This report proposes a novel framework for a rigorous robustness analysis of stochastic biochemical systems. The technique is based on probabilistic model checking. We adapt the general definition of robustness introduced by Kitano to the class of stochastic systems modelled as continuous time Markov Chains in order to extensively analyse and compare robustness of biological models with uncertain parameters. The framework utilises novel computational methods that enable to effectively evaluate the robustness of models with respect to quantitative temporal properties and parameters such as reaction rate constants and initial conditions. The framework is applied to gene regulation as an example of a central biological mechanism where intrinsic and extrinsic stochasticity plays crucial role due to low numbers of DNA and RNA molecules. Using our methods we have obtained a comprehensive and precise analysis of stochastic dynamics under parameter uncertainty. Furthermore, we apply our framework to compare several variants of two-component signalling networks from the perspective of robustness with respect to intrinsic noise caused by low populations of signalling components. We succeeded to extend previous studies performed on deterministic models (ODE) and show that stochasticity may significantly affect obtained predictions. Our case studies demonstrate that the framework can provide deeper insight into the role of key parameters in maintaining the system functionality and thus it significantly contributes to formal methods in computational systems biology.
In the domain of ODE models, there exist several analytic methods for effective analysis under parameter uncertainty. They build on static analysis (stoichiometric analysis, flux balance analysis) as well as dynamic numerical methods (simulation, monitoring by temporal formulae, sensitivity analysis) implemented in tools (e.g. @cite_20 @cite_30 @cite_7 ). Robustness analysis with respect to functionality specified in terms of temporal formulae has been introduced recently @cite_40 @cite_12 . There exist two major approaches how to define and analyse robustness. If only parameters of the model are perturbed, we speak of a to robustness. This approach has been explored by Fainekos & Pappas @cite_40 , further extended by A. Donz ' e et al @cite_10 and implemented in the toolbox Breach @cite_32 . Another option could be to perturb the model structure i.e. the reaction topology, as this is done in many gene knock-out biological experiments. Such changes are in principle discrete and the problem of robustness computation for such perturbations would reduce to solving many individual instances of the same problem for each discrete topology. However identifying model behaviour shared among individual perturbations can lead to more efficient analysis @cite_22 .
@cite_30 @cite_22 @cite_7 @cite_32 @cite_40 @cite_20 @cite_10 @cite_12
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Abstract Stochastic models such as Continuous-Time Markov Chains (CTMC) and Stochastic Hybrid Automata (SHA) are powerful formalisms to model and to reason about the dynamics of biological systems, due to their ability to capture the stochasticity inherent in biological processes. A classical question in formal modelling with clear relevance to biological modelling is the model checking problem, i.e. calculate the probability that a behaviour, expressed for instance in terms of a certain temporal logic formula, may occur in a given stochastic process. However, one may not only be interested in the notion of satisfiability, but also in the capacity of a system to maintain a particular emergent behaviour unaffected by the perturbations, caused e.g. from extrinsic noise, or by possible small changes in the model parameters. To address this issue, researchers from the verification community have recently proposed several notions of robustness for temporal logic providing suitable definitions of distance between a trajectory of a (deterministic) dynamical system and the boundaries of the set of trajectories satisfying the property of interest. The contributions of this paper are twofold. First, we extend the notion of robustness to stochastic systems, showing that this naturally leads to a distribution of robustness degrees. By discussing three examples, we show how to approximate the distribution of the robustness degree and the average robustness . Secondly, we show how to exploit this notion to address the system design problem , where the goal is to optimise some control parameters of a stochastic model in order to maximise robustness of the desired specifications. Abstract This paper describes a new approach for ascertaining the stability of stochastic dynamical systems in their parameter space by examining their time series using topological data analysis (TDA). We illustrate the approach using a nonlinear delayed model that describes the tool oscillations due to self-excited vibrations in turning. Each time series is generated using the Euler-Maruyama method and a corresponding point cloud is obtained using the Takens embedding. The point cloud can then be analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data. The contributions of this paper include presenting for the first time a topological approach for investigating the stability of a class of nonlinear stochastic delay equations, and introducing a new application of TDA to machining processes. Traditionally, traffic assignment models, both for within-day static and dynamic demand, have been formulated following an equilibrium approach in which a state ensuring internal consistency between demand (flows) and costs is sought. However, equilibrium analysis is significant under some assumptions on its “representativeness” (coincidence or closeness with the actual attractor of the system) and analytical properties, such as existence, uniqueness, and stability. Moreover, transients due to modifications of demand and or supply cannot be simulated through equilibrium models, nor can a statistical description of the state of the system, i.e. means, modes, moments and, more generally, frequency distributions of flows over time be obtained. In this paper, interperiodic (day-to-day) dynamic modeling of transportation networks is addressed following two different approaches, namely deterministic and stochastic processes. In both cases several theoretical results are shown by making use of a formal framework covering most models discussed in the literature as well as some possible extensions. Most of the results reported can be extended to cover within-day dynamic models but these models are not explicitly dealt with. Within the framework of deterministic processes the relevance of day-to-day dynamic models for demand supply interaction in comparison with the traditional user equilibrium approach is discussed, and conditions for coincidence of fixed-point attractors and equilibrium states are stated. Conditions for existence and uniqueness of fixed-point attractors are proposed, generalizing and extending those presented in the literature for user equilibrium. Conditions for stability of both fixed-points and equilibrium states were formulated by making use of results from non-linear dynamic system theory. Moreover, it is possible to devise a new family of “dynamic” algorithms which simulate the system convergence to a fixed-point in order to obtain an equivalent equilibrium state, as opposed to conventional “optimisation” algorithms. In this case the fixed-point stability analysis can be viewed as a convergence analysis for the algorithms specified this way. Conditions for stochastic process regularity are proposed ensuring, among other things, existence and uniqueness of a stationary probability distribution of system states. These conditions generalize and extend results presented in the literature to a wider class of possible dynamic models. Relationships between a deterministic process, together with corresponding fixed-points or equilibrium states, and stochastic probability distribution are also briefly addressed. Finally, some numerical examples confirming theoretical results are reported for a small test network. We consider temporal logic formulae specifying constraints in continuous time and space on the behaviors of continuous and hybrid dynamical system admitting uncertain parameters. We present several variants of robustness measures that indicate how far a given trajectory stands, in space and time, from satisfying or violating a property. We present a method to compute these robustness measures as well as their sensitivity to the parameters of the system or parameters appearing in the formula. Combined with an appropriate strategy for exploring the parameter space, this technique can be used to guide simulation-based verification of complex nonlinear and hybrid systems against temporal properties. Our methodology can be used for other non-traditional applications of temporal logic such as characterizing subsets of the parameter space for which a system is guaranteed to satisfy a formula with a desired robustness degree.