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Games, Genomes, and GraphsKatharine Adamyk1, Erik Holmes2, Georgia Mayfield3, Dennis J. Moritz4 and Dr. Marion Scheepers2

1Gordon College, 2Boise State University, 3Willamette University, 4Montana Tech

IntroductionCiliates are single-celled organisms with a scrambledcopy of their genome within their micronuclei. Decryp-tion of the micronuclear genome produces functionalmacronuclear DNA [3]. This process is modeled through"Context-Directed Swaps" (CDS) and "Context-DirectedReversals" (CDR) on signed permutations [4].

Figure 1: DNA rearrangement process in ciliates. [5]

Signed Permutations are arrangements of unique num-bers, each of which can be positive or negative.

α = [ (4,5)5 −(4,3) − 3−(3,2)(1,2)2(2,3)

−(5,4) − 4−(4,3) 1(1,2)]

Decryption Operations:I CDS uses two pointer pairs p and q occuring in the

order . . . p . . . q . . . p . . . q . . . to switch the blocksflanked by p and q.

α0 = [ (4,5)5 1(1,2) (1,2)2(2,3)−(5,4) − 4−(4,3) −(4,3) − 3−(3,2) ]

I CDR uses a pointer pair p and −p to reverseeverything between p and −p, e.g. a, b,−c becomesc,−b,−a.

α1 = [ (4,5)5 −(4,3)− 3−(3,2) −(3,2)− 2−(2,1) −(5,4)− 4−(4,3) 1(1,2)]

ObjectivesI Determine which permutations can be sorted using

CDR, CDS, or CDR and CDS together.I Determine all possible end configurations for a given

permutation using these operations.I Create games using these operations by defining

winning conditions.I Determine which player has a winning strategy for a

given permutation and winning condition.

MethodsTo examine the effects of the restricted operations on permutations, we defined several two-player games by choosinglegal moves (CDS, CDR, or CDS and CDR) and winning conditions. Since our games meet Zermelo’s conditions fordetermined games, some player always has a winning strategy [6].

Game I: Players ONE and TWO alternately choose CDR moves on a signed permutation until there are no legal movesleft. ONE wins if the final state contains only positive elements. Otherwise, TWO wins.Game II: Players ONE and TWO alternately choose CDS moves on an uninvertible permutation of positive numbersuntil there is no legal move left. ONE wins if k < n

2 in the final state α = [k + 1...n 1...k]. Otherwise, TWO wins.

061-

1+

2-

2+3-3+

4-

4+

5-

5+

Figure 2: Signed CycleGraph for α

01− 4− 3+ 5+ 6

1+

2−3−2+4+5−

Figure 3: The alternating cycle of α. The circles denote elements of the Strategic Pile.

The cycle-graph[1] or breakpoint-graph[2] of a permutation contains allinformation about the permutation. This permits the play of games defineddirectly on the graph. The graphs contain alternating cycles, formed bywalking a path that alternates between red and blue edges [2]. The graphof α in Figure 2 (left) includes the cycle in Figure 3 (above).

5 − 3 2 − 4 1

(2, 3)

(1, 2)

(3, 4)

E3

(4, 5)

(3, 4)

E1

(4, 5)

(1, 2)

(3, 4)

E2

(2, 3)

(3, 4)

E1

(3, 4)

(1, 2)

E2

(2, 3)

E1

(3, 4)(1, 2)

(2, 3)

E3

(4, 5)

E2

Figure 4: Game Tree of α

A game tree maps all outcomes of CDS and/or CDR moves on a given permutation. In α each path leads to [2 3 4 5 1],[−2 − 1 − 5 − 4 − 3], or [5 1 2 3 4] . The possible ending conditions are determined by the Strategic Pile.

Results

Invertibility Criterion

A permutation is invertible by CDS, CDR, or both if,and only if, → 0 · · · 1− and n+ · · · (n + 1) → are mem-bers of the same alternating cycle.

Winning Strategies for Game I

Player ONE has a winning strategy if:Iα is forward CDR-invertible.Player TWO has a winning strategy if:Iα is reverse CDR-invertible.

Winning Strategies for Game II

Consider a permutation whose m strategic pile ele-ments are of the form k+ · · · (k + 1)−.Player ONE has a winning strategy if:I The number of such k less than n

2 is at least(3/4)m.

Player TWO has a winning strategy if:I The number of such k larger than n

2 is at least(3/4)m + 2.

ImplicationsI The characterization of permutations invertible by

CDS/CDR provides an efficient criterion to decide if apermutation is sortable under the current model ofciliate operations [4].

I By identifying and examining the strategic pile wenarrowed down which end states are possible for apermutation uninvertible by CDS or CDR.

Future WorkI Characterize which player has a winning strategy in

our games.I Determine if the group theoretic order of permutations

are related to invertibililty by CDS or CDR.I Prove that the possible end states in CDR games

correspond bijectively to elements in the strategic pile.I Characterize what types of scrambled conditions

occur, in which permutations, and by which paths.

AcknowledgementsI 2012 REU/INBRE team for access to their prepublication findingsI Boise State University Mathematics REU ProgramI NSF - REU funding via grant DMS 1062857

References[1] V. Bafna and P.A. Pevzner, Sorting by transpositions, Proc. 6th Ann.

ACM-SIAM Symp. on Discrete Algorithms (1995) 614 - 623

[2] S. Hannenhalli, P. A. Pevzner. Transforming Cabbage into Turnip:Polynomial Algorithm for Sorting Signed Permutations by Reversals.Journal of the Association for Computing Machinery, Volume 46, Issue1, Jan. 1999. 1 - 27.

[3] D. Prescott. Genome Gymnastics: Unique Modes of DNA Evolution andProcessing In Ciliates. Nature Reviews Genetics, Volume 1, Issue 3, Dec.2000. 191 - 198.

[4] D.M. Prescott, A. Ehrenfeucht, G.Rozenberg Template-guidedrecombination for IES elimination and unscrambling of genes instichotrichous ciliates, Journal of Theoretical Biology 222 (2001), 323 -330

[5] E.C. Swart, J.R. Bracht, V. Magrini, P. Minx, X. Chen, et al. The Oxytrichatrifallax Macronuclear Genome: A Complex Eukaryotic Genome with 16,000Tiny Chromosomes, PLoS Biol (2013) 11(1): e1001473.doi:10.1371/journal.pbio.1001473

[6] E. Zermelo, Über eine Anwendung der Mengenlehre auf die Theorie desSchachspiels, Proc. Fifth Congress Mathematicians, CambridgeUniversity Press 1913, 501-504