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)

Recommend


More recommend