Boolean Dynamics

In general, Boolean dynamics are represented by a discrete time series of binary vectors (a state of the network is a binary vector, as explained above and in Figure 4.1(D)) in the state space of the network. This can be mathematically represented by an M * N matrix where M is the time steps considered and N is the number of nodes in the network. State space is a set of values (representing all possible states of a network) any of which can be assumed by a discrete

dynamical system. A Boolean system with N nodes can, theoretically, have a total of 2N _states

(finite state space)as each variable is represented by two states.

The temporal evolution of the network dynamics is a result of the Boolean functions deciding the future states of the network nodes and the updating scheme representing time delays of reactions.

4.1.1.1 Boolean Functions

A Boolean, or switching function, decides the output of the network, logically evaluating a set of Boolean inputs combined with specific Boolean rules. The dynamics of the network at time t are determined by the Boolean rules associated with its regulators and their states at time

t-1. If the state of a node iat timetisdenoted by 𝑥𝑥𝑖𝑖𝑡𝑡and the associated Boolean function by Bi

(Bi = B1, B2,…….BN)with n number of regulators, then the dynamics of the Boolean network

can be generated, as shown in Eq. 4.1.

xit=Bi(x1t-1, x2t-1,…….xnt-1) (4.1)

Further elaboration of Eq. 4.1 requires a set of logical operators to define the logical

expressions of the model by combining each of the regulators within each Bi. There are three

common Boolean operators widely used in logical modelling: ‘AND’, ‘OR’ and ‘NOT’. The operator ‘AND’ represents the association of two or more regulators in a dependent manner where all the regulators are needed for the activation of a node. If the activation of a node can be induced by any of its regulators, the Boolean function combines these independent regulators with an ‘OR’ operator. The operator ‘NOT’ represents negative regulation (inhibition) by an element of a regulatory node.

The use of Boolean operators can be further explained with the example provided in Figure 4.1. Section (B) of the figure demonstrates how ‘AND’, ‘OR’ and ‘NOT’ operators are used in Boolean functions. For example, if the regulation of factor B is considered, the corresponding Boolean function determines the future state of B in a way that the presence of factor A, with the absence of factor C at the same time, boosts the activity of B. When considering factor D, the presence of either B or C can enhance the activity of D.

In addition, there is a possibility to represent these Boolean functions with a threshold function for the network output, as shown in Eq. 4.2. These functions are defined in such a way

that they perform a logical operation based on one or more regulators (Xjj=1….n) of the relevant

node (Xi) and produce a single binary output of 0 or 1. The future state of the node (Xi (t+1))

is determined by the weighted (wji) sum of inputs Xi from the regulators (from nodes j to i).

The inputs are binary [0, 1] and weights (wji) are +1 for positive relationships and -1 for

inhibitory relationships. Changing the node to its active state requires the weighted sum to be above the threshold value (T) at time t. If the threshold value is set to 1; for example, the system requires the weighted sum of the regulators of the corresponding node to be net positive for its activation.

Xi (t+1)=�1, if _{0, if} ∑_∑j_wwjiXj(𝑡𝑡) ≥ T jiXj(𝑡𝑡) < T

j (4.2)

Truth tables associated with each Boolean function help to decide the functional output for each combination of regulator variables (Figure 4.1(C)). The truth table for a node having n regulators consists of n+1 columns assigning each regulator a column depicting their activity

at time t. There are 2n _{rows, one for each combination of regulator states. The last column states}

the future state of the output node for all possible combinations of the input nodes.

4.1.1.2 Updating Schemes

In Boolean modelling, time delays associated with each reaction are represented in the updating scheme. In most cases, a Boolean network state transitions can be produced through two types of update methods: synchronous and asynchronous. In a synchronous update, the state of all nodes are updated simultaneously at each time step; whereas, in an asynchronous

update, each node is updated according to its own unique timing. Updating large Boolean

networks with complex topology using different updating methods can result in different

qualitative behaviours(Squires et al., 2013). Both synchronous and asynchronous updates can

produce the same steady state dynamics (e.g., Ca2+ _{oscillations) but owing to the realistic}

temporal evolution in an asynchronous model to achieve the same outcome in the system, there could be differences in the nodes involved, trajectories traversed and attractors reached. Steady states are the final, persistent or unchanging states reached by the system. These states are called attractors. If there is a unique final state vector, the system has reached a point attractor. However, if the system has reached a steady state involving a finite number of state vectors that follow each other and reach the original state in a cycle, it is called a limit cycle attractor. Networks can also reach chaotic attractors without an identifiable pattern of node transitions.

Synchronous updating assumes that all the interactions within the system happen with equal time delays. In asynchronous updating, nodes can be either updated randomly (random asynchronous) or based on a series of discrete time points according to real time delays associated with system variables. It is believed that an asynchronous update produces rich and realistic dynamics of the system modelled, making the simple Boolean approach more biologically plausible.

As an alternative, several different asynchronous updating methods that are biologically more plausible have been defined in the literature by adding more variability to the updating scheme. These updating techniques have diverged into several extensions based on how the

updating order is defined.

A random order asynchronous update is the most common, where one node is updated at a time by keeping all others in their previous state and with the updating node being chosen

randomly with equal probability (Mesot & Teuscher, 2003). A generalized asynchronous

update is very close but with a mild extension to the random order asynchronous update method. This can be considered a semi-synchronous approach where in each time step, the model randomly picks up several nodes, which are then updated synchronously. However, the reliability of this type of nondeterministic asynchrony is questionable in real biological modelling. It is useful when very little, or no, knowledge is available about the system dynamics and an exploration of the system for potential steady states or patterns of behaviour from a large number of trials involving random updates is needed.

The sequential schedule updates each node in every step but in a predefined order. Block sequential schedules group the nodes of the network into disjoined blocks, which are updated sequentially while the states of the nodes within the blocks are updated synchronously (Goles & Noual, 2010).

A deterministic approach (i.e. with a predetermined order of events) is a biologically plausible updating method. This method updates one or more nodes at a time step according to a predefined criterion that is based on time delays associated with the network nodes. The incorporation of determinism with asynchrony in the model updating scheme appears helpful in understanding the real dynamics of the Boolean networks, because perfect synchrony is highly unlikely to be observed in the molecules involved, even within the same cellular physiological function. According to the literature, there may be some spatial reasons for the timing of regulatory activities (such as regulation may happen only at defined sites in the cell but the regulators may be produced in a different place; hence, they need to be transported to the reaction sites) or differences in delays may be due to the varied speeds of different reactions. However, it is difficult to know the order of some of the events happening inside biological systems because their speed exceeds the current experimental capability to capture them. It was believed that the integration of asynchrony in a deterministic manner in a model will capture these differences to define the true dynamics of the system through more precise information processing. Determinism is helpful in identifying periodic behaviours of a system because the attractors produced by synchronous or random order asynchronous updating may be just a part of the state space and may not represent or fully capture true attractors that correspond to the true system dynamics.

All these variations are accepted in the literature because modellers believe that even with an arbitrary updating scheme the principal properties of the network can be captured.