BACKGROUND: Protein-protein interaction networks (PPINs) provide a comprehensive view of the intricate biochemical processes that take place in living organisms. In recent years, the size and information content of PPINs have grown thanks to techniques that allow for the functional association of proteins. However, PPINs are static objects that cannot fully describe the dynamics of the protein interactions; these dynamics are usually studied from external sources and can only be added to the PPIN as annotations. In contrast, the time-dependent characteristics of cellular processes are described in Biochemical Pathways (BP), which frame complex networks of chemical reactions as dynamical systems. Their analysis with numerical simulations allows for the study of different dynamical properties. Unfortunately, available BPs cover only a small portion of the interactome, and simulations are often hampered by the unavailability of kinetic parameters or by their computational cost. In this study, we explore the possibility of enriching PPINs with dynamical properties computed from BPs. We focus on the global dynamical property of sensitivity, which measures how a change in the concentration of an input molecular species influences the concentration of an output molecular species at the steady state of the dynamical system. RESULTS: We started with the analysis of BPs via ODE simulations, which enabled us to compute the sensitivity associated with multiple pairs of chemical species. The sensitivity information was then injected into a PPIN, using public ontologies (BioGRID, UniPROT) to map entities at the BP level with nodes at the PPIN level. The resulting annotated PPIN, termed the DyPPIN (Dynamics of PPIN) dataset, was used to train a DGN to predict the sensitivity relationships among PPIN proteins. Our experimental results show that this model can predict these relationships effectively under different use case scenarios. Furthermore, we show that the PPIN structure (i.e., the way the PPIN is "wired") is essential to infer the sensitivity, and that further annotating the PPIN nodes with protein sequence embeddings improves the predictive accuracy. CONCLUSION: To the best of our knowledge, the model proposed in this study is the first that allows performing sensitivity analysis directly on PPINs. Our findings suggest that, despite the high level of abstraction, the structure of the PPIN holds enough information to infer dynamic properties without needing an exact model of the underlying processes. In addition, the designed pipeline is flexible and can be easily integrated into drug design, repurposing, and personalized medicine processes.

Abstract Reaction Systems (RSs) are a qualitative model inspired by biochemical processes, where the dynamics of complex systems is modelled by a collection of local reactions. Each reaction comprises a set of reactants that triggers a set of products unless hindered by the presence of some inhibitors. The use of inhibitors introduces non-monotonic behaviours that are difficult to analyze. This work focuses on the explainability of local phenomena, like the production of certain products or the reachability of certain attractors, by separating the causes responsible for reaching them from the irrelevant elements of a possibly much larger, global statespace. The main novelty of our approach is the ability to derive sufficient conditions that combine positive dependencies (e.g., requesting the presence of some entities at a certain stage, as already done in the literature) with negative ones (e.g., requesting the absence of some entities). This is achieved by combining and extending previous “static” constructions, like the transformation to Positive RSs and the minimization of RSs with “dynamic” techniques, like the process algebraic evolution of RSs, the slicing of computation and the on-the-fly generation of negative dependencies. We compare many different combinations of the above approaches, discussing their respective benefits and trade-offs in order to identify the most convenient analysis. We demonstrate our methodology on a case study involving T cell protein interactions, showing how it can reveal critical stimulus combinations and pinpoint potential drug targets by explaining phenotype emergence. Our analysis offers new insights and greater explanatory power than existing approaches.

Antibodies are indispensable components of the immune system, known for their specific binding to antigens. Beyond their natural immunological functions, they are fundamental in developing vaccines and therapeutic interventions for infectious diseases. The complex architecture of antibodies, particularly their variable regions responsible for antigen recognition, presents significant challenges for computational modeling. Recent advancements in deep learning have markedly improved protein structure prediction; however, accurately modeling antibody-antigen (Ab-Ag) interactions remains challenging due to the inherent flexibility of antibodies and the dynamic nature of binding processes. In this study, we examine the use of predicted Local Distance Difference Test (pLDDT) scores as indicators of residue and side-chain flexibility to model Ab-Ag interactions through a fingerprint-based approach. We demonstrate the significance of flexibility in different antibody-specific tasks, enhancing the predictive accuracy of Ab-Ag interaction models by 4%, resulting in an AUC-ROC of 92%. In addition, we showcase state-of-the-art performance in paratope prediction. These results emphasize the importance of accounting for conformational flexibility in modeling antibody-antigen interactions and show that pLDDT can serve as a coarse proxy for these dynamic features. By optimizing antibody flexibility using pLDDT, they can be engineered to improve affinity or breadth for a specific target. This approach is particularly beneficial for addressing highly variable pathogens like HIV and SARS-CoV-2, as greater flexibility enhances tolerance to sequence variations in target antigens.

Deep learning has achieved impressive results in various fields such as computer vision and natural language processing, making it a powerful tool in biology. Its applications now encompass cellular image classification, genomic studies and drug discovery. While drug development traditionally focused deep learning applications on small molecules, recent innovations have incorporated it in the discovery and development of biological molecules, particularly antibodies. Researchers have devised novel techniques to streamline antibody development, combining in vitro and in silico methods. In particular, computational power expedites lead candidate generation, scaling and potential antibody development against complex antigens. This survey highlights significant advancements in protein design and optimization, specifically focusing on antibodies. This includes various aspects such as design, folding, antibody-antigen interactions docking and affinity maturation.

Abstract This paper forges a strong connection between two well known computational frameworks for representing biological systems, in order to facilitate the seamless transfer of techniques between them. Boolean networks are a well established formalism employed from biologists. They have been studied under different (synchronous and asynchronous) update semantics, enabling the observation and characterisation of distinct facets of system behaviour. Recently, a new semantics for Boolean networks has been proposed, called most permissive semantics, that enables a more faithful representation of biological phenomena. Reaction systems offer a streamlined formalism inspired by biochemical reactions in living cells. Reaction systems support a full range of analysis techniques that can help for gaining deeper insights into the underlying biological phenomena. Our goal is to leverage the available toolkit for predicting and comprehending the behaviour of reaction systems within the realm of Boolean networks. In this paper, we first extend the behaviour of reaction systems to several asynchronous semantics, including the most permissive one, and then we demonstrate that Boolean networks and reaction systems exhibit isomorphic behaviours under the synchronous, general/fully asynchronous and most permissive semantics.

Abstract Cause/effect analysis of complex systems is instrumental in better understanding many natural phenomena. Moreover, formal analysis requires the availability of suitable abstract computational models that somehow preserve the features of interest. Our contribution focuses on the analysis of Reaction Systems (RSs), a qualitative computational formalism inspired by biochemical reactions in living cells. The primary challenge lies in dealing with inhibition mechanisms. On the one hand, inhibitors enhance the expressiveness of the computational abstraction; on the other hand, they can introduce nonmonotonic behaviors that can be computationally hard to deal with in the analysis. We propose an encoding of RSs into an equivalent formulation without inhibitors (called Positive RSs, PRSs for short) that is easier to handle, because PRSs exhibit monotonic behaviors. The effectiveness of our transformation is witnessed by its impact on two different techniques for cause/effect analysis. The first, called slicing, allows detecting the causes of some unforeseen phenomenon by reasoning backward along a given computation. Here, PRSs can be exploited to improve the quality of the analysis. The second technique, predictor analysis, is addressed by introducing a novel tool called MuMa, which is based on must/maybe sets, whence the tool name, an original abstraction for approximating ancestor formulas. MuMa exploits PRSs to improve the performance of the analysis.

MOTIVATION: Dynamical properties of biochemical pathways (BPs) help in understanding the functioning of living cells. Their in silico assessment requires simulating a dynamical system with a large number of parameters such as kinetic constants and species concentrations. Such simulations are based on numerical methods that can be time-expensive for large BPs. Moreover, parameters are often unknown and need to be estimated. RESULTS: We developed a framework for the prediction of dynamical properties of BPs directly from the structure of their graph representation. We represent BPs as Petri nets, which can be automatically generated, for instance, from standard SBML representations. The core of the framework is a neural network for graphs that extracts relevant information directly from the Petri net structure and exploits them to learn the association with the desired dynamical property. We show experimentally that the proposed approach reliably predicts a range of diverse dynamical properties (robustness, monotonicity, and sensitivity) while being faster than numerical methods at prediction time. In synergy with the neural network models, we propose a methodology based on Petri nets arc knock-out that allows the role of each molecule in the occurrence of a certain dynamical property to be better elucidated. The methodology also provides insights useful for interpreting the predictions made by the model. The results support the conjecture often considered in the context of systems biology that the BP structure plays a primary role in the assessment of its dynamical properties. AVAILABILITY AND IMPLEMENTATION: https://github.com/marcopodda/petri-bio (code), https://zenodo.org/record/7610382 (data).

Abstract Reaction systems (RSs) are a successful natural computing framework inspired by chemical reaction networks. A RS consists of a set of entities and a set of reactions. Entities can enable or inhibit each reaction and are produced by reactions or provided by the environment. In this paper, we define two quantitative variants of RSs: the first one is along the time dimension, to specify delays for making available reactions products and durations to protract their permanency, while the second deals with the possibility to specify different concentration levels of a substance in order to enable or inhibit a reaction. Technically, both extensions are obtained by modifying in a modular way the Structural Operational Semantics (SOS) for RSs that was already defined in the literature. Our approach maintains several advantages of the original semantics definition that were: (1) providing a formal specification of the RS dynamics that enables the reuse of many formal analysis techniques and favours the implementation of tools, and (2) making the RS framework extensible, by adding or changing some of the SOS rules in a compositional way. We provide a prototype logic programming implementation and apply our tool to three different case studies: the tumour growth, the Th cell differentiation in the immune system and neural communication.

Gene regulatory networks represent the interactions among genes regulating the activation of specific cell functionalities and they have been successfully modeled using threshold Boolean networks. In this paper we propose a systematic translation of threshold Boolean networks into reaction systems. Our translation produces a non redundant set of rules with a minimal number of objects. This translation allows us to simulate the behavior of a Boolean network simply by executing the (closed) reaction system we obtain. This can be very useful for investigating the role of different genes simply by “playing” with the rules. We developed a tool able to systematically translate a threshold Boolean network into a reaction system. We use our tool to translate two well known Boolean networks modelling biological systems: the yeast-cell cycle and the SOS response in Escherichia coli. The resulting reaction systems can be used for investigating dynamic causalities among genes.

Abstract In reaction systems, preimages and nth ancestors are sets of reactants leading to the production of a target set of products in either 1 or n steps, respectively. Many computational problems on preimages and ancestors, such as finding all minimum-cardinality nth ancestors, computing their size or counting them, are intractable. In this paper, we characterize all nth ancestors using a Boolean formula that can be computed in polynomial time. Once simplified, this formula can be exploited to easily solve all preimage and ancestor problems. This allows us to directly relate the difficulty of ancestor problems to the cost of the simplification so that new insights into computational complexity investigations can be achieved. In particular, we focus on two problems: (i) deciding whether a preimage/nth ancestor exists and (ii) finding a preimage/nth ancestor of minimal size. Our approach is constructive, it aims at finding classes of reactions systems for which the ancestor problems can be solved in polynomial time, in exact or approximate way.

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions. They describe the evolution of sets of objects representing biochemical molecules. One of the main characteristics of Reaction systems is the non-permanency of the objects, namely objects disappear if not pr oduced by any enabled reaction. Reaction systems execute in an environment that provides new objects at each step. Causality properties of reaction systems can be studied by using notions of formula based predictor. In this context, we define a notion of opacity that can be used to study information flow properties for reaction systems. Objects will be partitioned into high level (invisible) and low level (visible) ones. Opacity ensures that the presence (or absence) of high level objects cannot be guessed observing the low level objects only. Such a property is shown to be decidable and computable by exploiting the algorithms for minimal formula based predictors.

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozemberg introduced the idea of predictors. A predictor of a molecule s, for a given n, is the set of molecules to be observed in the environment to determine whether s is produced or not at step n by the system. We introduced the notion of formula based predictor, that is a propositional logic formula that precisely characterizes environments that lead to the production of s after n steps. In this paper we revise the notion of formula based predictor by defining a specialized version that assumes the environment to provide molecules according to what expressed by a temporal logic formula. As an application, we use specialized formula based predictors to give theoretical grounds to previously obtained results on a model of gene regulation.

Property preservation is investigated as an approach to modular verification, leading to reduction of the property verification time for formal models. For modelling purposes, formalisms with multi-way synchronisations are considered. For the modular verification technique to work, a specific type of synchronisation is required for which a sufficient and necessary condition is identified. It is a requirement on the semantics of the formalism, which is restricted to permit simultaneous execution only of component moves that make reference to each other. Implications for modular verification of several well-known formalisms for concurrent systems are investigated.

Maximally parallel semantics have been proposed for many formalisms as an alternative to the standard interleaving semantics for some modelling scenarios. Nevertheless, in the probabilistic setting an affirmed interpretation of maximal parallelism st

A formal model for diagnostics of biological systems modelled as P systems is presented. We assume the presence of some biologically motivated changes (frequently pathological) in the systems behavior and investigate when these changes could be diagnosed by an external observer by exploiting some techniques originally developed for reasoning on system security.

The Calculus of Looping Sequences is a formalism for describing evolution of biological systems by means of term rewriting rules. We propose to enrich this calculus by labelling elements of sequences. Since two elements with the same label are consid

P Systems are computing devices inspired by the structure and the functioning of a living cell. A P System consists of a hierarchy of membranes, each of them containing a multiset of objects, a set of evolution rules, and possibly other membranes. Evolution rules are applied to the objects of the same membrane with maximal parallelism. In this paper we present an extension of P Systems, called P Systems with Membrane Channels (PMC Systems), in which membranes are enriched with channels and objects can pass through a membrane only if there are channels on the membrane that enable such a passage. We show that PMC Systems are universal even if only the simplest form of evolution rules is considered, and we give two application examples.

To study systems whose dynamics changes with time, an extension of timed P systems is introduced in which evolution rules may vary with time. The proposed model is a timed automaton with a discrete time domain and in which each state is a timed P system. A result on expressive power and on features of the formalism sufficient for full expressiveness is proved and, as an application example, the model of an ecological system is given.

Abstract Bisimulations are well-established behavioural equivalences that are widely used to study properties of computer science systems. Bisimulations assume the behaviour of systems to be described as labelled transition systems, and properties of a system can be verified by assessing its bisimilarity with a system one knows to enjoy those properties. In this paper we show how semantics based on labelled transition systems and bisimulations can be defined for two formalisms for the description of biological systems, both capable of describing membrane interactions. These two formalisms are the Calculus of Looping Sequences (CLS) and Brane Calculi, and since they stem from two different approaches (rewrite systems and process calculi) bisimulation appears to be a good candidate as a general verification method. We introduce CLS and define a labelled semantics and bisimulations for which we prove some congruence results. We show how bisimulations can be used to verify properties by way of two examples: the description of the regulation of lactose degradation in Escherichia coli and the description of the EGF signalling pathway. We recall the PEP calculus (the simplest of Brane Calculi) and its translation into CLS, we define a labelled semantics and some bisimulation congruences for PEP processes, and we prove that bisimilar PEP processes are translated into bisimilar CLS terms.