

    silk is, it is important to note that failure occurs on approximately the order of a GPa.  

    Like in the third regime, the point of failure is determined entirely by the strength of

    the beta-sheet nanocrystals, which is directly proportional to their size.

Figure 2: Molecular Nanostructure of Silk Protein Domain

 

This image from a replica exchange simulation displays the twisted, hierarchical structure of spider silk on a nanoscale level.  The middle region makes up the beta-sheet nanocrystals, with the ancillary regions represent amorphous residues.  We can see the orderly stacking formation of the beta-sheet nanocrystals in the middle region of the molecule.  However, the other portion of the molecule displays the semi-amorphous regions of spider silk, with the much less organized structure (adapted from Keten et al., 2010 [2]).

Figure 3: Representation of spider silk nanostructure under increasing deformation

levels.

 

 

 

Figure 1: Idealized representation of spider silk nanostructure under tensile loading (from A. Nova et al., 2010 [1]); (a) Schematic of model setup and loading geometry of tensile testing; (b) Comparison of beta-sheet nanocrystal size effect on the stress-strain response.