Biomedical Engineering Reference
In-Depth Information
3
Mathematical and Geometric Models for Describing
Virus Phenomena
In this section, we explore how concepts borrowed from mathematics and geometry
may help in understanding structural features of virus capsids. Using these simplistic
models, we have addressed problems at various levels of capsid research, ranging
from understanding what is the optimal subunit shape for constructing a capsid,
to understanding the prevalence of certain T -number structures and the absence of
others.
3.1
The Canonical Capsid Model
The utility of simplistic models, which do not account for the specific atomic level
interactions (i.e., an all-atom force field) have been useful in explaining various cap-
sid phenomena such as capsid self-assembly [
27
-
33
], capsid morphology [
34
-
37
],
subunit stoichiometry [
38
-
40
], mechanical properties [
41
-
43
], and symmetry
[
37
,
44
]. We will be primarily discussing one such geometric model, the “canonical
capsid,” that has served as a useful platform for the elucidation of capsid design
principles [
36
,
37
,
40
].
The concept of the “canonical capsid,” which is a surprisingly simple construct, is
defined as a polyhedron whose faces, each representing a subunit, must be identical
in shape. This model is also known as a “monohedral tiling.” This simple model
is useful because a large number of capsids found in nature can be represented as
monohedral tilings [
40
]. In addition, these models can shed light on various physical
properties of virus capsids that can be described as canonical.
3.2
Prediction of the Optimal Subunit Shape
Given the construct of the canonical capsid, a key question for investigation is which
subunit shapes are permitted to exist within the confines of the canonical capsid.
Using simple geometry and polyhedral rules [
40
], we have shown that canonical
capsids can only accommodate
one
type of “prototile” (subunit design) consisting
of five interacting edges. The bisected trapezoid (Fig.
3
a) is one such acceptable
prototile design. It is the same subunit shape that appears in all the natural capsids
(Fig.
3
b) we find to be represented by the canonical capsid model [
40
]. It has indeed
been identified that many viruses share a common subunit protein fold (the double
ˇ-barrel), without sharing high sequence identity [
45
]. It is quite surprising that a
simple canonical capsid model predicts such a ubiquitous shape found in viruses
infecting almost all domains of life. Apparently, nature may be forcing viral capsid
proteins into adopting this very special shape. It is tempting to conjecture that there
is an overarching evolutionary pressure that may be acting on virus capsid's design.
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