formulas) [33]. In much the same vein, Ostrowski et al. [58] created a method to inferBNs from time series datasets, such that the inferred networks have a structure compatible with a given prior knowledge interaction graph and are capable of reproducing all the (experimentally) observed time series. ASPwas used to enumerate theBNs which best approximate the experimental data, while still satisfying the prior knowledge regarding the known dynamics of the network. When applied to datasets from networks composed of 13-40 nodes and 16-50 edges, the identification of one optimal BNtakes only a few seconds, showing remarkable efficiency. In addition, the rate of true positives is over 78%
for networks having less than 25 edges.
A study was also made in order to ascertain howASP-based approaches to determine possible logic models for a givenPKNfare against some of the existing approaches based on stochastic optimization algorithms (CellNOpt) [81]. While the stochastic approach had proven to be able to train networks of realistic size, it suffered from the lack of guarantee of finding an optimum, which is an intrinsic problem of stochastic search methods. Ad-ditionally, these methods scale poorly as the search space (and therefore computational time) increase exponentially with the network size. Results of this study demonstrated significant improvements with the ASPsolution, concerning both computational time and completeness in the search of optimal models, surpassing the stochastic approach by up to five orders of magnitude (according to the experiments presented).
3.2.2 Model Analysis
3.2.2.1 Attractor Determination
Model analysis has also shown to have benefited from the appropriate usage ofASP. Tarek Khaled and Belaïd Benhamou proposed to useASPto express and simulate the dynamics of gene regulatory networks, which are a type of BRN [38]. Their approach focused on the exhaustive enumeration of all the attractors of synchronous and asynchronous BNs. The methodology consisted of calculating and enumerating the stable models of the logic program expressing the interactions between the compounds of the network.
They considered all the possible configurations of the network and ran a simulation for each of them, thus managing to obtain all the possible state sequences of the STG, a process which ensures all attractors are detected. After running their algorithm on real life regulatory networks, the obtained results demonstrated to be promising, revealing room for improvement, with the additional benefit of exhaustive enumeration, thanks to leveraging the use ofASP. The same authors also made advances in the enumeration of all the attractors of asynchronousBNs having regulatory graphs (recall Fig.2.5) which are circuits (these networks are also known as circular networks) [39]. Intuitively, a circuit is a path on the regulatory graph that starts and ends in the same node. In this work, a connection was discovered between a previously studiedASPparadigm and the encoding of the attractors. More specifically, a correspondence was revealed between the attractors in anSTGand the stable models of the logic programs expressing the circular
3 . 2 . A S P I N S Y S T E M S B I O L O G Y
network, which made it possible for attractors to be detected without the need of going through the simulation ofBNs. The experiments conducted to demonstrate the validity of this approach were very promising, with results showing that the computation of the attractors of a certain type of circuit with 7000 nodes took less than 60 seconds.
The work of Fayruzov et al., already partially discussed in Section3.2.1, also dealt with model analysis, in particular with the problem of efficiently computingattraction basins [16]. The main issue here is related to the number of time steps to consider when simulating a network’s behavior. Too few steps, and we may not find the states that lead to an attractor. Too many steps, and we risk over-computing the number of states, which is computationally ineffective. To solve this, they introduce the notion of Markovian program, which is a program that enforces the rule that the next state of the model either depends only on the previous state, or is determined with respect to other components in the current state of the model. By building a framework that conforms to this notion of Markovian program, instead of solving theASPprogram for some long time interval {0, ..., tmax}, they are able to consecutively solve smaller programs for time intervals{0,1}, {1,2},{tmax−1, tmax}, which can be done more efficiently.
ASPhas also proven to succeed where previous state-of-the-art approaches have failed.
The work of Oliver Ray et al. in reaction networks, which are a general framework for representing and reasoning about complex biological interactions, demonstrated that a previously proposed approach to find stable behaviors in these networks was inadequate, but that correct results could be obtained by usingASP[62]. Past approaches employed weighted Boolean constraints to represent and reason about reaction networks, which revealed to perform inadequately in networks containing cycles. This is because the computation of ’unfounded’ results was not penalized, and reactions whose presence could not be justified by the existing network configuration would be obtained. ASP solved this problem, as it enforced that every result obtained had to be justified by some existing rule, therefore making it impossible that any of the computed reactions could be produced out of ‘thin air’. The authors noted, however, that the scalability of the proposed ASPapproach remained to be assessed, as well as its utility in real-world scenarios.
3.2.2.2 Consistency and Reachability Verification
In 2004, C. Baral et al. were among the first to use knowledge representation languages and reasoning methodologies to represent and reason about biological networks [2]. Up until then, most other qualitative systems biology approaches relied on simulation rather than reasoning, and had shown to handle certain activities poorly, mainly the dealing with incomplete network knowledge and the formulation of explanations, diagnosis and planning of network models. They proposed a system, implemented using AnsProlog (short for Answer Set Programming in Logic), which tackled the shortcomings of other approaches. This system, dubbedBioSigNet-RR(meaning ’biological signal networks’, with RR denoting ’representation and reasoning’), was able to perform various kinds of
reasoning such as planning, hypothetical reasoning and explaining observations. Addi-tionally, it could work with incomplete or partial information, while also allowing for easy updating of information when new knowledge became available. Their approach proved fruitful, as it was successfully able to reason in signalling networks, illustrating howASP could be leveraged to deal with elaboration tolerance and incomplete information.
Due to the nature of biological processes, it is not uncommon for a biological network in a given state to transition into another state that prevents the network from reaching certain configurations. These situations are called bifurcations. Louis Fitime et al. fo-cused on the identification of all such bifurcations in a network [17]. To achieve this, they first performed a static analysis of the networks, in an attempt to create abstractions that simplified and approximated the real-world counterparts. This was a necessary step in order to create an approach which was tractable on large biological networks. This static analysis was then combined withASP, in order to obtain an enumeration of the existing bifurcations. While their method was able to identify correct bifurcations only (no false positives), the nature of the approximation process made it so that the approach was incomplete (in other words, false negatives could exist).
3.2.3 Model Revision
Model revision has also seen several innovations with the employment ofASP, especially as far as automation is concerned. However, as we will see, not all of these approaches work with the Boolean logical model formalism, and the ones that do present shortcom-ings that do not allow for optimal, or even accurate model repairs in some cases.
Martin Gebser et al. addressed the problem of repairing large-scale biological net-works in order to predict unobserved variations [23]. To do so, they made use ofASPto define a program capable of performing a range of different operations in biological net-works, in order to re-establish consistency. Their approach built upon theSCMformalism, and consisted of defining a set of rules that would alter the model’s configuration until an acceptable version was found. Some of the ways these rules attempt to re-establish con-sistency was by introducing additional edges between vertices, treating certain vertices as input vertices, and flipping the sign of edges. Similarly, Lemos et al. developed a tool in C++(which also encapsulatedclingo) to suggest repairs over inconsistent Boolean models, based on a set of atomic repair operations [44]. These operations consisted of changing a regulator from inhibitor to activator and vice-versa, changing a Boolean operator in a node’s function from an AND to an OR and vice-versa, and removing all occurrences of a given regulator from a node’s function. The tool receives a model in theDNFformat and one or more time-series as matrices, and allows for the specification of the preferred up-dating scheme (stable state, asynchronous or synchronous). After executing, it provides the minimal repairs that render the model capable of generating dynamics coherent with the available time-series data. However, this approach does not take into account how changing the regulatory function of a node may impact its interactions with other nodes