Doctoral Thesis
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Institute
A common task in natural sciences is to
describe, characterize, and infer relations between discrete
objects. A set of relations E on a set of objects V can
naturally be expressed as a graph G = (V, E). It is
therefore often convenient to formalize problems in natural
sciences as graph theoretical problems.
In this thesis we will examine a number of problems found in
life sciences in particular, and show how to use graph theoretical
concepts to formalize and solve the presented problems. The
content of the thesis is a collection of papers all
solving separate problems that are relevant to biology
or biochemistry.
The first paper examines problems found in self-assembling
protein design. Designing polypeptides, composed of concatenated
coiled coil units, to fold into polyhedra turns out
to be intimately related to the concept of 1-face embeddings in
graph topology. We show that 1-face embeddings can be
canonicalized in linear time and present algorithms to enumerate
pairwise non-isomorphic 1-face embeddings in orientable surfaces.
The second and third paper examine problems found in evolutionary
biology. In particular, they focus on
inferring gene and species trees directly from sequence data
without any a priori knowledge of the trees topology. The second
paper characterize when gene trees can be inferred from
estimates of orthology, paralogy and xenology relations when only
partial information is available. Using this characterization an
algorithm is presented that constructs a gene tree consistent
with the estimates in polynomial time, if one exists. The
shown algorithm is used to experimentally show that gene trees
can be accurately inferred even in the case that only 20$\%$ of
the relations are known. The third paper explores how to
reconcile a gene tree with a species tree in a biologically
feasible way, when the events of the gene tree are known.
Biologically feasible reconciliations are characterized using
only the topology of the gene and species tree. Using this
characterization an algorithm is shown that constructs a
biologically feasible reconciliation in polynomial time, if one
exists.
The fourth and fifth paper are concerned with with the analysis
of automatically generated reaction networks. The fourth paper
introduces an algorithm to predict thermodynamic properties of
compounds in a chemistry. The algorithm is based on
the well known group contribution methods and will automatically
infer functional groups based on common structural motifs found
in a set of sampled compounds. It is shown experimentally that
the algorithm can be used to accurately
predict a variety of molecular properties such as normal boiling
point, Gibbs free energy, and the minimum free energy of RNA
secondary structures. The fifth and final paper presents a
framework to track atoms through reaction networks generated by a
graph grammar. Using concepts found in semigroup theory, the
paper defines the characteristic monoid of a reaction network. It
goes on to show how natural subsystems of a reaction network organically
emerge from the right Cayley graph of said monoid. The
applicability of the framework is proven by applying it to the
design of isotopic labeling experiments as well as to the
analysis of the TCA cycle.