Figure 1.
Illustration of multiscale hierarchical structure of protein materials, a summary of multiscale modeling and experimental tools, and an analogy to music (figure adapted from [14]).
In protein materials (left for the example of spider silk), multifunctional materials are created via the formation of hierarchical structures. The synergistic interaction of structures and mechanisms at multiple scales provides the basis for enhanced functionality of biological materials despite the reliance on few distinct building blocks. Similar to the case of protein materials is music (right), where universal elements such as basic wave forms or a set of available instruments are used in hierarchical assemblies to provide macroscale functionality, and eventually a particular orchestral sound (e.g. a symphony). Universality tends to dominate at smaller levels, whereas diversity is found predominantly at larger, functional levels. The integrated use of computational and experimental methods at multiple scales provides a powerful approach to elucidate the scientific concepts underlying the materiomics paradigm (center).
Figure 2.
Simple examples of transformations preserving structure in category theory.
Categories consist of objects and arrows which are closed under composition and satisfy certain conditions typical of functions. ψ is a structure-preserving transformation (or covariant functor, or morphism of categories) between the two categories. If the categorical objects in this example are considered as sets of instances, then each instance of the set ‘A man’ is mapped to an instance of the set ‘A tennis ball’. This concept applies to all objects and arrows in the categories.
Figure 3.
Visualization of protein filaments considered here, and abstraction of how key functional properties (here: mechanical properties under axial extension) can be understood based on the interplay of a set of “building blocks”.
A, Overview over fundamental building blocks of our protein materials. The protein materials considered here are composed of a linear arrangement of three elements, “bricks”, “glue”, and in some cases “lifeline”. Thereby as a design rule, brick and glue need to alternate in order to achieve a stable structure. That is, two brick or glue elements immediately next to each other would not stick together – the chemical reason is that bricks represent the protein's polypeptide backbone and glue represents H-bonding which can only occur between residues in the backbone. The “lifeline” is a third element introduced here, reflecting the situation when there is still a physical connection between bricks even after the glue breaks. Chemically, this resembles the existence of “hidden” polypeptide length such that there exists a “covalent” link between two brick elements even after the H-bond glue has broken. This hidden length is not “visible” before the glue is actually broken. B, Mechanical behavior of each of the building blocks characterized by a description of the failure extension. The hidden length of lifelines is reflected in the fact that the resting extension of the lifeline is roughly equal to the failure extension of the glue. Both the brick and the lifeline have large failure extensions relative to the glue. C, Model of a beta-sheet crystal under axial loading. This resembles a system without a lifeline since after breaking of the H-bond cluster ( = glue) between the layers formed by clusters of polypeptide ( = brick) no physical connection exists. D, Model of a beta-helical protein under axial loading. This resembles a system with a lifeline, as after breaking of the cluster of H-bonds ( = glue) that are formed between clusters of amino acids ( = brick) there still exists a physical connection due to the polypeptide backbone as shown in D ( = lifeline). As shown in E, the existence of a lifeline has major implications on the functional properties of the overall system. A system with a lifeline (D) shows a ductile response, where a connection can be sustained at large extension as compared to the glue alone. In contrast a system without a lifeline (C) shows a brittle response, where only a small extension can be sustained until the material breaks (which equals roughly the failure extension of the glue).
Figure 4.
Pictured here is an olog, which captures the semantic content of our situations, as described in Sections 3.1 – 3.3.
Each box represents an abstract type, and each arrow represents an aspect (or observable) of that type. Each type refers to a set of intended instances, which we think of as being contained in the box. For example, box E contains ductile sequences of bricks and glue (like a beta-helix or an alpha-helix), whereas box V contains real numbers (like 9.228). Each arrow from a source box to a target box refers to an observation one may make on things in the source box, for which the observed result is a thing in the target box. For example, arrow 11:E→O indicates that one can observe of any ductile material S a pair of numbers (R,r) where R is much greater than r. The meaning of these numbers R and r is enforced by a “commutative diagram” declared in Table 1 (line 6): the number R must refer to the failure extension of the system S and the number r must refer to the failure extension of its glue. This says that a ductile system fails at a much greater extension than its glue elements do. Perhaps a simpler but more mundane observation is made by arrow 37:Q→V which indicates that one can take any pair of real numbers (x,y) and observe the x-coordinate. So the pair (8.0, 3.2) is inside box Q, and it is observed by arrow 37 to have x-coordinate 8.0, which is in box V. Thus, each box is meant to contain an intended set of instances and each arrow is meant to functionally relate two such sets. The rest of the olog is recorded in Tables 1 and 2. Some are commutative diagrams which declare two paths through the olog to be equivalent and some are fiber products which define new types in terms of others.
Table 1.
Commutative diagrams in the olog. *
Table 2.
Fiber product diagrams in the olog.**
Figure 5.
Visual representation of the social network.
A, Overview over fundamental building blocks of our social networks. The social networks considered here are composed of a linear arrangement of three elements, “bricks”, “glue” and in some cases “lifeline”. Thereby, as a design rule, brick and glue need to alternate in order to achieve a stable structure. That is, two brick or glue elements immediately next to each other would not stick together; where the reason is that bricks represent participants with transceivers and glue represents wireless communication that, in our case, can only occur between neighboring participants. The “lifeline” is a third element that is introduced here, reflecting the situation when there is still a physical connection of bricks even after the glue breaks. This reflects the existence of a “hidden” connection in that there exists a physical passageway between two brick elements even after the communication over the wireless connection is no longer feasible. The hidden connection is not “visible” before the glue is actually broken because, for reasons of efficiency, participants will choose to communicate the simple messages wirelessly rather than verbally, as the latter requires much more effort. B, Mechanical behavior of each of the building blocks. The hidden length of lifelines is reflected in the fact that the resting extension of the lifeline is roughly equal to the failure extension of the glue. In other words, lifeline passageways are used only when wireless communication is no longer feasible. Both the brick and the lifeline have large failure extensions relative to the glue because participants and their verbal communication function perfectly well in the presence of noise on the wireless channels. C, Representation of a social network not allowing for face-to-face interaction under stress from wireless noise. This resembles a system without a lifeline, as after noise on the wireless line reaches a critical point, messages can no longer be correctly conveyed. D, Representation of a social network allowing for face-to-face interaction under high levels of wireless noise. This resembles a system with a lifeline, as after messages can no longer be conveyed wirelessly, communication can still take place, due to the physical passageways as shown in D. As shown in E, the existence of a lifeline has major implications on the functional properties of the system. A system with a lifeline (D) shows a ductile response, where a connection can be sustained at large displacements as compared to the glue alone. In contrast a system without a lifeline (C) shows a brittle response, where only a small displacement can be sustained until the material breaks (roughly the failure extension of the glue).
Table 3.
Analogy between protein and social network.***
Figure 6.
Schematic illustration of the approach discussed here, the representation of complex hierarchical systems such as biological materials (e.g. silk) and language in the same category theory space (olog).
The description of how functional properties emerge in different hierarchical systems can be rigorously described using this approach, and fundamental insight can be derived and compared between different systems. This finds immediate applications in the design of synthetic systems (e.g. novel fiber and bulk materials with tunable functional properties). The poem “The Road Not Taken” shown on the right written by Robert Frost (March 26, 1874 - January 29, 1963), published in 1916 in the collection Mountain Interval. Text from: http://en.wikisource.org/wiki/The_Road_Not_Taken.