detection of network motifs by local
play

Detection of network motifs by local Local Statistics - PowerPoint PPT Presentation

Detection of network motifs by local concentration Etienne Birmel e Context Detection of network motifs by local Local Statistics concentration A global statistic Motif detection Etienne Birmel e procedure Application to


  1. Detection of network motifs by local concentration Etienne Birmel´ e Context Detection of network motifs by local Local Statistics concentration A global statistic Motif detection Etienne Birmel´ e procedure Application to Yeast Laboratoire Statistique et G´ enome , Universit´ e d’Evry Conclusion Groupe SSB - ANR NeMo

  2. Detection of network motifs by local concentration Etienne 1 Context Birmel´ e Context 2 Local Statistics Local Statistics A global 3 A global statistic statistic Motif detection procedure 4 Motif detection procedure Application to Yeast 5 Application to Yeast Conclusion 6 Conclusion

  3. Detection of network Network motifs motifs by local concentration Etienne Birmel´ e Context A motif is a small graph which is over-represented in a network: Local Statistics it’s a candidate to be studied for a potential biological A global meaning. statistic Motif Example: the feed-forward loop detection procedure Y Application to �� �� Yeast ����� ����� �� �� ������� ������� ����� ����� ������� ������� ����� ����� Conclusion ������� ������� ����� ����� ������� ������� ����� ����� ������� ������� ����� ����� ������� ������� ����� ����� X ������� ������� Z ����� ����� ������� ������� ����� ����� ������� ������� ������������ ������������ � �

  4. Detection of network Network motif detection motifs by local concentration Etienne Birmel´ e Context Local Statistics All previous methods look for an overall over-representation: A global • U. Alon’s group (since 2002): simulations for size 3 and 4, statistic Motif Z -score detection procedure • J. Berg and M. L¨ assig (2004): probabilistic motifs by an Application to alignment heuristic Yeast Conclusion • F. Picard et al (2008): mixture model for the network and Polya-Aeppli distribution.

  5. Detection of network Leading ideas motifs by local concentration Etienne Birmel´ e Context Local • A small graph m may be over-represented because one of Statistics its subgraphs m ′ is over-represented. In that case, m ′ is A global statistic the relevant motif. Motif • Motifs in regulatory networks are known to be detection procedure concentrated on some places of the networks (Dobrin & al Application to Yeast 04). Conclusion • Z = f ( X 1 , . . . , X n ) is highly concentrated around its mean when the X i ’s are independent and changing the value of one of them does affect Z by less than a constant.

  6. Detection of network Changing the definition of a motif motifs by local concentration Etienne Consider a small graph m and a subgraph m ′ of m obtained by Birmel´ e the deletion of a vertex in m . Context �� �� m ′ � � Local ����� ����� �� �� � � m ����� ����� �� �� � � Statistics ����� ����� ����� ����� ����� ����� ����� ����� A global ����� ����� �� �� �� �� �� �� ����� ����� �� �� �� �� �� �� statistic �� �� �� �� �� �� ����� ����� ����� ����� ����� ����� ����� ����� Motif ����� ����� ����� ����� �� �� ����� ����� � � detection �� �� � � procedure m is a motif with respect to m ′ if there exist an occurence of Application to Yeast m ′ in the network which has a surprisingly high number of Conclusion extensions to occurences of m . m ′ �� �� �� �� �� �� �� �� � � � � �� �� �� �� � � � � �� �� �� �� � � � � �� �� �� �� �� ��

  7. Detection of network Random graph model motifs by local concentration Etienne Birmel´ e Context Local We fix the number n of nodes and the underlying random Statistics graph model is defined by a n × n matrix C: the edge indicators A global statistic ( X ij ) 1 ≤ i , j ≤ n are independent Bernoulli variables and Motif detection procedure P ( X ij = 1) = c ij Application to Yeast In particular, our theory is valid for: Conclusion • Edge probability proportional to d i d j . • Mixture models on graphs with fixed classes.

  8. Detection of network Random graph model motifs by local concentration Etienne Birmel´ e Context �� �� �� �� �� �� Local �� �� � � � � Statistics � � A global � � statistic � � � � Motif detection procedure P ( NN ) = 1 / 2 � � �� �� � � �� �� Application to Yeast �� �� �� �� Conclusion P ( RR ) = 1 / 4 �� �� �� �� �� �� P ( NR ) = 0 P ( RN ) = 1 / 16

  9. Detection of network motifs by local concentration Etienne 1 Context Birmel´ e Context 2 Local Statistics Local Statistics A global 3 A global statistic statistic Motif detection procedure 4 Motif detection procedure Application to Yeast 5 Application to Yeast Conclusion 6 Conclusion

  10. Detection of network Notations motifs by local concentration Etienne Birmel´ e Context Let m be a small graph on k vertices ( r 1 , . . . , r k − 1 , s ) and m ′ Local the subgraph obtained by deleting s . Statistics Let U = ( u 1 , . . . , u k − 1 ) be an ordered set of k − 1 vertices. A global statistic We define: Motif detection • N U ( m ) the number of occurrences of m which restriction procedure to U is isomorphic to m ′ ; Application to Yeast • Y U ( m ′ ) = I G [ U ] ∼ m ′ Conclusion • ext v U ( m ′ , m ) = 1 ⇔ ∀ i , X u i v = e r i s ext v U = 1 if adding the vertex v yields an occurence of m . ∈ U ext v • λ U = E ( � U ) the mean number of valid extensions. v /

  11. Detection of network Notations motifs by local concentration Etienne Birmel´ e Then Context Local � ext v Statistics N U ( m ) = Y U ( m ′ ) U ( m ′ , m ) A global v / ∈ U statistic and Y U and ext v Motif U are independent. detection procedure Application to Yeast U Conclusion �� �� �� �� �� �� � � �� �� � � �� �� � � �� �� � � �� �� �� �� �� ��

  12. Detection of network Example motifs by local concentration Etienne Birmel´ e r 2 r 2 m ′ m � � � � � � � � � � � � r 1 Context r 1 s �� �� �� �� �� �� �� �� �� �� �� �� Local Statistics � � � � r 3 r 3 � � � � � � � � A global statistic 7 Motif G �� �� �� �� detection �� �� procedure 6 Application to �� �� �� �� Yeast 1 2 3 4 5 Conclusion �� �� �� �� �� �� � � � � �� �� �� �� �� �� � � � � 8 �� �� �� �� �� �� 9 �� �� �� �� �� �� For U = (3 , 2 , 4), Y U ( m ′ ) = 1 and N U ( m ) = 3.

  13. Detection of network Poisson approximation motifs by local concentration Etienne Birmel´ e Context ∈ U ext v � U is a sum of independant Bernoulli r.v.’s and can v / Local therefore be approximated in total variation distance by a Statistics Poisson law of mean λ U : A global statistic ∀ A ⊂ Z + , Motif detection � p 2 | P ( N U ( m ) ∈ A | Y U ( m ′ )) − Po ( λ U )( A ) | ≤ min (1 , 1 /λ U ) procedure v Application to v Yeast Conclusion with p v = P ( ext v U = 1). In practice, p v ’s are small and that bound is quite sharp (between 1 . 8 e − 9 and 5 . 0 e − 3 for the different positions of the feed-forward loop in the Yeast regulatory network)

  14. Detection of network A local statistic motifs by local concentration Etienne Birmel´ e The upper bound approximation is even better for tail Context probabilities: Local If t = m − λ U > 1, Statistics λ U A global statistic t N U ( m ) ≥ m | Y U ( m ′ ) � � P ≤ t − 1 Po ( λ U )([ m , + ∞ )) Motif detection procedure t + 1 ≤ t − 1 Po ( λ U )( m ) Application to Yeast Conclusion which implies � N U ( m ) − λ U t + 1 e − ((1+ t ) ln(1+ t ) − � ≤ P ( Y U ( m ′ ) = 1) √ P > t λ U 2 π ( t − 1)

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend