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Temporal logics and model-checking techniques have proved successful to respectively express biological properties of complex bio-chemical systems, and automatically verify their satisfaction in both qualitative and quantitative models. In this paper, we propose a finite time horizon model-checking algorithm for the existential fragment of LTL with...
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Citations
... Model checking can be used to determine whether the behaviour of a model conforms to some desired properties specified in temporal logic. We use here Probabilistic Linear-time Temporal Logic with numerical constraints (PLTLc) [49], based on Linear-time Temporal Logic (LTL) [50], extended with probabilities [51] and numerical constraints over real value variables [52]. Several features of PLTLc facilitate the expression of the behaviour of biochemical networks, including the ability to express properties relative to an absolute time value or range, the use of functions which compare the concentration of a protein to its peak value, and the derivative function enabling the description of transient, sustained or oscillatory behaviour. ...
Background
Quorum sensing drives biofilm formation in bacteria in order to ensure that biofilm formation only occurs when colonies are of a sufficient size and density. This spatial behaviour is achieved by the broadcast communication of an autoinducer in a diffusion scenario. This is of interest, for example, when considering the role of gut microbiota in gut health. This behaviour occurs within the context of the four phases of bacterial growth, specifically in the exponential stage (phase 2) for autoinducer production and the stationary stage (phase 3) for biofilm formation.
Results
We have used coloured hybrid Petri nets to step-wise develop a flexible computational model for E.coli biofilm formation driven by Autoinducer 2 (AI-2) which is easy to configure for different notions of space. The model describes the essential components of gene transcription, signal transduction, extra and intra cellular transport, as well as the two-phase nature of the system. We build on a previously published non-spatial stochastic Petri net model of AI-2 production, keeping the assumptions of a limited nutritional environment, and our spatial hybrid Petri net model of biofilm formation, first presented at the NETTAB 2017 workshop. First we consider the two models separately without space, and then combined, and finally we add space. We describe in detail our step-wise model development and validation.
Our simulation results support the expected behaviour that biofilm formation is increased in areas of higher bacterial colony size and density. Our analysis techniques include behaviour checking based on linear time temporal logic.
Conclusions
The advantages of our modelling and analysis approach are the description of quorum sensing and associated biofilm formation over two phases of bacterial growth, taking into account bacterial spatial distribution using a flexible and easy to maintain computational model. All computational results are reproducible.
... 7 Although each of these approaches checks whether a model's behavior satisfies a certain requirement, the accessibility of these requirements varies, from being elusive (e.g., face validity) and difficult to retrieve as they depend on interactive explorations (e.g., visual analytics), 50 through being implicitly part of the method (e.g., sensitivity analysis), up to being explicitly and declaratively specified as properties in a formal language, as in model checking (e.g., in the work by Fages and Rizk). 51 In modeling and simulation, experimental model validation evaluates model validity through experimenting with the model, in which the model context is reflected by experimental configuration. 52 If the simulation experiments that are executed for validating models shall not only be annotated with models, but also be reused, we need a declarative and accessible description of these experiments. ...
... For deterministic modeling formalisms such as ordinary differential equations (ODEs), finite time modelchecking algorithms can be used to check the simulation trajectory against a defined property, e.g., in the work by Fages and Rizk. 51 For stochastic models, multiple replications are required and statistical model checking methods are usually employed to estimate the probability that an individual trajectory produced from a random simulation run satisfies the property, 58 as the approach used in our previous work. 5 The validation of models in our case study relies on checking simulation results against properties that are derived from observations in wet-lab experiments with human neural progenitor cells. ...
With the increasing size and complexity of models, developing models by composing existing ones becomes more important. We exploit the idea of reusing simulation experiments of individual models for composition to automatically generate experiments for the composed model. First, we illustrate the process of modeling based on composition and discuss the role simulation experiments can play in this process. Our focus is on semantic validation of the composed model. We explicitly specify simulation experiments in simulation experiment specification via a Scala layer, including the desired model behavioral properties and their required experiment set-ups. Models are annotated with experiment specifications, and upon composition, these specifications are adapted and automatically executed for the composed model. The approach is applied in a case study of developing a Wnt/β-catenin signaling pathway model by successively composing three individual models, where we exploit metric interval temporal logic to describe model behavioral properties and check averages of stochastic simulation results against these properties.
... Linear Temporal Logic (LTL) is widely used to check the properties of individual trajectories [10,47], and thus allows to express a broad range of dynamic model properties. ...
... To check an output trajectory π, we rely on a JAMES IIbased reimplementation of the model-checking algorithm introduced by Fages et al. [10]. The original algorithm captures the following operators of LTL: X (next), G (global), F (finally), and U (until). ...
... It is implemented on top of JAMES II as well. So far, this layer provides a re-implementation of the model checking approach followed in [10], as well as support for custom predicates. We consider the integration of additional model checking approaches to be future work. ...
Unambiguous experiment descriptions are increasingly required for model publication, as they contain information important for reproducing simulation results. In the context of model composition, this information can be used to generate experiments for the composed model. If the original experiment descriptions specify which model property they refer to, we can then execute the generated experiments and assess the validity of the composed model by evaluating their results. Thereby, we move the attention to describing properties of a model's behavior and the conditions under which these hold, i.e., its semantics. We illuminate the potential of this concept by considering the composition of Lotka-Volterra models. In a first prototype realized for JAMES II, we use ML-Rules to describe and execute the Lotka-Volterra models and SESSL for specifying the original experiments. Model properties are described in continuous stochastic logic, and we use statistical model checking for their evaluation. Based on this, experiments to check whether these properties hold for the composed model are automatically generated and executed.
... The integration of numeric functions and simple arithmetic expressions into temporal logic is not new and has, for example, been investigated in the context of LTL with constraints (LTLc) [7]. Its usefulness for the verification of complex quantitative and qualitative properties involving external data in the domain of biochemical systems has been shown by Fages and Rizk [8]. We concentrate here on a subset of PLTLc (the probabilistic extension of LTLc) which is restricted to bound variables and thus releases the model checker from having to solve constraint satisfaction problems. ...
... By 'recovering' we mean returning to a state in which at most 1% are infected.8 For example, a path could be labelled with multiple atomic propositions in a single iteration. ...
This paper focusses on the usefulness of approximate probabilistic model checking for the verification and validation (V&V) of large-scale agent-based simulations. We describe the translation of typical V&V questions into a variant of linear time logic as well as the formulation of properties that involve
external data. We further present a prototypical version of a highly customisable approximate model checker which we used in a range of experiments to verify properties of large scale models whose complexity prevents them from being amenable to conventional explicit or symbolic model checking.
... The integration of numeric functions and simple arithmetic expressions into temporal logic is not new and has, for example, been investigated in the context of LTL with constraints (LTLc) [9]. Its usefulness for the verification of complex quantitative and qualitative properties involving external data in the domain of biochemical systems has been shown by Fages and Rizk [10]. We concentrate here on a subset of PLTLc (the probabilistic extension of LTLc) which is restricted to bound variables and thus releases the model checker from having to solve constraint satisfaction problems. ...
This paper focusses on the usefulness of approximate probabilistic model checking for the internal and external validation of large-scale agent-based simulations. We describe the translation of typical validation criteria into a variant of linear time logic. We further present a prototypical version of a highly customisable approximate model checker
which we used in a range of experiments to verify properties of large scale models whose complexity prevents them from being amenable to conventional explicit or symbolic model checking.
... Another novel and promising technique integrates model-checking and trace analysis and can be applied even in the continuous domain to the results of numerical integrators. Fages [23], Donaldson [24] and others describe model checkers which inspect simulation outputs and evaluate quantified logical formulae over a single simulation trace (in contrast to the statistical model-checking approach, which requires an ensemble of traces generated from Monte Carlo simulations conducted in the discrete molecular regime). ...
Formal modeling approaches such as process algebras and Petri nets seek to provide insight into biological processes by using both symbolic and numerical methods to reveal the dynamics of the process under study. These formal approaches differ from classical methods of investigating the dynamics of the process through numerical integration of ODEs because they additionally provide alternative representations which are amenable to discrete-state analysis and logical reasoning. Backed by these additional analysis methods, formal modeling approaches have been able to identify errors in published and widely-cited biological models. This paper provides an introduction to these analysis methods, and explains the benefits which they can bring to ensuring the consistency of biological models.
... As might be expected, the exact inferencing problem for large CTMCs is also computationally infeasible. Analysis methods based on Monte Carlo simulations [23], [24], [21], [25], probabilistic model checking [15], [26] as well as numerically solving the Chemical Master Equation describing a CTMC [20], [27] are being developed. In these studies, the CTMCs are presented implicitly but the analysis is carried out on the full state space of the CTMCs. ...
... In our future work, an important goal will be to deploy HFF to perform tasks such as parameter estimation and sensitivity analysis. An equally important goal will be to develop approximate probabilistic verification methods for DBN models of biochemical networks and evaluate them with respect to approaches developed in related settings [23], [21], [24], [25]. ...
Dynamic Bayesian Networks (DBNs) can serve as succinct probabilistic dynamic models of biochemical networks. To analyze these models, one must compute the probability distribution over system states at a given time point. Doing this exactly is infeasible for large models; hence one must use approximate algorithms. The Factored Frontier algorithm (FF) is one such algorithm. However FF as well as the earlier Boyen-Koller (BK) algorithm can incur large errors.To address this, we present a new approximate algorithm called the Hybrid Factored Frontier (HFF) algorithm. At each time slice, in addition to maintaining probability distributions over local states -as FF does- HFF explicitly maintains the probabilities of a number of global states called spikes. When the number of spikes is 0, we get FF and with all global states as spikes, we get the exact inference algorithm. We show that by increasing the number of spikes one can reduce errors while the additional computational effort required is only quadratic in the number of spikes. We validated the performance of HFF on large DBN models of biopathways. Each pathway has more than 30 species and the corresponding DBN has more than 3000 nodes. Comparisons with FF and BK show that HFF is a useful and powerful approximate inferencing algorithm for DBNs.
... In Monte Carlo-based analysis, one must stochastically generate trajectories and check whether they pass a statistical test. This will, however, entail storing a good deal of information generated by the individual trajectories and multiple passes through this information in cases where the statistical test is based on a temporal property (Donaldson and Gilbert, 2008;Fages and Rizk, 2007). In these settings our mapping techniques will lead to powerful GPU implementations. ...
Biopathways are often modeled as systems of ordinary differential equations (ODEs). Such systems will usually have many unknown parameters and hence will be difficult to calibrate. Since the data available for calibration will have limited precision, an approximate representation of the ODEs dynamics should suffice. One must, however, be able to efficiently construct such approximations for large models and perform model calibration and subsequent analysis.
We present a graphical processing unit (GPU) based scheme by which a system of ODEs is approximated as a dynamic Bayesian network (DBN). We then construct a model checking procedure for DBNs based on a simple probabilistic linear time temporal logic. The GPU implementation considerably extends the reach of our previous PC-cluster-based implementation (Liu et al., 2011b). Further, the key components of our algorithm can serve as the GPU kernel for other Monte Carlo simulations-based analysis of biopathway dynamics. Similarly, our model checking framework is a generic one and can be applied in other systems biology settings. We have tested our methods on three ODE models of bio-pathways: the epidermal growth factor-nerve growth factor pathway, the segmentation clock network and the MLC-phosphorylation pathway models. The GPU implementation shows significant gains in performance and scalability whereas the model checking framework turns out to be convenient and efficient for specifying and verifying interesting pathways properties.
The source code is freely available at http://www.comp.nus.edu.sg/~rpsysbio/pada-gpu/
... As might be expected, the exact inferencing problem for large CTMCs is computationally infeasible. Analysis methods based on Monte Carlo simulations [8,9,12,14] as well as numerically solving the Chemical Master Equation describing a CTMC [7,13] are being developed. In these studies the CTMCs are presented implicitly while our DBNs are available explicitly. ...
... In terms of future work, an important goal will be to deploy HFF to perform tasks such as parameter estimation and sensitivity analysis. An equally important goal will be to develop approximate probabilistic verification methods for DBN models of biochemical networks and evaluate them with respect to approaches developed in related settings [8,9,12,14]. ...
Dynamic Bayesian Networks (DBNs) can serve as succinct models of large biochemical networks [19]. To analyze these models, one must compute the probability distribution over system states at a given time point. Doing this exactly is infeasible for large models and hence approximate methods are needed. The Factored Frontier algorithm (FF) is a simple and efficient approximate algorithm [25] that has been designed to meet this need. However the errors it incurs can be quite large. The earlier Boyen-Koller (BK) algorithm [3] can also incur significant errors.
To address this, we present here a novel approximation algorithm called the Hybrid Factored Frontier (HFF) algorithm. HFF may be viewed as a parametrized version of FF. At each time slice, in addition to maintaining probability distributions over local states -as FF does- we also maintain explicitly the probabilities of a small number of global states called spikes. When the number of spikes is 0, we get FF and with all global states as spikes, we get the - computationally infeasible- exact inference algorithm. We show that by increasing the number of spikes one can reduce errors while the additional computational effort required is only quadratic in the number of spikes. We have validated the performance of our algorithm on large DBN models of biopathways. Each pathway has more than 30 species and the corresponding DBN has more than 3000 nodes. Comparisons with the performances of FF and BK show that HFF can be a useful and powerful approximation algorithm for analyzing DBN models of biopathways.
... Post-hoc model-checkers complementary to this work are BioCham (Fages and Rizk 2007), BioNessie (Liu and Gilbert 2010) and MC 2 (Heiner, Gilbert, and Donaldson 2008), in that they can be applied to a continuous interpretation of the model (and in some cases to discrete-state simulations also). In contrast we are working exclusively with a discrete interpretation here. ...
Simulation modeling in systems biology embarks on discrete event simulation only for cases of small cardinalities of entities and uses continuous simulation otherwise. Modern modeling environments like Bio-PEPA support both types of simulation within a single modeling formalism. Developing models for complex dynamic phenomena is not trivial in practice and requires careful verification and testing. In this paper, we describe relevant steps in the verification and testing of a TNFα-mediated NF-κB signal transduction pathway model and discuss to what extent automated techniques help a practitioner to derive a suitable model.
