Canalyzation in mathematical modeling

1. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Canalyzation in mathematical modeling Elena Dimitrova Clemson University KnowEscape 2014 Elena Dimitrova Canalyzation in mathematical modeling

2. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean networks Canalyzing functions De

3. nitions Examples and applications Nested canalyzing functions De

4. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Partially NCFs De

5. nitions Depth and stability Derrida plots Conclusions Elena Dimitrova Canalyzation in mathematical modeling

6. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Networks Elena Dimitrova Canalyzation in mathematical modeling

7. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs What is a Boolean network? Elena Dimitrova Canalyzation in mathematical modeling

8. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs What is a Boolean function? I Boolean function (or switching function) is a function of the form f : Bn ! B, where B = f0; 1g, n is nonnegative integer. I Boolean functions are often written using the logical operators AND (^), OR (_), and NOT (:). Elena Dimitrova Canalyzation in mathematical modeling

9. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean function: Example f (x; y) = x ^ :y = x(y + 1) x y f 0 0 0 0 1 0 1 0 1 1 1 0 Facts I Every Boolean function can be represented as a polynomial over F2. I Every function f : Fn2 ! F2 can be represented as a Boolean function. Elena Dimitrova Canalyzation in mathematical modeling

10. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs What is a Boolean network? I F = (f1; : : : ; fn) : Bn ! Bn is a Boolean network (BN) on n variables if fi is a Boolean function in n variables for every i = 1; : : : ; n. I BNs are used for modeling networks in which the node activity/state can be described as a binary value: on-o, active-non active, 1-0, etc. Elena Dimitrova Canalyzation in mathematical modeling

11. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean network: Example f1(x1; x2) = x1 ^ :x2 f2(x1; x2) = x1 _ x2 Elena Dimitrova Canalyzation in mathematical modeling

12. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Applicatioins of Boolean networks Boolean network models have been used to examine the connections among diverse physical and engineered networks I Signal transduction networks [Kauman, Shmulevich et al., Helikar et al., Kochi et al.] I Biological networks [Klemm et al., Raeymaekers] I Social networks [Flache et al., Green et al., Moreira et al.] I Economic/prediction market networks [Jumadinova et al.] I Neural networks [Huepe et al.] I Complex networks in general [Wolfram] Elena Dimitrova Canalyzation in mathematical modeling

13. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean networks: Biology Segment polarity network in D. melanogaster R. Albert H. Othmer: J. Theor. Bio. 223 (2003), 1{18. Elena Dimitrova Canalyzation in mathematical modeling

14. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean model of segment polarity network R. Albert H. Othmer: J. Theor. Bio. 223 (2003), 1{18. Elena Dimitrova Canalyzation in mathematical modeling

15. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean networks: Electric circuits Elena Dimitrova Canalyzation in mathematical modeling

16. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Boolean networks: Sociology I Databases like Citebase harvest pre- and post-prints from various archives for dierent

17. elds. Nodes stand for published articles and a directed edge represents a reference to a previously published article. I The International Movie Database is analyzed in many studies. An actor collaboration network can be reconstructed, in which actors casted jointly are assumed to know each other. I The phenomenon of rhythms of social interactions was studied using Boolean networks. A clear weekend pattern was discovered while studying messaging within a massive online friendship site providing insight to the social life of the users. (www.hpl.hp.com=research=idlpapers=facebook) Elena Dimitrova Canalyzation in mathematical modeling

18. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Variations on Boolean networks I (Random/Probabilistic) Boolean networks (S. Kauman) I Polynomial dynamical systems (T. Thomas, D. Thiery, R. Laubenbacher, B. Stigler, B. Pareigis, A. Jarrah) I (Nested) canalyzing functions (S. Kauman, I. Shmulevich) I Cellular automata (von Neumann, J. Conway, S. Wolfram) I Graphical models (Barret, Mortveit, Reidys) I Logical models (R. Thomas) I Petri nets (Google `Petri Nets World') I Biologically meaningful functions (L. Raeymaekers) I ... Elena Dimitrova Canalyzation in mathematical modeling

19. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs Question Are all Boolean functions good for modeling?? Elena Dimitrova Canalyzation in mathematical modeling

20. Boolean networks Canalyzing functions Nested canalyzing functions Partially NCFs De

21. nitions Examples and applications Canalyzing functions Canalyzing function A Boolean function f (x1; : : : ; xn) is canalyzing if there exists an index i and a; b 2 f0; 1g such that f (x1; : : : ; xi1; a; xi+1; : : : ; xn) = b: That is, when xi is given the input value a, f evaluates to b regardless of the input values of the other variables. Here a is called the canalyzing value and b is called the canalyzed value. Elena Dimitrova Canalyzation in mathematical modeling

23. nitions Examples and applications Examples: Canalyzing function I The AND function x ^ y is canalyzing in x. Since 0 ^ y = 0 for any input value of y, 0 is a canalyzing value for x with canalyzed output value 0. I XOR(x; y) := (x _ y) ^ :(x ^ y) is not canalyzing in either variable. Elena Dimitrova Canalyzation in mathematical modeling

25. nitions Examples and applications Examples: Canalyzing function I The AND function x ^ y is canalyzing in x. Since 0 ^ y = 0 for any input value of y, 0 is a canalyzing value for x with canalyzed output value 0. I XOR(x; y) := (x _ y) ^ :(x ^ y) is not canalyzing in either variable. Elena Dimitrova Canalyzation in mathematical modeling

27. nitions Examples and applications Canalyzing functions: role and applications I Introduced by Kauman [Kau93] as appropriate rules in Boolean network models of gene regulatory networks. The de

28. nition is reminiscent of the concept of canalisation introduced by the geneticist C.H. Waddington [Wad42] to represent the ability of a genotype to produce the same phenotype regardless of environmental variability. I Play an important role in the study of random Boolean networks [Kau93, Lyn95, Sta87, Ste99]. I Have been used extensively as models for dynamical systems, such as gene regulatory networks [Kau93], evolution [Ste99], and chaos [Lyn95]. Elena Dimitrova Canalyzation in mathematical modeling

36. nitions Examples and applications But what if the input is NOT the canalyzing value?? Elena Dimitrova Canalyzation in mathematical modeling

38. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Nested Canalyzing Functions Let f (x1; : : : ; xn) be a Boolean function. For 2 Sn, f is a nested canalyzing function (NCF) in the variable order x(1); : : : ; x(n) with canalyzing values a1; : : : ; an and canalyzed values b1; : : : ; bn if it can be expressed in the form f = 8 : b1 if x(1) = a1 b2 if x(1)6= a1 and x(2) = a2 b3 if x(1)6= a1 and x(2)6= a2 and x(3) = a3 ... ... bn if x(1)6= a1 and and x(n1)6= an1 and x(n) = an :bn if x(1)6= a1 and and x(n)6= an Elena Dimitrova Canalyzation in mathematical modeling

40. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Examples: NCFs I f (x1; x2; x3) = x2 _ (:x1 ^ x3) is a NCF with b = [1; 0; 0], a = [1; 1; 0], and = [2; 1; 3]: f (x1; x2; x3) = 8 : 1 if x2 = 1 0 if x2 = 0 and x1 = 1 0 if x2 = 0, x1 = 0, and x3 = 0 1 if x2 = 0, x1 = 0, and x3 = 1: I f (x; y; z) = x(y 1)z is a NCF. I g(x; y; z;w) = x(y + z) is not a NCF. Elena Dimitrova Canalyzation in mathematical modeling

46. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Properties of NCFs I Introduced by Kauman et al. in [KPST03]. I [KPST04] showed that networks made from NCFs have stable dynamic behavior and might be a good class of functions to express regulatory relationships in biochemical networks. I Nested (hierarchically) canalyzing functions show ordered behavior [NFW07]. I Systems of NCFs have a smaller average cycle length and average height (number of time steps it takes to converge to an attractor) of the state space graph compare to general Boolean networks. Elena Dimitrova Canalyzation in mathematical modeling

54. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Representation of an NCF [JRL07] Let yi = x(i) + ai + bi , 1 i n and let g(x1; : : : ; xn) = y11(y22(: : : (yn1n1yn) : : :); where i = _ if bi = 1 ^ if bi = 0 and ai ; bi 2 f0; 1g for 1 i n. Then g is an NCF in the variable order x(1); : : : ; x(n) with canalyzing values a1; : : : ; an and canalyzed values b1; : : : ; bn. Further, any NCF can be represented in this form. Elena Dimitrova Canalyzation in mathematical modeling

56. nition Properties of NCFs Structure NCFs and unate cascade functions Other results NCFs as a toric variety I The ring of Boolean functions is isomorphic to the ring R = F2[x1; : : : ; xn]=hx2 i xi : 1 i ni. I Any polynomial in R can be represented as an point in F2n 2 corresponding to its coecients. I For 2 Sn, the set of points corresponding to NCFs in the variable order x(1); : : : x(n) form an algebraic variety VNCF . I VNCF = [2SnVNCF is an algebraic variety as well, the variety of nested canalyzing functions. I VNCF is a toric variety [JL07]. Elena Dimitrova Canalyzation in mathematical modeling

58. nition Properties of NCFs Structure NCFs and unate cascade functions Other results NCFs as a toric variety I The ring of Boolean functions is isomorphic to the ring R = F2[x1; : : : ; xn]=hx2 i xi : 1 i ni. I Any polynomial in R can be represented as an point in F2n 2 corresponding to its coecients. I For 2 Sn, the set of points corresponding to NCFs in the variable order x(1); : : : x(n) form an algebraic variety VNCF . I VNCF = [2SnVNCF is an algebraic variety as well, the variety of nested canalyzing functions. I VNCF is a toric variety [JL07]. Elena Dimitrova Canalyzation in mathematical modeling

60. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Binary decision diagrams I A binary decision diagram (BDD) is a directed, acyclic graph that corresponds to the evaluation of a Boolean function. I The average path length of a BDD is the length of the paths from the source node to a terminal node averaged over all possible inputs. Here, 2+2+3+3+3+3 8 = 2. Elena Dimitrova Canalyzation in mathematical modeling

62. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Unate cascade functions I The unate cascade functions (UCFs) make up the unique class of Boolean functions with the smallest average path length on a BDD [Mat05]. I This smallest average path length is 2 1 2n1 where n is the number of variables. I Thus UCFs can be evaluated more quickly than any other Boolean function. I The classes of UCFs and NCFs are identical [JRL07]. Elena Dimitrova Canalyzation in mathematical modeling

64. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Unate cascade functions I The unate cascade functions (UCFs) make up the unique class of Boolean functions with the smallest average path length on a BDD [Mat05]. I This smallest average path length is 2 1 2n1 where n is the number of variables. I Thus UCFs can be evaluated more quickly than any other Boolean function. I The classes of UCFs and NCFs are identical [JRL07]. Elena Dimitrova Canalyzation in mathematical modeling

66. nition Properties of NCFs Structure NCFs and unate cascade functions Other results BDD of a UCF/NCF Recall f (x1; x2; x3) = x2 _ (:x1 ^ x3) is a NCF with b = [1; 0; 0], a = [1; 1; 0], and = [2; 1; 3]: Average path length = 3+3+2+1 8 = 1:125. Elena Dimitrova Canalyzation in mathematical modeling

68. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Number of NCFs I There is a recursive formula for the number of UCFs. I Since UCFs are identical to NCFs, the formula can be used for counting the number of NCFs in n variables, NCF(n): NCF(n) = 2 E(n), where E(1) = 1; E(2) = 4 and, for n 3, E(n) = Xn1 r=2 n r 1 2r1 E(n r + 1) + 2n: Elena Dimitrova Canalyzation in mathematical modeling

70. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Other results on NCFs I NCFs were generalized to an arbitrary

71. eld by D. Murrugarra [Tom10] who also showed that some features of the dynamics of these functions is similar to their Boolean counterparts. I A.S. Jarrah and F. Hinkelmann developed an algorithm that identi

72. es all systems of NCFs that

73. t a given data set [JH10]. Elena Dimitrova Canalyzation in mathematical modeling

75. nition Properties of NCFs Structure NCFs and unate cascade functions Other results Other results on NCFs I NCFs were generalized to an arbitrary

76. eld by D. Murrugarra [Tom10] who also showed that some features of the dynamics of these functions is similar to their Boolean counterparts. I A.S. Jarrah and F. Hinkelmann developed an algorithm that identi

77. es all systems of NCFs that

78. t a given data set [JH10]. Elena Dimitrova Canalyzation in mathematical modeling

80. nitions Depth and stability Derrida plots Conclusions I Recall NCFs: f = 8 : b1 if x(1) = a1 b2 if x(1)6= a1 and x(2) = a2 b3 if x(1)6= a1 and x(2)6= a2 and x(3) = a3 bn if x(1)6= a1 and and x(n1)6= an1 and x(n) = an ... ... :bn if x(1)6= a1 and and x(n)6= an I Quite restrictive! Elena Dimitrova Canalyzation in mathematical modeling

82. nitions Depth and stability Derrida plots Conclusions I Recall NCFs: f = 8 : b1 if x(1) = a1 b2 if x(1)6= a1 and x(2) = a2 b3 if x(1)6= a1 and x(2)6= a2 and x(3) = a3 bn if x(1)6= a1 and and x(n1)6= an1 and x(n) = an ... ... :bn if x(1)6= a1 and and x(n)6= an I Quite restrictive! Elena Dimitrova Canalyzation in mathematical modeling

84. nitions Depth and stability Derrida plots Conclusions Nested canalyzing depth The nested canalyzing depth of a Boolean function f is the largest index (i) such that the variables x(1); : : : ; x(i) exhibit the nested canalyzing structure in the de

85. nition of a NCF. Remarks: I We make the convention that constant functions are NCFs and have depth n. I If a function has i nested canalyzing variables and the function in the remaining n i variables is constant, this function also has depth n. I Functions with no canalyzing variables have depth zero. Elena Dimitrova Canalyzation in mathematical modeling

87. nitions Depth and stability Derrida plots Conclusions Example: Nested canalyzing depth I x2 is canalyzing with canalyzing input 0 and output 0. I (1) = 2 and b1 = a1 = 0; (2) = 3, b2 = 1, and a2 = 0. I The remaining function in x1 and x4 is neither canalyzing nor constant, so f has depth 2. Elena Dimitrova Canalyzation in mathematical modeling

89. nitions Depth and stability Derrida plots Conclusions Activity I Kauman and Shmulevich explored the in uence of the function's variables on its output [SK04]. I The partial derivative of a function f (x1; : : : ; xn) with respect to the variable xj is de

90. ned as @f (x) @xj = f (xj ;0) f (xj ;1): Here, xj ;i = (x1; : : : ; xj1; i ; xj+1; : : : ; xn) and denotes the XOR function. I Activity of a variable: The average of the partial derivatives of j = 1 a variable over all possible inputs, f 2k P x2f0;1gk @f (x) @xj : Elena Dimitrova Canalyzation in mathematical modeling

98. nitions Depth and stability Derrida plots Conclusions Sensitivity I The sensitivity of a function quanti

99. es the sensitivity of the output to variations in the function inputs. PI It is given by sf (x) = k i=1 [f (x ei )6= f (x)] ; where ei denotes the ith unit vector and is an indicator function. I Essentially computing the number of inputs with Hamming distance one from an input x that gives a dierent function output than x. I The average sensitivity of a function, sf , is the expected value of sf (x), sf = E sf (x) = Pk i=1 f i : Elena Dimitrova Canalyzation in mathematical modeling

110. nitions Depth and stability Derrida plots Conclusions Sensitivity I Shmulevich and Kauman suggest that networks created using functions that are less sensitive will be more dynamically ordered (stable) than those with higher sensitivity. I They show in [SK04] that a random Boolean function in n variables has average sensitivity n2 . I Also, for a Boolean function with depth at least one, the 14 14 expected activities of the variables (x1; : : : ; xn) are given by ; ; ; : : : ; , and hence the average sensitivity is n+1 . 14 124 Elena Dimitrova Canalyzation in mathematical modeling

116. nitions Depth and stability Derrida plots Conclusions Depth and sensitivity I When reverse engineering gene networks, a nested canalyzing structure may apply to some of the variables, while the behavior of the remaining variables may be ambiguous or unknown. I We say that a function has depth at least d if d of the variables are known to be nested canalyzing, while the remaining n d variables can take on any Boolean function, canalyzing or otherwise. I We'll show that the larger the value of d is, the less sensitive the function will be on average. Elena Dimitrova Canalyzation in mathematical modeling

122. nitions Depth and stability Derrida plots Conclusions Depth and sensitivity Extending Kauman and Shmuleviche's result to functions of depth at least d, we have the following result. Result (Layne, D., Macauley [LDM12]) Let fd be a Boolean function in n variables with nested canalyzing depth at least d. Renumbering the variables if necessary, assume that x1; : : : ; xd are the nested canalyzing variables with (i) = i . Then the expected activities of the variables (x1; : : : ; xn) are given by E fd = 1 2 ; 1 4 ; : : : ; 1 2d ; 1 2d+1 ; : : : ; 1 2d+1 : Further, the average sensitivity of fd is E[sfd ] = n d 2d+1 + Xd i=1 1 2d = n d 2d+1 + 1 1 2d : Elena Dimitrova Canalyzation in mathematical modeling

124. nitions Depth and stability Derrida plots Conclusions Depth and sensitivity Now, we see that E[sfd ] E[sfd+1] = 1 1 2d + n d 2d+1 1 1 2d+1 + n d 1 2d+2 = n d 1 2d+2 ; which rapidly goes to zero, so each subsequent canalyzing variable has a much smaller impact on the sensitivity. I Thus, the dierence in sensitivity between fully nested canalyzing functions and partially nested canalyzing functions of sucient depth is very slight. Elena Dimitrova Canalyzation in mathematical modeling

126. nitions Depth and stability Derrida plots Conclusions Moral Using strictly NCFs for modeling when stability is a desirable property is an overkill. PNCFs of sucient depth are good enough. Elena Dimitrova Canalyzation in mathematical modeling

128. nitions Depth and stability Derrida plots Conclusions Derrida plots x1(t) and x2(t): two states in random Boolean network. (t) = 1 n jjx1(t) x2(t)jj1 (normalized Hamming distance), where jj jj1 is the standard `1 metric. A Derrida curve is a plot of (t + 1) vs. (t) arranged uniformly over dierent states and networks [DP86]. Elena Dimitrova Canalyzation in mathematical modeling

130. nitions Depth and stability Derrida plots Conclusions Phases Consider the curve for small values of (t). I Frozen phase: Curve lies below y = x. The phase space of such networks consists of many

131. xed points and small attractor cycles. I Chaotic phase: Curve lies above y = x. Small perturbations propagate throughout the network. I Critical phase: The boundary threshold between the frozen and chaotic phases. There is evidence that many biological and social networks lie in the critical phase, as they must be stable enough to endure changes in the environment, yet exible enough to adapt when necessary [BABC+08, NPA+08, NPL+08, SKA05]. Elena Dimitrova Canalyzation in mathematical modeling

133. nitions Depth and stability Derrida plots Conclusions Derrida curves Elena Dimitrova Canalyzation in mathematical modeling

135. nitions Depth and stability Derrida plots Conclusions Observations I Networks of larger depth show more orderly dynamics than those of smaller depth. I The curves move closer together as depth increases, i.e., marginal bene

136. t of stability as depth increases. I Observations from Derrida plots match theoretical results. Elena Dimitrova Canalyzation in mathematical modeling

138. nitions Depth and stability Derrida plots Conclusions Conclusions I PNCFs are a reasonable generalization of NCFs. I Stability increases as depth increases, however the marginal gain in stability drops o quickly. I Just a few degrees of canalyzation are necessary to drop the network into critical regime. Elena Dimitrova Canalyzation in mathematical modeling

140. nitions Depth and stability Derrida plots Conclusions Thank you! Elena Dimitrova Canalyzation in mathematical modeling

142. nitions Depth and stability Derrida plots Conclusions [BABC+08] E. Balleza, E. Alvarez-Buylla, A. Chaos, S.A. Kauman, I. Shmulevich, and M. Aldana. Critical dynamics in genetic regulatory networks: Examples from four kingdoms. PLoS ONE, 3:e2456, 2008. [DP86] B. Derrida and Y. Pomeau. Random networks of automata: a simple annealed approximation. Europhys. Lett., 1:45{49, 1986. [JH10] Abdul Salam Jarrah and Franziska Hinkelmann. Inferring biologically relevant models: Nested canalyzing functions. In Algebraic and Numeric Biology 2010, 2010. [JL07] Abdul Salam Jarrah and Reinhard C. Laubenbacher. Discrete models of biochemical networks: The toric variety of nested canalyzing functions. In AB, pages 15{22, 2007. [JRL07] Abdul Salam Jarrah, Blessilda Raposa, and Reinhard Laubenbacher. Nested canalyzing, unate cascade, and polynomial functions. Physica D. Nonlinear phenomena, 233:167{174, 2007. [Kau93] Stuart Kauman. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, 1993. [KPST03] Stuart Kauman, Carsten Peterson, Bjorn Samuelsson, and Carl Troein. Random boolean network models and the yeast transcriptional network. PNAS, 100(25):14796{14799, 2003. [KPST04] Stuart Kauman, Carsten Peterson, Bjorn Samuelsson, and Carl Troein. Genetic networks with canalyzing boolean rules are always stable. Proceedings of the National Academy of Sciences of the United States of America, 101(49):17102{17107, December 2004. Elena Dimitrova Canalyzation in mathematical modeling

144. nitions Depth and stability Derrida plots Conclusions [LDM12] L. Layne, E.S. Dimitrova, and M. Macauley. Nested canalyzing depth and network stability. Bulletin of Mathematical Biology, 74:422{433, 2012. [Lyn95] J. F. Lynch. On the threshold of chaos in random boolean cellular automata. Random Struct. Algorithms, 6(2{3):239{260, 1995. [Mat05] Munehiro Matsuura. Average path length of binary decision diagrams. IEEE Trans. Comput., 54(9):1041{1053, 2005. Fellow-Butler, Jon T. and Fellow-Sasao, Tsutomu. [NFW07] S. Nikolajewa, M. Friedel, and T. Wilhelm. Boolean networks with biologically relevant rules show ordered behavior. BioSystems, 90:40{47, 2007. [NPA+08] M. Nykter, N.D. Price, M. Aldana, S.A. Ramsey, S.A. Kauman, L.E. Hood, O. Yli-Harja, and I. Shmulevich. Gene expression dynamics in the macrophage exhibit criticality. Proc. Natl. Acad. Sci., 150:1897{1900, 2008. [NPL+08] M. Nykter, N.D. Price, A. Larjo, T. Aho, S.A. Kauman, O. Yli-Harja, and I. Shmulevich. Critical networks exhibit maximal information diversity in structure-dynamics relationships. Phys. Rev. Lett., 100:058702, 2008. [SK04] I. Shmulevich and S.A. Kauman. Activities and sensitivities in boolean network models. Phys Rev Lett, 93(4):048701, 2004. Elena Dimitrova Canalyzation in mathematical modeling

146. nitions Depth and stability Derrida plots Conclusions [SKA05] I. Shmulevich, S.A. Kauman, and M. Aldana. Eukaryotic cells are dynamically ordered or critical but not chaotic. Proc. Natl. Acad. Sci., 102:13439{13444, 2005. [Sta87] D. Stauer. On forcing functions in kaumans random boolean networks. Journal of Statistical Physics, 46(3{4):789794, 1987. [Ste99] M. D. Stern. Emergence of homeostasis and noise imprinting in an evolution model. PNAS, 96(19):10746{10751, 1999. [Tom10] D. M. Tomairo. Multi-states nested canlyzing functions. Preprint, 2010. [Wad42] C. H. Waddington. Canalisation of development and the inheritance of acquired characters. Nature, 150:563{564, 1942. Elena Dimitrova Canalyzation in mathematical modeling

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