A Semi-tensor Product Approach for Probabilistic Boolean Networks

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1 A Semi-tesor Product Approach for Probabilistic Boolea Networks Xiaoqig Cheg, Yusha Qiu, Wepi Hou ad Wai-Ki Chig Advaced Modelig ad Applied Computig Laboratory Departmet of Mathematics The Uiversity of Hog Kog, Hog Kog s: Abstract Modelig geetic regulatory etworks is a importat issue i systems biology Various models ad mathematical formalisms have bee proposed i the literature to solve the capture problem The mai purpose i this paper is to show that the trasitio matrix geerated uder semi-tesor product approach (Here we call it the probability structure matrix for simplicity) ad the traditioal approach (Trasitio probability matrix) are similar to each other Ad we shall discuss three importat problems i Probabilistic Boolea Networks (PBNs): the dyamic of a PBN, the steady-state probability distributio ad the iverse problem Numerical examples are give to show the validity of our theory We shall give a brief itroductio to semi-tesor ad its applicatio After that we shall focus o the mai results: to show the similarity of these two matrices Sice the semi-tesor approach gives a ew way for iterpretig a BN ad therefore a PBN, we expect that advaced algorithms ca be developed if oe ca describe the PBN through semi-tesor product approach Keywords: Boolea Networks (BNs), Semi-tesor Product Approach, Iverse Problem, Probabilistic Boolea Networks (PBNs), Similar Matrices, Steady-state Distributio I INTRODUCTION Modelig geetic regulatory etworks is oe of the importat topics i systems biology [7], [11] A umber of models ad mathematical formalisms have bee proposed to explai the geetic itersectios, icludig liear models [19], Bayesia etworks [16] ad its extesios, differetial equatios model [14], Boolea Networks (BNs) ad its extesio Probabilistic Boolea Networks (PBNs) [17], [18] BN ad PBN models are some of the most attractive models BN was first itroduced by Kauffma [1], [13] I a BN, the expressio states of the gee are categorized ito levels, either o (1) or off (0) The dyamics of a BN ca be viewed as a process that each gee is govered by a fuctio (called Boolea fuctio) BN is called a determiistic model sice the target gee oly depeds o the iitial state ad the set of Boolea fuctios Ad a BN will evetually eter ito a attractor cycle, whose legth could be either 1 (sigleto attractor) or more tha 1 (periodic attractor) Fidig the attractor cycles ad their features are importat topics for a BN The attractor cycles i a BN may reveal some cacer cells or abormality i a cell Thus fidig the attractor cycles ad their features are of importat topics i BN Other research problems ad developmets related to BNs ca be foud i [1], [], [10], [1], [13] Shmulevich [17] poited out that the holistic behavior of the etwork should be studied because it is believed that gees are ot idepedet of each other Based o a couple of reasos (eg the limitatio that BN is a determiistic model, BN may oly reveal part of the iformatio while geeratig to the ext state, the desire for a ope system ad so o), a stochastic versio of BNs, amely, Probabilistic Boolea Networks (PBNs) was proposed [17], [18] It is based o the appealig rule-based property of BN, but it also icorporates with stochastic features PBN ows a couple of advatages over a BN, for example, it ca cope with the ucertaity i the data ad the Boolea fuctios due to its stochastic ature The proportio of steady-state probabilistic distributio provides a holistic picture of the etwork It ca also reveal whether the gees are iteractig with each other, ad how they iteract Cheg et al [3], [4], [5] proposed a algebraic approach called the semi-tesor product approach Ad they successfully applied their theory to BN problems ad BN cotrol problems, see for istace [6] Based o Cheg s works, Yag ad Li also applied semi-tesor approach to PBN cotrol problems [][3], however, they did ot discuss much about the theoretical support of applyig semi-tesor product to PBN cotrol problems I the semi-tesor product approach theory, a mappig is defied from the gee expressio state to the colum of idetity matrix I, where true equals to the first colum ad false equals to the secod colum Therefore there is o logical fuctios ad logical expressios i each iteratio step The, they defie a kid of operatio called semi-tesor product, which is based o Kroecker product ad primitive product of matrices The semi-tesor product shares all the appealig properties with the primitive matrix product This ca be easily show uder its defiitio Hece BN ca be trasformed ito a algebraic form by multiplyig all the BN equatios together The most saliet limitatio of the semitesor approach is it will take much effort i trasformig a BN ito that form But the flaws do ot detract from the jade s essetial beauty Semi-tesor approach is a powerful mathematical method ad it also provides a ew way for dealig with geetic regulatory etworks The mai cotributio of this paper is that we proved the probability trasitio matrix ad probability structure matrix are similar matrices Thus, semi-tesor product theory is 014 The 8th Iteratioal Coferece o Systems Biology (ISB) /14/$ IEEE 85 Qigdao, Chia, October 4 7, 014

2 applicable to PBN problems For a give PBN, the trasitio probability matrix geerated from the two ways (the traditioal oe ad usig the semi-tesor techique) are differet Actually, the trasitio matrix of a BN geerated by the semitesor product approach is called a structure matrix [5] So we call the probability trasitio matrix of a PBN geerated from the semi-tesor product approach probability structure matrix Here we try to fid the relatioship betwee them We ca show that they are similar matrices, which is oe of the mai results i this paper Based o the similarity property of these two matrices, we discuss three importat problems i studyig a PBN: (i) the dyamics of a PBN, (ii) the steadystate probability distributio ad (iii) the iverse problem of costructig a PBN The remaider of this paper is structured as follows Sectio gives a review o some importat cocepts of BNs ad PBNs Sectio 3 presets about the mai results o semi-tesor product for PBNs, ad we show that the two matrices are similar We propose three importat problems i studyig a PBN i Sectio 4 ad discuss their relatioships i these two approaches The fial sectio cocludes the paper II PRELIMINARIES A Boolea Networks ad Probabilistic Boolea Networks 1) Boolea Networks (BNs): BN G(V, F ) is a special case of a sequetial dyamic system [15], cosistig of a set of biary odes V (also called Boolea variables) such that each of which has a Boolea fuctio assiged to it Suppose there are gees i the BN, F is the set of the Boolea fuctios where F = {f 1, f,, f, f i : {0, 1 {0, 1 Ad V is the set of all the vertices, V = {v 1, v,, v The value of v i represets the state of gee i, either 0 (o) or 1 (off) The dyamics of the BN ca be expressed as v i (t + 1) = f i (v 1 (t), v (t),, v (t)) = f i (v(t)) Here v(t) is called Gee Activity Profile (GAP) Sice we kow that v i {0, 1, the value of v(t) ca be take from S = {00 {{ 0, 00 {{ 1,, 11 {{ 1 The size of set S is ) Probabilistic Boolea Networks (PBNs): BN is a determiistic model, the oly radomess comes from its iitial state However, i a biological system, oise ad radomess are usually uavoidable, ad there always exists oise i experimetal data, so a stochastic model is more appropriate The cocept ad idea of a Probabilistic Boolea Network (PBN) are itroduced i order to capture the stochastic ature of the biological system A PBN is a ope model where the data ad the Boolea fuctios ca be chaged i differet cases The PBN model shares similar rules with a BN except that more tha a BN fuctio is assiged to each gee Suppose l i Boolea fuctios are assiged to gee v i, deoted by fi 1, f i,, f li i Ad the probability of choosig the jth Boolea fuctio is c j i This implies that l i j=1 c j i = 1, 0 < cj i < 1, for i = 1,,, If we choose the j i th Boolea fuctio for gee v i, the the BN ca be expressed as BN j1 j j, where j i {1,,, l i It ca be see that there are totally N = l i BNs Ad we assume that it is idepedet to choose the Boolea fuctio for each gee, so we have the probability of choosig BN j1 j j give by P {f 1 = f j1 1, f = f j,, f = f j = c ji i = q j1 j j We use A j1 j,j to deote the trasitio probability matrix for BN j1j j Ad we use the compact form of j 1 j j, the the BNs ca be deoted by BN 1, BN,, BN N The probability of choosig the jth BN is q j ad the trasitio probability matrix for the jth BN is A j, j {1,,, N The the probability trasitio matrix for the PBN is give by A = N j=1 q ja j Sice = P {V(t + 1) = a V(t) = b N j=1 {V(t + 1) = a V(t) = b the jth etwork is choseq j Example 1: This is a example of a PBN ad its BNs The truth table of the PBN is give by State f1 1 f 1 f1 f c i j Based o the truth table, we have four BNs ad they are listed as follows: A 1 = A = A 3 = A 4 = Ad we have q 1 = c 1 1c 1 = 004, q = c 1 1c = 036, q 3 = c 1 c 1 = 006, q 4 = c 1 c = 054, so the trasitio probability matrix A is give by A = 4 q i A i = The 8th Iteratioal Coferece o Systems Biology (ISB) /14/$ IEEE 86 Qigdao, Chia, October 4 7, 014

3 B Semi-tesor product The semi-tesor product is defied i the followig Defiitio 1: (Cheg et al [5]) Give a m matrix A ad a p q matrix B, the semi-tesor product of A ad B is give by A B = (A I l/ )(B I l/p ) where l is the least commo multiple of ad p Ad for ay matrix M ad N m 11 N, m 1 N,, m 1s N m 1 N, m N,, m s N M N = m t1 N, m t N,, m ts N The size of matrix M is t s By deletig the largest commo factor of ad p, the size of the semi-tesor product matrix ca be determied For example, if the size of A is m ax, the size of B is ay q, ad a is the largest commo factor of ax ad ay, the the size of A B is my xq The proof ca be easily derived from the defiitio The followig two defiitios are usually adopted i semi-tesor product theory Defiitio : (Cheg et al [5]) δ j was defied as the jth colum of matrix I This is the defiitio of logical matrix Defiitio 3: (Cheg et al [5]) A matrix L M m is called a logical matrix if each colum of L is i the form of δ, j j {1,,, It is obvious that the trasitio probability matrix of ay BN is a logical matrix Cheg et al [5] proposed a mappig from {0, 1 to set {δ, 1 δ i the semi-tesor approach, the they ca forbid usig logical expressios i the comig steps The mappig is defied as follows: [ ] [ ] 1 0 T 1 ad F After givig the most importat cocepts i semi-tesor theory, we have oe of the most importat theorems i semi-tesor product theory Theorem 1: (Cheg et al [5]) Give a logical fuctio f(p 1, p,, p r ), there exist a uique r matrix M f, such that f(p 1, p,, p r ) = M f p 1 p p r Moreover, M f is a logical fuctio The followig example shows us how to express the logical fuctio ito the algebraic form Before the example, some of the most importat matrices are give as follow: Structure matrix logical fuctio algebraic form M c = δ [1,,, ] f(p, q) = p q f(p, q) = M c pq M d = δ [1, 1, 1, ] f(p, q) = p q f(p, q) = M d pq M = δ [, 1] f(p) = p f(p) = M p M r = δ 4 [1, 4] f(p) = p f(p) = M r p W [] = δ 4 [1, 3,, 4] f(p, q) = qp f(p, q) = W [] pq Example : Cosider the logical fuctio f(p, q, r) = (p q) (r p) We shall rewrite it i the algebraic form ad compute the structure matrix M f Here f ca be expressed as follow: f(p, q, r) = (p q) (r p) = M d M c pm qm c rp = M d M c (I M )pqm c rp = M d M c (I M )(I 4 M c )pqrp = M d M c (I M )(I 4 M c )pw [,4] pqr = M d M c (I M )(I 4 M c )(I W [,4] )p qr = M d M c (I M )(I 4 M c )(I W [,4] )M r pqr := M f pqr The, we have M f = M d M c (I M )(I 4 M c )(I W [,4] )M r = δ [1 1 1 ] III METHODS I this sectio, we shall itroduce a importat matrix T, which is defied as follows: T = δ [, 1,, 1] Propositio 1: We have T = I Proof: For ay matrix A, we deote A j the jth colum of A If A = A T the A ca be obtaied from this way: A j = A j, where deotes the total colums i A Thus obviously we have T = I Give the defiitio of T, we ca give the mai theorem i this paper i the followig Assume there are gees i the PBN, we ca use T for T for simplicity Theorem : If we deote A the trasitio probability matrix of a PBN, ad A semi deote the probability structure matrix, the we have A semi = T AT, where T is defied above Sice we kow that T = I, we also have A = T A semi T This meas that A ad A semi are similar matrices Proof: First of all, we show that it is true for ay BN Suppose we have N A = q i A i q i deote the probability choosig the ith BN Ad if for ay i, A i semi = T A i T holds, the A = T A semi T is satisfied We eed to prove that for each BN, A i semi = T Ai T holds We ca ifer from the defiitio of A i ad A i semi that their size are the same, amely, We deote the BN state at time t by V(t), ad the BN state at time t obtaied from the semi-tesor approach by V semi (t) I order to show the equality, we have to fid the relatioship betwee V(t) ad V semi (t) I the semi-tesor product, there is a mappig from {0, 1 to {δ, 1 δ, where 0 δ, 1 δ 1 Ad we kow that if there are gees, the there are totally gee states I the costructio of A i, a ij deote whether the BN state j 014 The 8th Iteratioal Coferece o Systems Biology (ISB) /14/$ IEEE 87 Qigdao, Chia, October 4 7, 014

4 ca be trasferred to state i, where state 1 to state equals to 00 {{ 0, 00 {{ 1,, 11 {{ 1, respectively However, i the semi-tesor product, the dyamic of a BN ca be expressed as V semi (t + 1) = A semi V semi (t) Similar to A, A semi is also a logical matrix Thus A semi δ j equals to the jth colum of A semi I the semi-tesor product, state 1 to state equal to 11 {{ 1, 11 {{ 0,, 00 {{ 0 Hece A i semi = T Ai T is proved The followig example shows how our mai theorem works Example 3: Cosider the same PBN i Example 1, we try to solve the structure matrix ad apply the above theorem to it As for the semi-tesor approach, we eed to solve out the logical equatio for each BN first, it is easy to fid out the logical equatio for each BN, which is give by, f 1 f BN 1 q (p q) BN q (p q) BN 3 p (p q) BN 4 p (p q) Here p ad q are logical variables Next step we eed to figure out A i semi, i {1,, 3, 4 Take BN1 as a example, we have f(p, q) = q M (M d pq) = (I M )(I M d )qpq = (I M )(I M d )W [] pq = (I M )(I M d )W [] (I M r )pq The structure matrix equals to δ 4 [, 4,, 3] Ad T = δ 4 [4, 3,, 1], we ca easily cofirm that A 1 semi = T A 1 T, similarly, we ca prove A = T A semi T A Dyamics of a PBN IV THE THREE PROBLEMS Suppose there are gees ad l i, i {1,,, Boolea fuctios are assiged for gee v i So we have total N = l i Boolea etwork, ad q i is the probability of choosig the ith BN The dyamic of the PBN ca be expressed as f1 1 (x 1 (t), x (t),, x (t)) with probabilityp 1 1 f1 (x 1 (t), x (t),, x (t)) with probabilityp 1 x 1 (t) = f l 1 1 (x 1(t), x (t),, x (t)) with probabilityp l 1 1 f 1 (x 1 (t, x (t),, x (t))) with probabilityp 1 f (x 1 (t), x (t),, x (t)) with probabilityp x (t) = f l (x 1(t), x (t),, x (t)) with probabilityp l f(x 1 1 (t, x (t),, x (t))) with probabilityp 1 f1 (x 1 (t), x (t),, x (t)) with probabilityp x (t) = f l (x 1 (t), x (t),, x (t)) with probabilityp l Here x i (t) deotes the ith gee state at time t ad f j i is the jth Boolea fuctio for gee i as we state i the itroductio part, we have l i j=1 p j i = 1 We kow that the dyamic of the PBN ca be expressed as V(t + 1) = AV(t), ad accordig to Theorem, we have V(t + 1) = T A semi T V(t), it is obvious that T V(t) = V semi (t) (I a BN, V(t) is i the form of δ j, where is the total umber of gees ad j meas it is the jth states from 00 {{ 0 to 11 {{ 1 ) Lemma 1: The dyamic of PBN usig probability structure matrix ca be expressed as B Steady State Aalysis V semi (t + 1) = A semi V semi (t) It is kow that there is a limitatio of usig BN to describe the real biological system A PBN based o the fudametal idea of a BN ca better capture the ucertaity characteristic of the biological system Ad it has bee foud a PBN model is a stochastic process with the Markov property Statioary distributio is a importat factor i Markov Chai Semitesor approach provides a ew view for describig the PBN, therefore there may arise a lot of ways dealig with the steady state distributio problems based o the semi-tesor product approach I this subsectio, we shall fid the relatioship betwee the steady-state distributio with which is foud based o the semi-tesor approach A statioary distributio is defied as follows: Defiitio 4: For a time-homogeeous Markov chai, which meas that the Markov chai ca be described by a sigle, time-idepedet matrix P The the statioary distributio π = (π 1, π,, π ) exists if the solutio of the equatio P π = π subject to j=1 π j = 1 exists We remark that if the steady-state probability distributio of a PBN exists the it must be the statioary probability distributio but ot vice versa Thus if we deote π as the statioary distributio ad π semi as the statioary distributio regardig to probability structure matrix It is atural that we ca defie π semi as follows: Defiitio 5: Noted that the PBN with A semi as its probability trasitio matrix is a Markov chai, so we defie π semi = (πsemi 1, π semi,, π semi )T as the statioary distributio, which ca be give by π semi = A semi π semi subject to πi semi = 1 Here A semi is a matrix The we kow that π = Aπ, which meas π = T A semi T π, similarly, we ca obtai π = T π semi 014 The 8th Iteratioal Coferece o Systems Biology (ISB) /14/$ IEEE 88 Qigdao, Chia, October 4 7, 014

5 Example 4: The trasitio probability matrix ad probability structure matrix are give, respectively, by A = ad A semi = By solvig the equatios: Example 5: We costruct the trasitio probability matrix ad the probability structure matrix as follows: A = A semi = Ad we use the projectio based gradiet method to compute the iverse problem [0] Thus the problem becomes a least squares problem: we have Aπ = π ad A semi π semi = π semi mi q Uq p st q 1 = 1 ad mi q semi U semi q semi p semi st q semi 1 = 1 ad π = (00667, 03333, 03333, 0667) T π semi = (0667, 03333, 03333, 00667) T Ad it is obvious that C The Iverse Problem π = T π semi The iverse problem is to fid a appropriate PBN from a prescribed trasitio matrix [8], [9], [1] Suppose A is the give trasitio probability matrix Suppose there are l i ozero elemets i each colum, the we have l i feasible BNs at most, they are labeled as BN1, BN,, BN m, where m = Π l i Ad the trasitio matrix assiged to BN i is A i Thus the iverse problem ca be expressed i the form of fidig the appropriate set of q j miimize the followig fuctio: m f(q 1, q,, q m ) = A i q i A subject to m q i = 1 Similarly, we ca defie the iverse problem of costructig a PBN by semi-tesor product approach Defiitio 6: Give the probability structure matrix as A semi, we try to fid a appropriate q semi = (qsemi 1, q semi, ) ad Ai semi accordig to qi semi, such that i qi semi Ai semi = A semi subject to i qi semi = 1 Accordig to the previous defiitio of qsemi i, it ca be easily obtaied that qsemi i = q i, which meas that all the algorithms solvig the iverse problem ca be applied to the PBN regardig to a give probability structure matrix A semi Here q semi 1 meas the L 1 orm of q semi ad the elemets of q ad q semi are oegative Here p = F (A) ad U = [F (A 1 ), F (A ),, F (A N )], where for ay matrix B, F (B) = (b 11, b 1,, b 1, b 1, b,, b,, b 1, b,, b ) T b ij is the (i, j)th elemet i B ad the same defiitio holds for U semi ad p semi From the give A ad A semi, the value of U, U semi, p, p semi are listed as U = U semi = p = [015, 085, 0, 0, 03, 0, 07, 0, 0, 05, 05, 0, 03, 0, 0, 077] T p semi = [077, 0, 0, 03, 0, 05, 05, 0, 0, 07, 0, 03, 0, 0, 085, 015] T Usig the projectio-based gradiet algorithm, we have q = q semi = [01506, 0, 0, 0, 0, 0, 0, 0003, 00394, 00039, 00857, 00, 0674, 0039, 03776] T The graph of the distributios of q ad q semi are show i the followig two figures We ca ifer from the graphs that they are exactly same 014 The 8th Iteratioal Coferece o Systems Biology (ISB) /14/$ IEEE 89 Qigdao, Chia, October 4 7, 014

6 Fig 1 The Distributio of q Fig The Distributio of q semi V CONCLUSIONS This paper studies PBNs by usig semi-tesor approach We show the relatioship betwee the probability trasitio matrix ad probability structure matrix Various of algorithms have bee developed to solve the BN problems through semi-tesor approach This gives a broader fields of visio to dealig with the PBN problems As we have show i previous sectios, the PBN built from semi-tesor approach ad the origial oe are equivalet, by equivalet here we meas they ca be trasformed to each other uder all coditios ad they share may same properties The reaso is because A ad A semi are similar matrices Thus all the theories, algorithms for the origial PBN ca be used for the PBN geeralized from semi-tesor approach Ad the time complexity ad sample complexity for solvig the PBN geerated from semi-tesor approach is at least o worse tha that of the origial PBN The theories for BN geerated from semi-tesor approach ca be applied to the origial BN trasitio matrix For example, the theory about sigleto attractor It also provides evidece that the semi-tesor theory ad the origial theory about BN (or PBN) are equivalet ACKNOWLEDGMENT The authors would like to thak the referees ad the editor for their helpful commets ad suggestios This research work was supported by the Research Grats Coucil of Hog Kog, uder grat No ad HKU Hug Hig Yig Physical Research Grat ad Natioal Natural Sciece Foudatio of Chia Grat Nos ad S REFERENCES [1] R Albert ad A Barabasi, Dyamics of complex systems: scalig laws for the period of Boolea etworks, Phys Rev Lett, 000, 84, 5660 [] M Aldaa, Boolea dyamics with scale-free topology, Physica D, 003, 185 (1), [3] D Cheg ad H Qi, Cotrollability ad observability of Boolea cotrol etworks, Automatica, 009, 45 (7), [4] D Cheg ad H Qi, A liear represetatio of dyamics of Boolea etworks, IEEE Tras Auto Cotr, 010, 55 (10), [5] D Cheg, H Qi ad Z Li, Aalysis ad cotrol of Boolea etworks: a semi-tesor approach, Spriger, 011 [6] D Cheg, H Qi ad Z Li, Model costructio of Boolea etworks via observed data, IEEE Tras Neural Networks, 011, (4), [7] J Celis, M Kruhofferm, I Gromova, C Frederikse, M Ostergaard ad T Ortoft, Gee expressio profilig: Moitorig trascriptio ad traslatio products usig DNA microarrays ad proteomics, FEBS Letters, 000, 480(1), 16 [8] X Che, W Chig, XS Che, Y Cog ad N Tsig, Costructio of probabilistic Boolea etworks from a prescribed trasitio probability matrix: A maximum etropy rate approach, East Asia Joural of Applied Mathematics, 011, 1, [9] W Chig, X Che ad N Tsig, Geeratig probabilistic Boolea etworks from a prescribed trasitio probability matrix, IET Systems Biology, 009, [10] D Drossel, T Mihaljev, F Greil, Number ad legth of attractors i a critical Kauffma model with coectivity oe, Phys Rev Lett, 005, 94, [11] S Huag, Gee expressio profilig, geetic etworks, ad cellular states: a itegratig cocept for Tumorigeesis ad drug discovery, Joural of Molecular Medicie, 1999, 77, [1] The origis of order: self-orgaizatio ad selectio i evolutio, Oxford Uiversity Press, New York, 1993 [13] Metabolic stability ad epigeesis i radomly costructed geetic ets, J Theor Biol, 1969, (3) [14] T Mestl, E Plahte ad S Omholt, A mathematical frame work for describig ad aalysig gee regulatory etworks, J Theor Biol, 1995, 176 (), [15] H Mortveit ad C Reidays, A itroductio to sequetial dyamic systems, Spriger, 008 [16] K Murphy ad S Mia, Modellig gee expressio data usig dyamic Bayesia etworks, available at murphyk/papers/ismb99pdf, 1999 [17] I Shmulevich, E Dougherty, S Kim ad W Zhag, Probabilistic Boolea etworks: a rule-based ucertaity model for gee regulatory etwork, Bioiformatics, 00, 18 () [18] I Shmulevich ad E Dougherty, Probabilistic Boolea etworks: The modelig ad cotrol of gee regulatory etworks, SIAM Press, 009 [19] E va Somere, L Wessels ad M Reider, Liear modelig of geetic etworks from experimetal data, ISMB, Sa Digeo, CA, August 19-3, 000 [0] Y We, Z Zhag, X Cheg, W Chig ad V Vassiliadis, Sparse solutio of o-egative Least Squares problems with projectio with applicatio i the costructio of probabilistic Boolea etworks, submitted, 014 [1] S Zhag, W Chig, X Che ad N Tsig, Geeratig probabilistic Boolea etworks from a prescribed statioary distributio, Iformatio Scieces, 010, 180, [] M Yag, R Li, T Chu, Optimal cotrol of steady-state probability distributios of probabilistic Boolea etwork, Proceedigs of the 3d Chiese Cotrol Coferece, July 6-8, Xi a, Chia [3] F Li, J Su Staility ad stabilizatio issue of probabilistic Boolea etwork Proceedigs of the 30th Chiese Cotrol Coferece, July -4, 011, Yatai, Chia 014 The 8th Iteratioal Coferece o Systems Biology (ISB) /14/$ IEEE 90 Qigdao, Chia, October 4 7, 014