Modelling the Sphereactor

Structure of a Pdu Microcompartment

One approach to modelling we decided to take, was to investigate the structure of a Pdu microcompartment and its size relative to an E.coli chassis. Pdu microcompartments are thought to be very similar structurally to carboxysomes, the microcompartments involved in carbon fixation in autotrophic cyanobacteria and some chemoautotrophic bacteria. Comparisons of shell proteins of carboxysomes and Pdu microcompartments have shown considerable homology, and it is extremely likely that they assemble to form a similar structure. Studies of carboxysomes have shown that they are regular icosahedral structures and it is thought that Pdu microcompartments also assume roughly this shape, with approximate diameter 120nm. [3]

A regular icosahedron is a polyhedron with twenty equilateral triangular faces, 30 edges, and 12 vertices.[14]

Figure 1: A regular icosahedron and its net [14]

For a regular icosahedron, it is possible to calculate values for the circumradius, (Rc, radius of a sphere touching all of an icosahedron’s vertices), inradius (Ri, radius of a sphere inscribed in an icosahedral that touches all of its faces) and midradius (Rm, radius that touches the middle of each edge) as follows:

where ‘α’ is the edge length of an icosahedron.

Say the radius = 60nm. Then using the above formula for the midradius, edge length can be calculated as α = 74.1641nm.

The total external surface area of an icosahedron, At, can be found by calculating the area of one equilateral triangular face, Ae, then multiplying by the number of faces.

An icosahedron is composed of 20 pyramids, each with a height Ri and a base of area Ae. The total volume of the icosahedron, Vt, can be found by calculating the volume of one of these pyramids, Vp, then multiplying by the number of pyramids.

The size of the sphereactor relative to its E. coli chassis was then investigated. The approximate length, L, and diameter, D, of an E. coli cell are 2μm and 0.8μm respectively [15]. If we assume an E. coli cell has a roughly cylindrical structure with two hemispherical ends, the approximate volume of our chassis can be calculated.

As the total volume of our sphereactor, Vt, is approximately 889,970nm3 or 8.8897x10-4μm3, we can therefore say that it is roughly 1/979 the size of its E. coli chassis.

Next, the proportions of the individual proteins that assemble into the shell of the BMC were investigated. It is thought that five molecules of the protein pduN form pentamers at each of the icosahedron’s twelve vertices. Therefore, theoretically there should be 60 pduN protein molecules present in a single microcompartment. PduA, -B, -B’, -J, -K, -T and –U all form hexagonal building blocks which associate into flat sheets to form the triangular faces of the icosahedron. While the single-BMC-domain proteins PduA, -J, -K, and -U form protein hexamers, PduB, -B’, and -T have tandem BMC domains so instead form hexagonal protein trimers. [4] The centre-to-centre length of two hexagonal sub-units varies but is on average 68Å or 6.8nm. [3] Using this value, the surface area of the external face of an individual hexagonal building block, Ah, can be calculated. The total external surface area of the icosahedron, At, can then be used to give the approximate number of hexagonal sub-units, Ht, present in the BMC shell.