【latex版】水贴

In formal terms, a Boolean network can be represented as a graph G = (V, E) consisting of a set of n nodes V = {v1, …, vn} and a set of k edges E = {e1, …, ek} between the nodes. For every time point t, each node vi has a state vi(t) ∈ {0, 1} denoting either no expression or expression of a gene or absence or presence of activity of a regulatory property, respectively.

In a non-probabilistic Boolean network, the state vector, or simply the state S(t) of the network at time t corresponds to the vector of the node states at time t, i.e., S(t) = (v1(t), …, vn(t)). Thus, since every vi(t) can take only 2 possible values 0 or 1, the number of all possible states is 2n. In probabilistic Boolean networks (PBNs), as we will outline below, we are dealing with a probability distribution over several states at each time point. This is why, in order to extend the definition of states to probabilistic Boolean Networks, we will refer to a specific state as Si from now on where i ∈ {0, …, 2n}, independent of the time of its appearance. Every node is updated at every time point by application of a set of update functions F = {F1, …, Fn} that integrate the input information of edges on one node. In other words, the function Fi assigns a new state value to the node vi at time t + 1, i.e., vi(t + 1). They depend on the state of k input nodes with k ∈ {0, …, n} at time t.

In a non-probabilistic Boolean network, the state vector, or simply the state S(t) of the network at time t corresponds to the vector of the node states at time t, i.e., S(t) = (v1(t), …, vn(t)). Thus, since every vi(t) can take only 2 possible values 0 or 1, the number of all possible states is 2n. In probabilistic Boolean networks (PBNs), as we will outline below, we are dealing with a probability distribution over several states at each time point. This is why, in order to extend the definition of states to probabilistic Boolean Networks, we will refer to a specific state as Si from now on where i ∈ {0, …, 2n}, independent of the time of its appearance. Every node is updated at every time point by application of a set of update functions F = {F1, …, Fn} that integrate the input information of edges on one node. In other words, the function Fi assigns a new state value to the node vi at time t + 1, i.e., vi(t + 1). They depend on the state of k input nodes with k ∈ {0, …, n} at time t.

In a non-probabilistic Boolean network, the state vector, or simply the state S(t) of the network at time t corresponds to the vector of the node states at time t, i.e., S(t) = (v1(t), …, vn(t)). Thus, since every vi(t) can take only 2 possible values 0 or 1, the number of all possible states is 2n. In probabilistic Boolean networks (PBNs), as we will outline below, we are dealing with a probability distribution over several states at each time point. This is why, in order to extend the definition of states to probabilistic Boolean Networks, we will refer to a specific state as Si from now on where i ∈ {0, …, 2n}, independent of the time of its appearance. Every node is updated at every time point by application of a set of update functions F = {F1, …, Fn} that integrate the input information of edges on one node. In other words, the function Fi assigns a new state value to the node vi at time t + 1, i.e., vi(t + 1). They depend on the state of k input nodes with k ∈ {0, …, n} at time t.

There have been different approaches to address uncertainty and stochasticity in the Boolean framework (Shmulevich, 2002; Garg et al., 2009; Twardziok et al., 2010).

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In formal terms, a Boolean network can be represented as a graph G = (V, E) consisting of a set of n nodes V = {v1, …, vn} and a set of k edges E = {e1, …, ek} between the nodes. For every time point t, each node vi has a state vi(t) ∈ {0, 1} denoting either no expression or expression of a gene or absence or presence of activity of a regulatory property, respectively. In a non-probabilistic Boolean network, the state vector, or simply the state S(t) of the network at time t corresponds to the vector of the node states at time t, i.e., S(t) = (v1(t), …, vn(t)). Thus, since every vi(t) can take only 2 possible values 0 or 1, the number of all possible states is 2n. In probabilistic Boolean networks (PBNs), as we will outline below, we are dealing with a probability distribution over several states at each time point. This is why, in order to extend the definition of states to probabilistic Boolean Networks, we will refer to a specific state as Si from now on where i ∈ {0, …, 2n}, independent of the time of its appearance. Every node is updated at every time point by application of a set of update functions F = {F1, …, Fn} that integrate the input information of edges on one node. In other words, the function Fi assigns a new state value to the node vi at time t + 1, i.e., vi(t + 1). They depend on the state of k input nodes with k ∈ {0, …, n} at time t.

There have been different approaches to address uncertainty and stochasticity in the Boolean framework (Shmulevich, 2002; Garg et al., 2009; Twardziok et al., 2010). We will follow the probabilistic Boolean network (PBN) approach proposed by (Shmulevich, 2002), but apply it in a slightly different way. Originally, probabilistic Boolean networks were designed to represent the uncertainty in knowledge about regulatory functions. If there is experimental data showing that both transcription factors A and B activate gene C, but it is unclear whether they can act separately or only in combination, there is not only one determined logical function that can describe their interaction. In probabilistic Boolean networks this uncertainty is taken into account by relaxing the constraint of fixed update rules Fi and by permitting instead one or more functions per node. Thus, function Fi is replaced by a set of functions Fi={fji} with j∈{1, …, l(i)}, where fji is a Boolean logic function and l(i) the total number of functions for node vi. In each update step the functions are chosen randomly according to their probability which we assign.

Our model uses this feature of probabilistic Boolean networks to represent two kinds of stochasticity. The first is the aforementioned uncertainty about the correct function to apply. By employing different possible functions and varying their probability we can see which of them fits the known data better. The second way how we use the probabilistic functions is to model dynamic features of the system. The single processes that influence a variable are split into different functions and we assign each of them a probability. That way we can adjust the probability of the activation of a variable under certain conditions instead of assigning it one fixed value. We can also easily split activation and inactivation into different functions. Depending on the influencing variables the probabilities of the state of the variable change.

The implemented model is based on the probabilistic Boolean networks approach (Shmulevich, 2002). The complete model consists of n different variables, which are updated by Boolean rules in each time step. The rules all consist of AND, OR, and NOT connections between the different variables. In each time step one of the functions is chosen with its assigned probability to determine the next state of the variable. Probabilistic Boolean models can be simulated in different ways. Either one can simulate single trajectories of the model and analyze the results like the outcome of a stochastic experiment, or one can analyze the resulting Markov chain. All simulations were carried out using the R-Package BoolNet (Müssel et al., 2010).