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Computing preimages of Boolean networks
BMC Bioinformatics volume 14, Article number: S4 (2013)
Abstract
In this paper we present an algorithm based on the sumproduct algorithm that finds elements in the preimage of a feedforward Boolean networks given an output of the network. Our probabilistic method runs in linear time with respect to the number of nodes in the network. We evaluate our algorithm for randomly constructed Boolean networks and a regulatory network of Escherichia coli and found that it gives a valid solution in most cases.
Introduction
In systems and computational biology Boolean networks (BN) are widely used to model regulative dependencies of organisms [1, 2]. We consider networks, which map a set of environmental conditions to the presence of proteins and finally to actual chemical reactions, which are often modeled as fluxes of a fluxbalance analysis [3]. Hence, these networks are used to make in silico predictions of behavior of organisms in a certain environment [4].
In this paper we address the inverse problem, i.e., we want to predict environmental conditions that allow certain reactions to take place, and others not. Hence, in general, we need to find a set of possible inputs that lead to a given output. This so called predecessor problem or preimage problem has been addressed by Wuensche in [5] and has been shown to NPhard in general [6], which makes it infeasible to solve it for large networks. In [7] an algorithm with reduced complexity for BNs with canalizing Boolean functions has been introduced. However, the problem is infeasible under certain conditions. Both algorithms are designed to find the whole set of preimages, i.e., all inputs to the BN with lead to a certain, desired, output.
In some applications, knowledge of the whole preimage set is not important, merely it can be sufficient to know a subset of the preimage set. Here, we propose a probabilistic algorithm, which solves this problem in linear time with respect to the number of nodes in the network, based on a variation of the well known SumProduct Algorithm (SPA) [8], which is used for a variety of tasks, including decoding error correction codes in communication engineering [9].
Methods
Boolean networks and main idea
We consider networks like shown in Figure 1, mapping the values of the N innodes $\mathbb{I}=\left\{1,\phantom{\rule{2.77695pt}{0ex}}2,\phantom{\rule{2.77695pt}{0ex}}3\right\}$ to the M outnodes $\mathbb{O}=\left\{12,13,14,15,16\right\}$, i.e., we can represent this BN as a function mapping the N input values uniquely to the M output values:
The network itself consists of n nodes, and a set of directed edges connecting these nodes. Each node i has a certain state, which can be either zero or one, represented by a variable x_{ i }. Its value is determined by evaluating a Boolean function (BF) f_{ i }. Further, lets define the set ñ(f_{ j }) as the incoming nodes of node j. For example in Figure 1, ñ(f_{5}) = {1, 3}. The BF f_{ j } is a function mapping k_{ j } = ñ(f_{ j }) values of {0, 1}^{k} to {0, 1}, where k is also called the indegree of node j. The number of edges emerging from a node is called outdegree.
Given a vector of input values x ∈ {0, 1}^{N} , x = (x_{1}, x_{2}, . . . , x_{ N } ) the corresponding output of f is y = f (x), y ∈ {0, 1}^{M}. In general there does not exist a unique inverse function f^{−1}. Instead the cardinality of the set Ω_{ y } := {x : f(x) = y} will be larger one. We call Ω_{ y } the set of preimages of y. In this paper we are interested to find at least parts of Ω_{ y }. Suppose there is a probability distribution P_{ y } on {0, 1}^{N} such that
If we knew the probability distribution P_{ y }, we would have solved the problem. But as explained, this is too difficult in general. Our main idea now is to approximate P_{ y } by the product of the marginal distributions P_{ i } on the individual x_{ i }, i.e.,
as the wellknown SPA can be used to compute the marginals efficiently. If the approximation is good enough sampling out the product of the marginals will yield an element in Ω_{ y } with reasonable probability.
Proposed algorithm
In this section we will first discuss the basic principles of factor graphs and the SPA. Then we will describe the BN as factor graph and will formulate the actual algorithm to find the marginals. Finally, the sampling is described.
Factor graphs and sumproduct algorithm
Assume some function g(x_{1}, . . . , x_{ n }) defined on some domain ${\mathbb{A}}^{n}$, which can be factorized in m local functions h_{ j }, j ∈ [m] := {1, 2, . . . , m}, i.e.,
where X_{ j } is the subset of [n] containing the argument of h_{ j } . We can then define a factor graph [8] as a bipartite graph consisting of n nodes representing variables {x_{1}, . . . , x_{ n }} (variable nodes) and of m nodes representing functions {h_{1}, . . . h_{ m }} (function node). Edges exist between a function node and a variable node if and only if x_{ i } is an input to function h_{ j } .
The marginal function g_{ i }(x_{ i }) is defined as [8]
where ${\sum}_{~\left\{{x}_{j}\right\}}g\left({x}_{1},\phantom{\rule{0.3em}{0ex}}\dots \phantom{\rule{0.3em}{0ex}},{x}_{n}\right)$ is defined as
In general the computation of the g_{ i } is difficult, but due to the factorization of g the task can be efficiently solved using the the so called SumProduct Algorithm (SPA) [8]. The algorithm iteratively passes messages between the nodes of the graph. At each iteration the messages µ are sent from the function nodes to the variable nodes, containing the corresponding marginal function of the local function. These messages are computed as follows [8]:
Function to variable node
where n(i) give the set of neighboring nodes of node i.
At the variable nodes, these messages are then combined to a marginal function λ and sent back to the function nodes [8]:
Variable to function node
The Boolean network as factor graph
We apply the concept of factor graphs to BNs. Each node in the network represents one variable x_{ i } ∈ {0, 1}, i ∈ [n] of the factor graph, hence we have n variable nodes. Each BF f_{ j } of the BN $\left(j\in \left[n\right]\backslash \mathbb{I}\right)$ is a function node and is connected to the node j and the incoming nodes ñ(f_{ j }). Lets to define ${\tilde{X}}_{j}$ as the variables of the incoming nodes of node j, i.e. the argument of the BN f_{ j }. Further, we define ${\tilde{X}}_{j}^{\left(i\right)}$ as ${\tilde{X}}_{j}$ without the node i.
Finally, if we consider the variables as each node as random variables, we have a common distribution of all variables nodes described by the density function,
For sake of readability we will omit the subscripts of the density function, if they are obvious from context. We are interested in finding the marginal distributions of the innodes, which can be described by the density functions
This problem is an instance of the problem described in Section Factor Graphs and SumProduct Algorithm, hence we apply the same methods here.
Update rule: function to variable node
If we focus on one function node $j\in \left[n\right]\backslash I$ there exists a common distribution of all variables relevant for this node. Namely, these relevant variables are the ones located in ${\tilde{X}}_{j}$ of the BF f_{ j }, and the value of node j. We can write the density of this distribution as:
Lets define ñ(f_{ j }) as the set of indices of the input nodes of the BF f_{ j }.
We need to send the local marginal distribution of each variable i ∈ {j} ∪ ñ(f_{ j }) back to the variable node, or more formally:
If i = j, i.e. if the message is designated for the node containing the output of the BF, the density of the marginal distribution becomes:
which is the probability distribution of the functions output. We can assume that the elements of ${\tilde{X}}_{j}$ are pairwise independent, hence we can write:
where λ_{ l } is the probability distribution of variable node l and is defined in Eq. 3.
In the other cases, i.e., i ≠ j, Eq. (1) becomes:
We still can assume that the elements of ${\tilde{X}}_{j}^{\left(i\right)}$ are pairwise independent, hence we can write:
If the Boolean functions output ${x}_{j}={f}_{j}\left({\tilde{X}}_{j}\right)$ is already completely determined by ${\tilde{X}}_{j}^{\left(i\right)}$, i.e., if the variable x_{ i } has no influence on the output for this particular choice of the other variables, we assume x_{ i } to be uniformly distributed:
and since x_{ j } is completely determined by ${\tilde{X}}_{j}^{\left(i\right)}$
Otherwise, x_{ i } is totally determined by x_{ j } and the other variables, i.e., x_{ i } is 0 or 1 depending on BF. Hence, we can write
where ${p}_{{x}_{j}}\left(f\left({\tilde{X}}_{j}^{\left(i\right)},\phantom{\rule{2.77695pt}{0ex}}{x}_{i}\right)={x}_{j}\right)$ is either 0 or 1. Further we can assume x_{ j } independent of ${\tilde{X}}_{j}^{\left(i\right)}$, hence
Finally, we can summarize for i ≠ j:
with
Update rule: variable to function node
The update rule is the same for all variable nodes j ∈ [n] and is independent of the function node to which they are directed.
where ${S}_{j}$ is the set of all function nodes, which have node j as input.
Finding the input distributions
In our algorithm, we use the well known loglikelihood ratio (LLR) to represent the probability distribution of binary variables [10]. It is defined as:
A scheme of the algorithm is given in Algorithm 1.
The probability distribution of each node j ∈ [n] at iteration t is given as ${L}_{j}^{\left(i\right)}$ and are initialized with ${L}_{j}^{\left(0\right)}=0$, which is equivalent to the uniform distribution. Then we set the LLRs for the outnodes to either −∞ or +∞ depending on the desired output y of the BN. At each iteration the algorithm can be split in two steps. The first step iterates over all function nodes $\left(j\in \left[n\right]\backslash \mathbb{I}\right)$ and all input variables i ∈ ñ(f_{ j } ) calculating the LLR ${L}_{j\to i}^{\left(t\right)}$ using Eq. (2) and Eq. (4).
In the second step we update all variablesnodes, where the LLRs L_{ j } represents the distributions λ_{ j } and, hence the product of Eq. 3 becomes a summation. Please note, that the LLR of the previous iteration is also added to the sum, in order to prevent rapid changes of the distributions.
After performing a certain number of iterations t_{ max }, the desired marginal distributions of the input variables are found.
Algorithm 1
Initialize ${L}_{j}^{\left(0\right)}=0$ for all nodes
Set the desired LLRs of the outnodes, i.e., ${L}_{j}^{\left(0\right)}$ is either −∞ or +∞, for all outnodes $j\in O$.
t = 0
repeat
t=t+1
for each noninnode $\left(j\in \left[n\right]\backslash I\right)$do
for each input variable i ∈ ñ(f_{ j }) do
calculate L_{ j→i } using Eq. (2) and Eq. (4)
end for
end for
for each nonoutnode v do
${L}_{j}^{\left(t\right)}={L}_{j}^{\left(t1\right)}+{\sum}_{l\in {S}_{j}}{L}_{l\to j}^{\left(t\right)}$
end for
until maximum number of iterations reached
Sampling
The sampling part of our approach is straightforward. Using the marginal distributions ${L}_{j}^{\left({t}_{max}\right)}$, $j\in \mathbb{I}$ we randomly draw vectors x and check if they fulfill y = f(x). If so, they are added to the set ${\tilde{\text{\Omega}}}_{y}$. This procedure is repeated for a certain number of samples.
Simulation results and discussion
We tested our algorithm with randomly generated networks and the regulatory network of Escherichia coli (Ecoli) [2]. The random networks consist of 2400 nodes with N = 200 and M = 1200. We have chosen the BFs from:
· all functions with k ≤ 15 (Type A)
unate, i.e. locally monotone, functions with k ≤ 15 (Type B)
After generating a network we draw a certain number T of uniformly distributed input vectors x and obtain y = f(x). For each y we applied then Algorithm 1 to obtain the marginal distributions${L}_{j}^{\left({t}_{max}\right)}$,$j\in \mathbb{I}$ . To investigate the convergence behavior with respect to t_{ max } we first apply harddecision to evaluate a good choice for t_{ max }, i.e., we generate an estimate $\tilde{x}$ by setting
Then we evaluate the network $\u1ef9=\text{f}\left(\tilde{x}\right)$, and measure the similarity between y and $\u1ef9$ by counting the equal entries and divide them by the length of y. We did so for 100 networks of Type A and B, and set T = 100. The averaged results can be seen in Figure 2.
One can see, that for t_{ max } ≥ 14 there is almost no improvement in the similarity. This number is equal to two times the number of nodes between input and output, i.e., it seems to be sufficient that the messages travel once through the network and back. Thus, the following simulations have been perform setting t_{ max } = 14.
Next, we apply sampling as described in Section Sampling. We did so for 100 different networks of Type A and B, and the Ecoli network. For each random network we did T = 100 runs, for Ecoli T = 1000. The results can be viewed in Table 1. We depict the percentage of solved networks, i.e. the portion of networks we found at least one valid x ∈ Ω_{ y } . Further, we give the average number of valid x and the average number of unique x.
One can see from the results, that in general for most networks and y s at least one preimage can be found. It is worth mentioning, that for the Ecoli network every sampled solution was unique. This is due to the fact, that there exist a few inputs, who completely determine the output. The other input variables have then no influence and hence a marginal distribution of 0.5. Further, the results for the network of type B are much better than for type A. It seems that the marginal distributions for unate functions give better estimation of the actual distribution than the marginal distributions for nonunate functions.
Conclusions
In this work, we proposed a probabilistic algorithm to address the preimage problem of Boolean networks. This is of interest when designing experiments, in which certain regulators are supposed to be in a specific state. Performing a series of simulations with Random networks we showed, that the algorithm works not only for unate functions, of which most biologically motivated networks consist, but for any kind of Boolean functions. By replacing the fixed output values of the network by probabilities one can simply apply the algorithm to networks, whose designated output is described by probability distributions. Further, the algorithm may be easily adjusted to work on stochastic, e.g. Bayesian, networks, where the nodes contain only transition probabilities instead of Boolean function. Therefore, it is needed to adapt the update rules accordingly. It remains an open question, which influence topographic properties, such as number of layers and number of nodes in these layers, have to the performance of the proposed algorithms, since we only investigated networks which are similar to the regulatory network of Ecoli.
Abbreviations
 Ecoli :

Escherichia coli
 BF:

Boolean Function
 BN:

Boolean Network
 Eq:

Equation
 LLR:

LogLikelihood Ratio
 SPA:

SumProduct Algorithm
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Acknowledgements
The authors would like to thank Shrief Rizkalla for implementing parts of the simulation.
Declarations
This work was supported by the German research council " Deutsche Forschungsgemeinschaft" (DFG) under Grant Bo 867/252 and the Ulm University.
This article has been published as part of BMC Bioinformatics Volume 14 Supplement 10, 2013: Selected articles from the 10th International Workshop on Computational Systems Biology (WCSB) 2013: Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/14/S10
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Authors' contributions
Idea and Concept: JK, SS. Design of the overall project: MB. Scientific mentor of JK and SS: MB. Implementation and Evaluation: JK. Wrote Paper: JK and SS. All authors discussed the results and implications and commented on the manuscript at all stages.
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Klotz, J.G., Bossert, M. & Schober, S. Computing preimages of Boolean networks. BMC Bioinformatics 14, S4 (2013). https://doi.org/10.1186/1471210514S10S4
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Keywords
 Boolean Function
 Marginal Distribution
 Random Network
 Function Node
 Boolean Network