Robustness of the ABA Signalling Network

Robustness is the ability of a system to maintain its original functions despite small perturbations in the elements or regulatory mechanisms of the system. Biologically, it is the tolerance against random attacks. Robustness can be determined from the effect of the removal of nodes (gene/protein knockouts) or removal/addition of edges (functional perturbations) that connect the elements of the network or changes in state of a node (environmental perturbation) on the behaviour of the system. Functional perturbations and node removals, analogous to genetic mutations, can be considered as permanent alterations to the system; these are crucial in genetic engineering for developing biological insights.

Perturbation of the state does not disturb the steady state properties of the network because state changes have no influence on the structure of the network. This kind of perturbation may let the system eventually reach the original attractors. However, if a state of an attractor is perturbed, the system may, or may not, reach the original attractor. If there are more than one attractor in the system, perturbation of a state on one attractor may converge the network on a new attractor. If the state perturbed network reaches its original attractors, the network is said to be robust (Xiao & Dougherty, 2007).

In most studies, the robustness of Boolean networks was evaluated in terms of the perturbing states (environmental perturbation) or functions of the nodes (removal/addition of edges) in the system. In this study, we define the robustness as the probability that a random permutation to a chosen node or transition function of a node results in the same network behaviour as in the original. We assume that state perturbations are temporary as they represent environmental fluctuation induced changes in the system, and are not examined in this study.

In the last section involving perturbation to elements, we explained how system performance is affected by removing secondary and hub elements of the ABA signalling. We found that the performance of the ABA signalling system is unaffected by the removal of most

of the sparsely connected nodes but sensitive to hub removal. Thus, the ABA signalling system is robust against perturbation to a large number of secondary elements (67% of 43) and sensitive to system hub elements that constitutes 23% (13) of the total network nodes. Now we ask the question, how robust is the system to perturbation of links representing functional perturbations. In this section, we perturbed the ABA signalling network by altering one

randomly selected Boolean transition function at a time (transition functions are written as 2n

bit binary numbers where n is the number of regulators. We shuffled the order of the binary bits in the selected function). We simulated 500 copies of functionally perturbed networks and compared them with the steady state dynamics of the original network.

The results reveal that nearly 60% of the perturbed networks settled in exactly the same attractors as the original network. With reference to the results of the output node ‘CLOSURE’, 85% of the perturbed networks functioned in the same way as the original network, having an 85% agreement in the length of the attractors and attractor basins.

The model discovers that functional perturbations influence the global output to different degrees. Random functional perturbations on a particular node sometimes completely change the Boolean function of the selected node. If these complete changes occur on the functions of important network nodes such as PYR, SnRK2, PP2C, SLAC1, ACTIN, GORK and ROS, the system becomes insensitive to the ABA signal. Even if the perturbation occurs on an important Boolean function, it may not seriously affect the system output if there are other links available to compensate for the effect of the damaged link. For example, if the functional link between

the K+ _{outward channel, GORK, and NO is impaired (the transition function for GORK is (!}

NO | ROS| pH) & DEPOLAR)), it does not cause any damage to the activity of GORK as either ROS or pH can minimize that effect. If the perturbation occurs on a critical link, however, the degree of the influence is severe (e.g., if the regulatory link between depolarization and GORK

is impaired, it will seriously affect the K+ _{efflux of the system). In general, we observe that the}

nodes found to be critical (explained in the last section) are more sensitive to functional perturbations.

Functional perturbations are identified to be not as severe as removing a particular node because removing a node from the system disturbs all the related regulatory functions. We realize that the ABA system benefits from having extra reinforcing regulatory links to maintain important elements of the network active and thereby make them the hubs in the network. In some situations, the ABA signalling system responds to functional perturbations by generating new system attractors. In general, perturbation derived system attractors are simple limit cycles

governed by [Ca2+_]

cyt oscillations with minor changes in the length of the attractors or state of

a frozen node. We found that 85% of perturbation derived attractors preserve the global outcome of the network - stomatal closure. These results indicate that the ABA signalling system is a functionally robust system.

So far we have discussed the results of our synchronous Boolean model, which is an extension of the existing Boolean model for the ABA signalling network defined by Li et al.

(2006)b_{. The next section compares both models to evaluate the extra knowledge generated by}

extending the existing ABA model using the novel experimental findings. The discussion proceeds by highlighting the similarities and differences between the two models and

advancements of our model over the Li et al. (2006)b _model.

5.7 Comparison of Current Model Results with the Li et al.