We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.
Deep Dive into Nested canalyzing depth and network stability.
We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are
A large influx of biological data on the cellular level has necessitated the development of innovative techniques for modeling the underlying networks that regulate cell activities. Several discrete approaches have been proposed, such as Boolean networks [8], logical models [17], and Petri nets [5]. In particular, Boolean networks have emerged as popular models for gene regulatory networks [1,12]. However, not all Boolean functions accurately reflect the behavior of biological systems, and it is imperative to recognize classes of functions with biologically relevant properties. One such notable class is the canalyzing functions, introduced by Kauffman [8], whose behavior mirrors biological properties described by Waddington [20]. The dynamics of Boolean networks constructed using these functions are of considerable interest when determining their modeling potential. Random Boolean networks constructed using such functions have been shown to be more stable than networks using general Boolean functions, in the sense that they are insensitive to small perturbations [10]. Karlssona and Hörnquist [7] explore the relationship between the proportion of canalyzing functions and network dynamics. In [10], the authors further expand the canalyzation concept and introduce the class of nested canalyzing functions (NCFs). In [11], networks of NCFs are shown to exhibit stable dynamics. Also, Nikolajewa, et al. [13] divide NCFs into equivalence classes based on their representation and show how the network dynamics are influenced by choice of equivalence class. Nested canalyzing functions have a very restrictive structure and become increasingly sparse as the number of input variables increases [6]. Also, it is possible that not all variables exhibit canalzying behavior. Hence, a relaxation of the nested canalyzing structure is necessary.
In this article, we further explore canalyzation by analyzing functions that retain a partially nested canalyzing structure. We quantify the degree to which a function exhibits this canalyzing structure by a quantity we call the nested canalyzing depth. Functions of depth d generalize the nested canalyzing functions, because NCFs are the special case of when d = k, where k is the number of Boolean variables. In Section 3, we demonstrate notable properties of these partially nested canalyzing functions, and show that their representation is unique. This leads to a theorem about the structure of functions of depth d, which generalizes a result in [6] about NCFs. In Section 4, we compute the expected activities and sensitivities of functions given their canalyzing depths, which are extensions of results of Shmulevich and Kauffman [18] about activities and sensitivities of canalyzing functions. We prove that as canalyzing depth increases, functions become less sensitive to perturbations in the input; however, the marginal benefit incurred by adding further canalyzing variables sharply decreases. As a result, functions of larger depth provide an improvement in sensitivity over general canalyzing functions, but imposing a fully nested canalyzing structure provides little benefit over functions of sufficient canalyzing depth. Finally, in Section 5, we use Derrida plots to show that dynamics of networks constructed using more structured functions rapidly approach the well-known critical regime, whereas networks with functions of relatively few nested canalyzing variables remain in the chaotic phase. This is in contrast to the findings of Kauffman et al. [11], but in agreement with recent work of Peixoto [16], and it further supports the biological utility of certain canalyzing functions.
. In this case, x i is a canalyzing variable, the input a i is its canalyzing value, and the output value b i when x i = a i is the corresponding canalyzed value. Note that if f is constant, then every variable is trivially canalyzing.
If a canalyzing variable x i does not receive its canalyzing input a i , then the output is some function g i (x i ), where xi = (x 1 , . . . , x i-1 , x i+1 , . . . , x k ). If g i is constant, x i is called a terminal canalyzing variable of f . Note that for each i = j, x j is then trivially canalyzing in g i . If g i is not constant, we ask whether it too is canalyzing. If so, there is a canalyzing variable x j with canalyzing input a j , and when x j = a j , the output of f is a function g ij (x ij ), which may or may not be canalyzing. Here, xij denotes x with both x i and x j omitted. Eventually, this process will terminate when the function g is either constant or no longer canalyzing. Definition 1. Let f (x 1 , . . . , x k ) be a Boolean function. Suppose that for a permutation σ ∈ S k , an integer d > 0 and a Boolean function g(x σ(d+1) , . . . , x σ(k) ),
where either x σ(d) is a terminal canalyzing variable (and hence g is constant), or g is nonconstant and none of the variables x σ(d+1) , . . . , x σ(k) are canalyzing in g. Then f is said to be a partially nested c
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
