Applicability�of�Loop� Applicability�of�Loop� Recombination�in�Ciliates�using� Recombination�in�Ciliates�using� the�Breakpoint�Graph the�Breakpoint�Graph Robert�Brijder,�Hendrik Hendrik Jan� Jan�Hoogeboom Hoogeboom,� ,� Robert�Brijder,� and�Michael�Muskulus Muskulus and�Michael� Leiden�Institute�of�Advanced� Leiden�Institute�of�Advanced� Computer�Science, Computer�Science, Leiden�University Leiden�University CompLife�'06 Sept.�27-29,�2006
Overview Overview • Brief�overview�of�gene�assembly�in�ciliates. Brief�overview�of�gene�assembly�in�ciliates. • • Brief�overview�of�a�formal�model. Brief�overview�of�a�formal�model. • • Recall�reduction�graph. Recall�reduction�graph. • • Introduce�graph�`on�top�of Introduce�graph�`on�top�of’ ’ reduction� reduction� • graph,�and�show�its�uses. graph,�and�show�its�uses. Sept.�27-29,�2006 CompLife�'06
Gene�Assembly�in�Ciliates Gene�Assembly�in�Ciliates A�gene�of�the�Sterkiella Sterkiella nova nova A�gene�of�the� ��� ��� � � � � � � � � � ������������� ������������� ������������ ������������ ��� ��� � � � � � � � � � ������������ ������������ ������� ������� ���� ���� Sept.�27-29,�2006 CompLife�'06
Gene�Assembly Gene�Assembly • The�gene�assembly�process�is�done�using� The�gene�assembly�process�is�done�using� • molecular�operations. molecular�operations. • With�the�aid�of� With�the�aid�of� pointers pointers these�operations� these�operations� • `know’ ’ how�these�parts�need�to�be�glued� how�these�parts�need�to�be�glued� `know together. together. 3 1 2 4 Actually 3 4 2 3 2 4 b e Sept.�27-29,�2006 CompLife�'06
Gene�Assembly Gene�Assembly The�gene�assembly�process�is�accomplished� The�gene�assembly�process�is�accomplished� using�three�molecular�operations: using�three�molecular�operations: • Loop�recombination Loop�recombination • • Hairpin�recombination Hairpin�recombination • • Double Double- -loop�recombination loop�recombination • Sept.�27-29,�2006 CompLife�'06
Gene�Assembly Gene�Assembly • Loop�recombination Loop�recombination • • Hairpin�recombination Hairpin�recombination • Sept.�27-29,�2006 CompLife�'06
Gene�Assembly Gene�Assembly • Double Double- -loop�recombination loop�recombination • Sept.�27-29,�2006 CompLife�'06
Gene�Assembly Gene�Assembly The�process�is�irreversible: The�process�is�irreversible: when�a�molecular�operation�is�applied�on�a� when�a�molecular�operation�is�applied�on�a� pointer,�then�this�pointer�cannot�be�used� pointer,�then�this�pointer�cannot�be�used� again. again. Sept.�27-29,�2006 CompLife�'06
Overview Overview • Brief�overview�of�gene�assembly�in�ciliates. Brief�overview�of�gene�assembly�in�ciliates. • • Brief�overview�of�a�formal�model. Brief�overview�of�a�formal�model. • • Recall�reduction�graph. Recall�reduction�graph. • • Introduce�graph�`on�top�of Introduce�graph�`on�top�of’ ’ reduction� reduction� • graph,�and�show�its�uses. graph,�and�show�its�uses. Sept.�27-29,�2006 CompLife�'06
Modeling�Gene�Assembly Modeling�Gene�Assembly 3 4 2 3 2 4 b e SPRS legal�string 342 3 2 4 Π = { 2 , 2 , 3 , 3 ,...}. � Sept.�27-29,�2006 CompLife�'06
Modeling�Gene�Assembly Modeling�Gene�Assembly Loop�recombination�- - String�negative�rule String�negative�rule Loop�recombination� ∈ Π ∈ Π * For � p and � x , z : = snr p ( xppz ) xz 45 3 3 45 4545 = e.g., � snr ( ) . 3 Sept.�27-29,�2006 CompLife�'06
Modeling�Gene�Assembly Modeling�Gene�Assembly Hairpin�recombination�- - String�positive�rule String�positive�rule Hairpin�recombination� = spr p ( xpy p z ) x y z = where p p = � = � � u x x x u x x x − 1 2 n n n 1 1 Sept.�27-29,�2006 CompLife�'06
Modeling�Gene�Assembly Modeling�Gene�Assembly Double- -loop�recombination� loop�recombination�- - String�double�rule String�double�rule Double = sdr ( xpyqzpuqw ) xuzyw p , q Sept.�27-29,�2006 CompLife�'06
Modeling�Gene�Assembly Modeling�Gene�Assembly • Successful�reductions: 3434 spr sdr 2 3 , 4 λ 342 3 2 4 spr spr 2 4 2 4 2 2 3 2 spr 4 λ • corresponds�to�a�successful�assembled�gene.� • Non-deterministic�process. b e • No�deadlocks. • Finite�process.�Pointers�are�removed�in�every�step. Sept.�27-29,�2006 CompLife�'06
Overview Overview • Brief�overview�of�gene�assembly�in�ciliates. Brief�overview�of�gene�assembly�in�ciliates. • • Brief�overview�of�a�formal�model. Brief�overview�of�a�formal�model. • • Recall�reduction�graph. Recall�reduction�graph. • • Introduce�graph�`on�top�of Introduce�graph�`on�top�of’ ’ reduction� reduction� • graph,�and�show�its�uses. graph,�and�show�its�uses. Sept.�27-29,�2006 CompLife�'06
Reduction�Graph Reduction�Graph Concept�of�breakpoint�graph:� reality-and-desire Sept.�27-29,�2006 CompLife�'06
Reduction�Graph�Example Reduction�Graph�Example Double�edges�are�the�reality�edges,�single�edges�the� • desire�edges There�is�a�cyclic�component. • This�is�the�less�general�version�of�reduction�graph. • Sept.�27-29,�2006 CompLife�'06
A�known�result A�known�result Let � u � be � a � legal � string, � and � let � N � be � the � number � of � cyclic� components � in � R(u) . � Then � every� successful � reduction � of � u � has � exactly� N � string � negative � rules. Complexity: O (| u |) Sept.�27-29,�2006 CompLife�'06
Example Example 23 2 4 34 u = Let � � be � a � legal � string. � Then � R(u) � has � one � cyclic� component. � Thus, � every� successful � reduction � of � u � has � exactly� one � string � negative � rule. Successful � reductions : snr �spr �spr , 2 3 4 snr �spr �spr , 3 2 4 snr �spr �spr , 3 4 2 snr �spr �spr . 4 3 2 Sept.�27-29,�2006 CompLife�'06
Overview Overview • Brief�overview�of�gene�assembly�in�ciliates. Brief�overview�of�gene�assembly�in�ciliates. • • Brief�overview�of�a�formal�model. Brief�overview�of�a�formal�model. • • Recall�reduction�graph. Recall�reduction�graph. • • Introduce�graph�`on�top�of Introduce�graph�`on�top�of’ ’ reduction� reduction� • graph,�and�show�its�uses. graph,�and�show�its�uses. Sept.�27-29,�2006 CompLife�'06
Motivation Motivation • Now�we�know� Now�we�know� how�many how�many Snr Snr rules�are� rules�are� • needed,�can�we�also�determine� on�which� on�which� needed,�can�we�also�determine� pointers� these� these�Snr Snr rules�are�applied�in� rules�are�applied�in� pointers� successful�reductions? successful�reductions? • Previous�example:� Previous�example:�Snr Snr domains�are�{2},� domains�are�{2},� • {3},�{4}. {3},�{4}. • If�so,�is�this�`information If�so,�is�this�`information’ ’ retained�in�the� retained�in�the� • reduction�graph? reduction�graph? Sept.�27-29,�2006 CompLife�'06
More�complicated�Example More�complicated�Example Sept.�27-29,�2006 CompLife�'06
More�complicated�Example More�complicated�Example � Which�is: Sept.�27-29,�2006 CompLife�'06
Pointer�Component�Graph Pointer�Component�Graph 7 5 6 3 4 2 • A�multigraph.�Notice�that�loops�and�parallel� edges�are�allowed. Sept.�27-29,�2006 CompLife�'06
Pointer�Component�Graph Pointer�Component�Graph � Which�is: � R is�the�linear�connected�component,�and�the� others�are�cyclic�components. Sept.�27-29,�2006 CompLife�'06
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