Abstract We investigate the utility of a mathematical
framework based on discrete geometry to model
biological and synthetic self- assembly. Our primary
biological example is the self-assembly of icosahedral
viruses; our synthetic example is
surface-tension-driven self-folding polyhedra. In both
instances, the process of self-assembly is modeled by
decomposing the polyhedron into a set of partially
formed intermediate states. The set of all
intermediates is called the configuration space,
pathways of assembly are modeled as paths in the
configuration space, and the kinetics and yield of
assembly are modeled by rate equations, Markov chains,
or cost functions on the configuration space. We review
an interesting interplay between biological function
and mathematical structure in viruses in light of this
framework. We discuss in particular: (i) tiling theory
as a coarse- grained description of all-atom models;
(ii) the building game—a growth model for the
formation of polyhedra; and (iii) the application of
these models to the self-assembly of the bacteriophage
MS2. We then use a similar framework to model
self-folding polyhedra. We use a discrete folding
algorithm to compute a configuration space that
idealizes surface-tension-driven self-folding and
analyze pathways of assembly and dominant
intermediates. These computations are then compared
with experimental observations of a self-folding
dodecahedron with side 300 Am. In both models, despite
a combinatorial explosion in the size of the
configuration space, a few pathways and intermediates
dominate self-assembly. For self-folding polyhedra, the
dominant intermediates have fewer degrees of freedom
than comparable intermediates, and are thus more rigid.
The concentration of assembly pathways on a few
intermediates with distinguished geometric properties
is biologically and physically important, and suggests
deeper mathematical structure.
%0 Journal Article
%1 kaplan-polyhedra-self-assembly-2014
%A Kaplan, Ryan
%A Klobusický, Joseph
%A Pandey, Shivendra
%A Gracias, David H.
%A Menon, Govind
%D 2014
%I MIT Press - Journals
%J Artificial Life
%K alife polyhedra self-assembly
%N 4
%P 409--439
%R 10.1162/artl_a_00144
%T Building Polyhedra by Self-Assembly: Theory and
Experiment
%U http://dx.doi.org/10.1162/ARTL_a_00144
%V 20
%X Abstract We investigate the utility of a mathematical
framework based on discrete geometry to model
biological and synthetic self- assembly. Our primary
biological example is the self-assembly of icosahedral
viruses; our synthetic example is
surface-tension-driven self-folding polyhedra. In both
instances, the process of self-assembly is modeled by
decomposing the polyhedron into a set of partially
formed intermediate states. The set of all
intermediates is called the configuration space,
pathways of assembly are modeled as paths in the
configuration space, and the kinetics and yield of
assembly are modeled by rate equations, Markov chains,
or cost functions on the configuration space. We review
an interesting interplay between biological function
and mathematical structure in viruses in light of this
framework. We discuss in particular: (i) tiling theory
as a coarse- grained description of all-atom models;
(ii) the building game—a growth model for the
formation of polyhedra; and (iii) the application of
these models to the self-assembly of the bacteriophage
MS2. We then use a similar framework to model
self-folding polyhedra. We use a discrete folding
algorithm to compute a configuration space that
idealizes surface-tension-driven self-folding and
analyze pathways of assembly and dominant
intermediates. These computations are then compared
with experimental observations of a self-folding
dodecahedron with side 300 Am. In both models, despite
a combinatorial explosion in the size of the
configuration space, a few pathways and intermediates
dominate self-assembly. For self-folding polyhedra, the
dominant intermediates have fewer degrees of freedom
than comparable intermediates, and are thus more rigid.
The concentration of assembly pathways on a few
intermediates with distinguished geometric properties
is biologically and physically important, and suggests
deeper mathematical structure.
@article{kaplan-polyhedra-self-assembly-2014,
abstract = {Abstract We investigate the utility of a mathematical
framework based on discrete geometry to model
biological and synthetic self- assembly. Our primary
biological example is the self-assembly of icosahedral
viruses; our synthetic example is
surface-tension-driven self-folding polyhedra. In both
instances, the process of self-assembly is modeled by
decomposing the polyhedron into a set of partially
formed intermediate states. The set of all
intermediates is called the configuration space,
pathways of assembly are modeled as paths in the
configuration space, and the kinetics and yield of
assembly are modeled by rate equations, Markov chains,
or cost functions on the configuration space. We review
an interesting interplay between biological function
and mathematical structure in viruses in light of this
framework. We discuss in particular: (i) tiling theory
as a coarse- grained description of all-atom models;
(ii) the building game—a growth model for the
formation of polyhedra; and (iii) the application of
these models to the self-assembly of the bacteriophage
MS2. We then use a similar framework to model
self-folding polyhedra. We use a discrete folding
algorithm to compute a configuration space that
idealizes surface-tension-driven self-folding and
analyze pathways of assembly and dominant
intermediates. These computations are then compared
with experimental observations of a self-folding
dodecahedron with side 300 Am. In both models, despite
a combinatorial explosion in the size of the
configuration space, a few pathways and intermediates
dominate self-assembly. For self-folding polyhedra, the
dominant intermediates have fewer degrees of freedom
than comparable intermediates, and are thus more rigid.
The concentration of assembly pathways on a few
intermediates with distinguished geometric properties
is biologically and physically important, and suggests
deeper mathematical structure.},
added-at = {2015-02-02T12:20:07.000+0100},
author = {Kaplan, Ryan and Klobu{\v{s}}ick{\'{y}}, Joseph and Pandey, Shivendra and Gracias, David H. and Menon, Govind},
biburl = {https://www.bibsonomy.org/bibtex/244c0e570844ec996bbddececdbfc8d37/mhwombat},
doi = {10.1162/artl_a_00144},
interhash = {3c14494d18766d39e72418c65a8a071f},
intrahash = {44c0e570844ec996bbddececdbfc8d37},
journal = {Artificial Life},
keywords = {alife polyhedra self-assembly},
month = oct,
number = 4,
pages = {409--439},
publisher = {{MIT} Press - Journals},
timestamp = {2016-07-12T19:25:30.000+0200},
title = {Building Polyhedra by Self-Assembly: Theory and
Experiment},
url = {http://dx.doi.org/10.1162/ARTL_a_00144},
volume = 20,
year = 2014
}