# Team:Slovenia/Coiled-coil polyhedra.html

Modelling

## Topology of two-dimensional lattice made of single type of polypeptide chain

Our goal was to investigate the possible topologies that can lead to 2D or 3D assemblies consisting of three coiled-coil forming segments as the simplest building block. We also restricted the analysis in the first stage to the assemblies composed of one single type of polypeptide.

Legend:
a and a’ – parallel heterodimer and its pair
A and A’ – antiparallel heterodimer and its pair
A – antiparallel homodimer (pairs with itself)
a – parallel homodimer (pairs with itself)

Two-dimensional lattice can form through many different ways as well it can be assembled from different primitive cells.
Let us first consider only options using only one type of polypeptide chain comprising three coiled-coil segments, which is the simplest type of building block.
It turns out that trigonal lattice is geometrically easily obtainable as seen in the example below, where we use three segments of different antiparallel homodimers.

Figure 1

The same structure assembles if we use a-b-a’, a-B-a’, A-b-A’, A-b-C, a-b-c or any of their cyclic permutations.

Some of the above polypeptide chains can generate also hexagonal lattice, for example: a-b-a’, a-B-a’ and a-B-a and some cyclic permutations.

Figure 2

We could also use any of the following polypeptide chains: A-b-C, a-b-c and a-B-a or certain permutations. Hexagonal lattice seems to be the most common among the inspected lattices, since it can obviously be assembled of many different chains in a few different ways depending on the location of parallel and antiparallel elements within the chain.

For general overview of all inspected combinations see #Enumeration of all possible combinations for the assembly of structures made of single polypeptide chain and three coiled-coil forming segments

Note that in the examples above we have inspected only assemblies composed of one polypeptide chain type composed of three coiled-coil-forming elements. A whole lot more structures can be assembled if we allow the use of two, three,.. different chains with different number of elements. For more information #Extension to self-assemblies made of several different polypeptide chains

## Creating three-dimensional polygons from a single type of polypeptide chain

Note: here we show all possible assemblies, again restrict our discussion only to cases with one-type-three-element chains.

The simplest polyhedron that we can form is tetrahedron or triangular pyramid (a polyhedron composed of four triangular faces, three of which meet at each vertex). Tetrahedron is formed from four polypeptide chains. It can be assembled using any of the polypeptide chains with structures a-B-a’, A-B-C or a-B-a as seen on the animation below:

[click on image to play]

It should be noted that all of the cyclic permutations are also possible to form this type of polyhedra.

The next structure that can be obtained is n-sided prism, where the n depends on the polypeptide concentration, temperature and length of the linker (among other parameters). Some types of chains (combinations of coiled-coil-forming segments) even pre-define whether n can be even or odd (i.e. a-b-a’ forms prisms with even number of sides, contrary to a-B-a’ where there are no limitations on the accessible number of sides). This assembly is possible to obtain from different ways – below is presented example a-b-a’, which can form a box (parallelepiped) as the smallest polyhedron of this type of polypeptide chain:

Figure 3

It seems that these are all 3D structures one can form using only above stated type of chains. For generalization follow the link #Extension to self-assemblies made of several different polypeptide chains
For discussion about how to encourage the formation of 3D structures follow the link #Extension to self-assemblies made of several different polypeptide chains

[click on image to play]

Table 1

## Extension to self-assemblies made of several different polypeptide chains

Here is where real diversity begins. Let us loot at some examples:

Example A

Example B

Example C

Example D

All of these examples derive from the same three-armed motif but can assemble into topologically very different structures. Here we also present an idea how to provide greater control of the selectivity between forming 2D lattice or 3D polyhedra by altering the AA sequence and making the link between elements more or less flexible.
As shown previously on the example of DNA (He et al., 2008) it can be expected that the more rigid the link the more likely it is to create planar lattice. Therefore it is crucial that there is no flexible linker between elements P6-P2, P3-P5 and APH-NZ (as presented in example below) and that one helix continues into the other, creating one super-element. The advantage of this design is that we control the angles and can therefore assemble lattice with higher symmetry.

Figure 4

In this manner we can expect creation of the following planar structures:

Figure 5: Example A and B

Figure 6: Example C and D

In order to allow the extension into the third dimension we need the linkers between elements to allow flexibility. In this way we can form theoretically any regular polyhedron with three-armed motifs in the vertexes. Below you can see the formation of a cube, formed from the elements in the Example D:

Figure 7: Cube