Miimum Domiatig Set Approach to Aalysis ad Cotrol of Biological Networks Tatsuya Akutsu Bioiformatics Ceter Istitute for Chemical Research, Kyoto Uiversity Joit work with Jose Nacher i Toho Uiversity
Motivatio: Cotrol Theory for Biological Systems Oe of the mai targets of Systems Biology Though cotrol theory is well established for liear systems, biological systems have o-liear compoets ad are very complex (large-scale) May lead to ew drugs ad treatmet methods Practical cotrol methods exist, but o useful theory Itroductio of 4 gees turs ormal cells ito iduced pluripotet stem cells (ips cells) Abormal がん細胞 Cell Cotrol 制御 Normal 正常細胞 Cell
Cotets Scale-free Networks Cotrollability i Scale-free Networks Miimum Domiatig Set (MDS) Relatio to Structural Cotrollability Theoretical Aalysis of MDS Size Computer Simulatio Database Aalysis Applicatios to Aalysis of Biological Networks Extesios Coclusio
Scale-free Networks
Scale-Free Network [Barabasi & Albert, 999] Degree of a ode The umber of adjacet odes P(k) Degree distributio Frequecy of odes with degree k Scale-free etwork P(k) follows power law Differet from radom etworks degree=2 degree=5 P ( k) degree =3 k
Metabolic Network, Graph ad Degree A B C D E F G H I J Degree Node with degree : J Nodes with degree 2: B, C, D, F, G, H Nodes with degree 3: A,E, I P(k) (degree distributio): P()=., P(2)=.6, P(3)=.3, P(4)=P(5)=P(6)= =
Scale-Free Distributio P ( k) k Power laws are scale free because if k is rescaled (multiplied by a costat), the P(k) is still proportioal to k May real etworks (e.g., geetic etworks, metabolic etworks, protei-protei iteractio etworks) are reported to have the scale-free property
Poisso Distributio ad Power-Law Distributio Poisso distributio (radom graph) Power-law distributio (scale-free graph) e -λ λ k /k! k - P (k) log P (k) k log(k)
Cotrollability i Scale-free Networks [Liu, Slotie, Barabasi: Nature, 2]
Cotrollability of Liear Systems () Iput: Liear System: Iitial state: x Fial state: x F Output: u(t) (fuctio of t) which drives the system from x to x F i fiite time ) ( ) ( ) ( t B t A dt t d u x x + = + = M N N u u u B x x x A x x x 2 2 2 x(t): N-dim. real vector (iteral odes) u(t): M-dim. real vector (cotrol odes) A: N N real matrix B: N M real matrx
Cotrollability of Liear Systems (2) Fact. System is cotrollable iff N NM matrix C=(B,AB,A 2 B,,A N- B) has full rak (i.e., rak(c)=n). = = 22 4 34 3 2 b b B a a a a A = 4 4 34 3 2 22 b a b a a b a b a b b C rak(c)=4 for most parameters structural cotrollability
Structural Cotrollability G B (V L,V R ;E B ) : bipartite graph costructed from G(V,E) by B R j L i j i i R i R i L i L E x x E x x V x x V V x x V = = ), ( ), ( }, { }, { Thm. [Liu et al. 2] The miimum umber of odes eeded to fully cotrol the system is max {N-M *,}, where M * is the size of the maximum matchig of G B.
Cotrollability of Scale-free Networks The umber of eeded driver odes [Liu et al. 2] Radom etworks: N D Scale-free etworks e < k >/ 2 [ ( ) < > ] exp k N D 2 if <2, may odes must be cotrolled <k>: average degree : umber of odes i a etwork
Miimum Domiatig Set ad Its Relatio to Structural Cotrollability
Miimum Domiatig Set () V D is a domiatig set of udirected graph G(V,E) ( v V-V D )( u V D )({u,v} E) Miimum domiatig set: domiatig set with the smallest umber of odes
Miimum Domiatig Set (2) Well-kow cocept i graph theory ad computer sciece NP-hard, but ca be solved exactly by usig Iteger Liear Programmig (ILP) to some extet Has bee applied to desig/cotrol of mobile ad-hoc etworks (MANET) trasportatio routig computer commuicatio etworks
Relatio betwee MDS ad Cotrollability MDS Thm. Suppose that every edge i a etwork is bi-directioal ad every ode i MDS ca cotrol all of its outgoig liks separately. The, the etwork is structurally cotrollable by selectig the odes i MDS as the driver odes. [Nacher & Akutsu: New J. Phys. 22]
ILP-based Method for MDS Very simple, but works for etworks with a few thousads of odes i may cases {,},,,.. mi } }, { { = = = i E v v i j j j i i x i x s t x j i x i = x i i MDS
Theoretical Aalysis of MDS Size
Estimatio of MDS Size i Scale-free Networks >2 Upper boud: trivially O() Lower boud: Ω() <2 Upper boud: O( -(2-)(-) ) takig the miimum order O(.75 ) whe =.5 Based o a kid of mea-field approximatio [Nacher & Akutsu: J. Phys.: Cof. Ser. 23]
Lower Boud for >2 () Assumig αk -, we have The followig is well kow, where C(S) is the set of edges betwee S ad V-S if S + C(S) <, S is ot a domiatig set If we select all odes with degree > K, we have ) ( = = + α α α dk k 2 2 2 2 2 ) ( ) ( + < = < α K K dk k dk k k S C K K
Lower Boud for >2 (2) Sice we ca assume S </2, we should have The, we estimate a lower boud of S by This meas that the umber icreases as icreases 2 / 2 2 K > K K dk k S K > 2 2 2 α
Upper Boud for <2 () We select all odes with degree greater tha K= β as DS The, N DS =#odes i DS (domiatig set) is give by O the other had, the total umber of edges E G is E DS (=the umber of edges covered by DS) is The, prob. that a arbitrary edge is NOT covered by DS is ) ( ) ( ) ( ) ( ) ( = = = β β β α O dk k N DS ) ( 2 2 = = α dk k k E G ) ( ) (2 2 2 β β α = = dk k k E DS
Upper Boud for <2 (2) Sice a ode is covered by DS if at least oe edge coectig to the ode is covered by DS, the expected umber (N G-DS ) of odes ot covered by DS is N G β )(2 ) DS O( ) = Here, we balace N G-DS with N DS by lettig β ( ) = + ( β )(2 ) which results i β=2-. Therefore, a upper boud of the size of DS is estimated as which is o() for <<2 O( ( + ( β )(2 ) O( (2 )( ) It is iterestig that it takes the miimum (O(.75 )) whe =.5 ) )
Computer Simulatio
MDS size vs. Scalig Expoet () MDS size decays as decays (especially aroud =2)
Database Aalysis
Data
Why Not Cotradictig [Liu et al.]? Liu et al. assumed oly driver ode values ca be directly cotrolled through exteral sigals. Coversely, MDS approach assumed each driver ode ca cotrol its liks idividually. a ode with degree k is regarded as k driver odes.
Applicatios to Aalysis of Biological Networks
MDS for Aalyzig Biological Networks Applyig cotrol to real cells is far from easy However, MDS may be useful to fid importat proteis, gees, ad other molecules Aalysis of PPI etworks [Milekovic et al. PLoS Oe, 2] (before our work) [Wuchty, PNAS, 24] [Khuri & Wuchty, BMC Bioiformatics, 25] [Wag et al., BIBM 24] Aalysis of metabolic cacer etworks [Asgari et al., PLoS ONE, 23]
Applicatio to Aalysis of PPI Networks Wuchty foud that MDS is useful to fid importat proteis [Wuchty, PNAS 24] Proteis i MDS are eriched with essetial, cacer-related, ad virus-targeted gees. These proteis are highly ivolved i regulatory fuctios, showig high erichmet i trascriptio factors ad protei kiases, ad participate i regulatory liks, phosphorylatio evets, ad geetic iteractios. [Wuchty, PNAS 24]
Extesios
Cotrol of Bipartite Networks May real etworks have bipartite structure (left/right odes) Drug-target, researcher-paper, gee-disease Oly left odes ca be driver odes MDS approach eeds much smaller umber of driver odes [Nacher & Akutsu: Sci. Rep. 23]
Results o Bipartite Networks New feature: Itroductio of degree cutoff (P(k)= for k > H) For <2, the umber of driver odes is ) )( (2 2 H m O
Critical/Redudat Nodes i MDS We applied the cocepts of critical/redudat odes [Jia et al.: Nat. Comm. 23] to MDS because MDS is ot ecessarily uiquely determied Critical ode: appears i every MDS Redudat ode: ever appears i ay MDS Critical odes are expected to be more importat tha MDS [Nacher & Akutsu: J. Comp. Net. 25]
Robust MDS Robust MDS (RMDS): each ode is domiated by at least C odes (C= MDS) Robust agaist deletio of arbitrarily C- edges Upper boud of RMDS size (for <2): (D: miimum degree) RMDS size correspods to MDS size with miimum degree D-C+ ( DC+ )(2 )( ) O ( DC+ )(2 ) + [Nacher & Akutsu: PRE, 25]
Related Work by Molar et al. Aalysis of MDS size with degree cutoff [Sci. Rep. 23] Aalysis of MDS size with degree correlatio [Sci. Rep. 24] Damage-resiliet domiatig sets agaist radom ad targeted attacks [Sci. Rep. 25]
Coclusio
Coclusio Establishmet of a coectio betwee MDS ad structural cotrollability MDS size is small (o()) if <2 Heterogeeous etworks are ot difficult to cotrol This tedecy was verified (to some extet) by computer simulatio ad database aalysis Several extesios Bipartite etworks, Critical/Redudat odes, Robust MDS MDS is useful for idetifyig importat proteis i PPI etworks
Future Work Developmet of a framework/theory which makes cotrol of biological systems easy More rigorous theoretical aalysis o MDS size (our aalyses are based o a kid of mea-field approximatio) More biological applicatios Thak you!