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John R. Jungck Professor of Biological Sciences and Mathema:cal Sciences
Computa:onal Biology and Bioinforma:cs Interdisciplinary Science and Engineering Lab
University of Delaware
IMA Annual Program Year Workshop: BIOLOGICAL SYSTEMS AND NETWORKS
Graph Theory for Systems Biology: Interval Graphs, Mo:fs, and PaJern Recogni:on
Triad Profile Analysis of three species: “EvoluConary design principles and funcConal characterisCcs based on kingdom-‐specific network moCfs.” Tae-‐Hwan Kim, Junil Kim, Pat Heslop-‐Harrison, and Kwang-‐Hyun Cho. (2011) BioinformaCcs 27 (2): 245-‐251.
‘Two-‐node feedback’ is the most significant mo:f in all three species. By considering the sign of each two-‐node feedback interac:on, we examined the enrichment of the three types of two-‐node feedbacks:
(1) posi:ve–posi:ve (PP), (2) nega:ve–nega:ve (NN), and (3) posi:ve–nega:ve (PN).
We found that PN is enriched in the network of A.thaliana, NN in the network of S.cerevisiae and PP and NN in the network of H.sapiens. Each feedback type has characteris:c features of robustness, mul:stability and homeostasis.
Triad Profile Analysis of three species: “EvoluConary design principles and funcConal characterisCcs based on kingdom-‐specific network moCfs.” Tae-‐Hwan Kim, Junil Kim, Pat Heslop-‐Harrison, and Kwang-‐Hyun Cho. (2011) BioinformaCcs 27 (2): 245-‐251.
PNAS (2003)
Could a liJle graph theory inform Rives & Galitski’s work?
Jungck, John R., and Rama Viswanathan. (2015).
Graph Theory for Systems Biology: Interval Graphs, Mo:fs, and PaJern Recogni:on.
In Raina Robeva, Editor, Algebraic And Discrete Mathema8cal
Methods for Modern Biology. Elsevier: Chennai, India, pages 1-‐27 plus electronic supplement.
Interval graphs:
An interval graph G = (V,E) is an undirected graph obtained from a collec:on C of intervals on the real
line. To each interval in C there corresponds a vertex in G. The edge (u,v) is in E if and only if their
corresponding intervals intersect.
Benzer divided the region of the chromosome into subsec:ons with Ordered overlapping dele:ons and then mapped point muta:ons in Standard three-‐point crosses to determine order within subsec:ons
Benzer’s map was en:tled the first
“Fine Structure of the Gene”
It clearly demonstrated that the gene
was not the unit of func:on, recombina:on, and func:on.
Muta:on was not uniform; there were hot spots.
Recombina:on could occur between adjacent nucleo:des in DNA.
Graph Theory approach
Fulkerson & Gross 1964
IntersecCon Graph & Complementary Graph
No “Z4” s
TransiCvely Oriented Complementary Graph
Maximal Cliques
A
C
B
D
Hamiltonian Path of Maximal Cliques
outDegree b(3) -‐> a(2) -‐> c(1) -‐> d(0)
Interval Graph SoluCon
Steps in the graph theory solution of "Benzer's problem”
1. Convert each deletion mutant (restriction fragment) into a vertex. 2. Construct the intersection graph by placing an edge
between each pair of vertices which represent overlapping deletions.
3. Construct the complement of the intersection graph. 4. Check for absence of Z4's in the intersection graph. 5. Determine whether the complement of the intersection graph
can be made transitive. 6. Find all the maximal cliques in the intersection graph. 7. Order these maximal cliques in the same way as in the transitive
complementary graph. 8. Find the Hamiltonian path of all the ordered maximal cliques. 9. Construct the interval graph by assigning deletions to each
interval of the line, which sequentially orders the maximal cliques, for all the cliques to which the deletion vertex belongs.
Thus, the algorithm is capable of processing the original recombination matrix data through each of these nine steps.
Set TheoreCc approach
Shkurba 1965
original form
1 2 3 4 5 6 7
Move 5 one to the right
1 2 3 4 6 5 7
Move 1 three to the right
2 3 4 1 6 5 7
“Shkurba form”
2 3 4 1 6 5 7
Interval Graph SoluCon
Two Alterna:ve Approaches
Visualiza:on maJers!
BioGrapher
javaBenzer
Network Design
The field of Network Design comprises a large class of problems:
• Low cost • High Capacity • Fault Tolerant • Highly Connected (the remaining nodes will s:ll be able to communicate)
• Low conges:on
One-‐dimensional logic gate assignment and interval graphs
• Ohtsuki, T., Mori, H., Khu, E.S., Kashiwabara, T., and Fujisawa, T. One-‐dimensional logic gate assignment and interval graphs. Circuits and Systems, IEEE Transac:ons on 26 (9 ): 675 – 684 (1979)
• This paper gives a graph-‐theore:c approach to the design of one-‐dimensional logic gate arrays using MOS or units. The incidence rela:on between gates and nets is represented by a graph , and a possible layout of gates and nets is characterized by an interval graph , where is called an augmenta:on. It is shown that the number of tracks required for between-‐gate wiring is equal to the clique number (chroma:c number) of , and hence the op:mum placement problem is converted to that of minimum clique number augmenta:on. This turns out to be an NP-‐complete problem. Instead a polynomial-‐:me algorithm for finding a minimal augmenta:on is presented, where an augmenta:on is minimal if no proper subset of it is an augmenta:on. An algorithm for gate sequencing with respect to a given augmenta:on is also presented.
“This follows from a classic interval-‐graph coloring result: if at most L subintervals of a line segment contain any point of the segment, then the subintervals can be colored with at most L colors so that overlapping subintervals have dis:nct colors. ...”
194 No. 96 Tores and Barillot
Optimizing Pedigree Drawing Using Interval GraphTheory
Frederic Tores1 Emmanuel Barillot1,2
[email protected] [email protected] INFOBIOGEN, 7 rue Guy Moquet, BP8, 94800 Villejuif, France2 GENETHON, 1 rue de l’Internationale, 91000 Evry, France
Keywords: pedigree drawing, interval graph, interval graph sandwich, graph augmentation, graphdiminut
1 Introduction
A pedigree is a set of individuals that are related by four types of relations: mate, parent, child andsib. Conventionally, in a pedigree drawing: (1) Mates are linked to a mating unit. (2) Sibs are linkedto a sibship unit. (3) The sibship is linked to its parental mating (4) people from the same generationare drawn on the same horizontal line, and the older generations are at the top (see Figure 1).
Naive strategies to pedigree drawing may lead to poor readability of the representation becauseof numerous link crossings. Pedigrees including consanguinity loops, individuals with multiples matesor several related families are particularly problematic to draw neatly. To our knowledge, none of theexisting pedigree drawing softwares draws pedigree perfectly in all possible cases.
We propose to use interval graph theory to find a perfect representation of the pedigree, thatis a representation with no link crossing, if such a representation exists. If not, we propose to useline-crossing elimination in directed graph to find the best layout.
2 Method and Results
We have defined the rules of readability to whom a pedigree drawing should conform to be perfectlymeaningful: (a) No overlap is allowed between individuals. (b) Mates must be adjacent. (c) Sibs mustbe adjacent, but orphan spouses may be inserted within a sibship. (d) Parents are above their childsibship. (e) There are no link crossing. A pedigree for which a drawing verifying these five rules andthe four conventions given in introduction exists is said to be a perfectly drawable pedigree (PDP).
Starting from a pedigree, we now define the following set V of vertices: a vertex per individual, avertex per mating, and a vertex per sibship. We have shown that establishing if a pedigree is PDP is
SDD
F
mating
sibship
M
S
MF
Figure 1: Example of simple pedigree drawing and its associated interval graph. Right, solid lines aremandatory edges, dotted with a cross are forbidden, dashed have been added to get an interval graph.
