Computational Methods in Systems Biology

We address the sequential reprogramming of gene regulatory networks modelled as Boolean networks. We develop an attractor-based sequential reprogramming method to compute all sequential reprogramming paths from a source attractor to a target attractor, where only attractors of the network are used as intermediates. Our method is more practical than existing reprogramming methods as it incorporates several practical constraints: (1) only biologically observable states, viz. attractors, can act as intermediates; (2) certain attractors, such as apoptosis, can be avoided as intermediates; (3) certain nodes can be avoided to perturb as they may be essential for cell survival or difficult to perturb with biomolecular techniques; and (4) given a threshold k, all sequential reprogramming paths with no more than k perturbations are computed. We compare our method with the minimal one-step reprogramming and the minimal sequential reprogramming on a variety of biological networks. The results show that our method can greatly reduce the number of perturbations compared to the one-step reprogramming, while having comparable results with the minimal sequential reprogramming. Moreover, our implementation is scalable for networks of more than 60 nodes.

A major challenge in precision medicine consists in finding the appropriate network rewiring to induce a particular reprogramming of the cell phenotype. The rewiring is caused by specific network action either inhibiting or over-expressing targeted molecules. In some cases, a therapy abides by a time-scheduled drug administration protocol. Furthermore, some diseases are induced by a sequence of mutations leading to a sequence of actions on molecules. In this paper, we extend previous works on abductive-based inference of network reprogramming [3] by investigating the sequential control of Boolean networks. We present a novel theoretical framework and give an upper bound on the size of control sequences as a function of the number of observed variables. We also define an algorithm for inferring minimal parsimonious control sequences allowing to reach a final state satisfying a particular phenotypic property.

Stochastic simulation is a widely used method for estimating quantities in models of chemical reaction networks where uncertainty plays a crucial role. However, reducing the statistical uncertainty of the corresponding estimators requires the generation of a large number of simulation runs, which is computationally expensive. To reduce the number of necessary runs, we propose a variance reduction technique based on control variates. We exploit constraints on the statistical moments of the stochastic process to reduce the estimators’ variances. We develop an algorithm that selects appropriate control variates in an on-line fashion and demonstrate the efficiency of our approach on several case studies.

At the scale of the individual cell, protein production is a stochastic process with multiple time scales, combining quick and slow random steps with discontinuous and smooth variation. Hybrid stochastic processes, in particular piecewise-deterministic Markov processes (PDMP), are well adapted for describing such situations. PDMPs approximate the jump Markov processes traditionally used as models for stochastic chemical reaction networks. Although hybrid modelling is now well established in biology, these models remain computationally challenging. We propose several improved methods for computing time dependent multivariate probability distributions (MPD) of PDMP models of gene networks. In these models, the promoter dynamics is described by a finite state, continuous time Markov process, whereas the mRNA and protein levels follow ordinary differential equations (ODEs). The Monte-Carlo method combines direct simulation of the PDMP with analytic solutions of the ODEs. The push-forward method numerically computes the probability measure advected by the deterministic ODE flow, through the use of analytic expressions of the corresponding semigroup. Compared to earlier versions of this method, the probability of the promoter states sequence is computed beyond the naïve mean field theory and adapted for non-linear regulation functions.Availability. The algorithms described in this paper were implemented in MATLAB. The code is available at Zenodo. https://doi.org/10.5281/zenodo.3251708 .

One goal of synthetic biology is to implement useful functions with biochemical reactions, either by reprogramming living cells or programming artificial vesicles. In this perspective, we consider Chemical Reaction Networks (CRN) as a programming language, and investigate the CRN program synthesis problem. Recent work has shown that CRN interpreted by differential equations are Turing-complete and can be seen as analog computers where the molecular concentrations play the role of information carriers. Any real function that is computable by a Turing machine in arbitrary precision can thus be computed by a CRN over a finite set of molecular species. The proof of this result gives a numerical method to generate a finite CRN for implementing a real function presented as the solution of a Polynomial Initial Values Problem (PIVP). In this paper, we study an alternative method based on artificial evolution to build a CRN that approximates a real function given on finite sets of input values. We present a nested search algorithm that evolves the structure of the CRN and optimizes the kinetic parameters at each generation. We evaluate this algorithm on the Heaviside and Cosine functions both as functions of time and functions of input molecular species. We then compare the CRN obtained by artificial evolution both to the CRN generated by the numerical method from a PIVP definition of the function, and to the natural CRN found in the BioModels repository for switches and oscillators.

Cell classifiers are decision-making synthetic circuits that allow in vivo cell-type classification. Their design is based on finding a relationship between differential expression of miRNAs and the cell condition. Such biological devices have shown potential to become a valuable tool in cancer treatment as a new type-specific cell targeting approach. So far, only single-circuit classifiers were designed in this context. However, reliable designs come with high complexity, making them difficult to assemble in the lab. Here, we apply so-called Distributed Classifiers (DC) consisting of simple single circuits, that decide collectively according to a threshold function. Such architecture potentially simplifies the assembly process and provides design flexibility. We present a genetic algorithm that allows the design and optimization of DCs. Breast cancer case studies show that DCs perform with high accuracy on real-world data. Optimized classifiers capture biologically relevant miRNAs that are cancer-type specific. The comparison to a single-circuit classifier design approach shows that DCs perform with significantly higher accuracy than individual circuits. The algorithm is implemented as an open source tool.

We present a model of excitability in larval Drosophila muscles. Our model was initially based on modified Hodgkin-Huxley equations, adapted to represent variable, regenerative depolarisations (action potentials) we have occasionally observed in intracellular recordings and that can be triggered by excitatory junction potentials at neuromuscular synapses. We modified several kinetic equations describing voltage sensitive $$Ca^{2+}$$ C a 2 + and $$K^{+}$$ K + ionic currents, previously used to predict excitability in muscle cells of the mammalian cardiac atrioventricular node. The resulting nonlinear differential equations had multiple unknown parameters. Thus, to fit the model to experimental observations of variable excitability, we developed a new implementation of particle swarm optimisation. This GPU-based implementation allows us to adopt an ensemble model approach in which each experimental observation is used to find a plausible parameterisation, resulting in a set of models accounting for cell-to-cell variability of muscle excitability in Drosophila larvae, and with potential applications to population-based modeling of other excitable cell types.

Stochastic effects in cell growth and division drive variability in cellular volumes both at the single-cell level and at the level of growing cell populations. Here we consider a simple and tractable model in which cell volumes grow exponentially, cell division is symmetric, and its rate is volume-dependent. Consistently with previous observations, the model is shown to sustain oscillatory behaviour with alternating phases of slow and fast growth. Exact simulation algorithms and large-time asymptotics are developed and cross-validated for the single-cell and whole-population formulations of the model. The two formulations are shown to provide similar results during the phases of slow growth, but differ during the fast-growth phases. Specifically, the single-cell formulation systematically underestimates the proportion of small cells. More generally, our results suggest that measurable characteristics of cells may follow different distributions depending on whether a single-cell lineage or an entire population is considered.

The expression of a gene is characterised by its transcription factors and the function processing them. If the transcription factors are not affected by gene products, the regulating function is often represented as a combinational logic circuit, where the outputs (product) are determined by current input values (transcription factors) only, and are hence independent on their relative arrival times. However, the simultaneous arrival of transcription factors (TFs) in genetic circuits is a strong assumption, given that the processes of transcription and translation of a gene into a protein introduce intrinsic time delays and that there is no global synchronisation among the arrival times of different molecular species at molecular targets.In this paper, we construct an experimentally implementable genetic circuit with two inputs and a single output, such that, in presence of small delays in input arrival, the circuit exhibits qualitatively distinct observable phenotypes. In particular, these phenotypes are long lived transients: they all converge to a single value, but so slowly, that they seem stable for an extended time period, longer than typical experiment duration. We used rule-based language to prototype our circuit, and we implemented a search for finding the parameter combinations raising the phenotypes of interest.The behaviour of our prototype circuit has wide implications. First, it suggests that GRNs can exploit event timing to create phenotypes. Second, it opens the possibility that GRNs are using event timing to react to stimuli and memorise events, without explicit feedback in regulation. From the modelling perspective, our prototype circuit demonstrates the critical importance of analysing the transient dynamics at the promoter binding sites of the $${\mathsf {DNA}}$$ DNA , before applying rapid equilibrium assumptions.

Type I Diabetes (T1D) is a chronic disease in which the body’s ability to synthesize insulin is destroyed. It can be difficult for patients to manage their T1D, as they must control a variety of behavioral factors that affect glycemic control outcomes. In this paper, we explore T1D patient behaviors using a Signal Temporal Logic (STL) based learning approach. STL formulas learned from real patient data characterize behavior patterns that may result in varying glycemic control. Such logical characterizations can provide feedback to clinicians and their patients about behavioral changes that patients may implement to improve T1D control. We present both individual- and population-level behavior patterns learned from a clinical dataset of 21 T1D patients.

Time-bounded reachability problems are concerned with assessing whether a model’s trajectories traverse a given region of the state-space within given time-bounds. In the case of stochastic models reachability is associated with a measure of probability which depends on the model’s parameters. In this paper we propose a methodology that, given a reachability specification (for a parametric stochastic model), allows for computing a reachability related probability distribution on the parameter space, i.e. a distribution that allows for identifying regions of the parameter space for which there is a non-null probability to match the considered reachability specification. The methodology relies on the characterisation of distance between a model’s trajectory and a reachability specification which we show being assessable by using a hybrid automaton as a monitor of a model’s trajectory. An automata-based adaptation of the Approximated Bayesian Computation method is then introduced to estimate the reachability distribution on the parameter space.

Chemical reaction networks (CRNs) have been applied successfully to model a wide range of phenomena and are commonly used for designing molecular computation circuits. Often, CRNs with specific properties (oscillations, Turing patterns, multistability) are sought, which entails searching an exponentially large space of CRNs for those that satisfy a property. As the size of the CRNs being considered grows, so does the frequency of isomorphisms, by up to a factor $$N!$$ N ! , where $$N$$ N is the number of species. Accordingly, being able to generate sets of non-isomorphic CRNs within a class can lead to large computational savings when carrying out global searches. Here, we present a bijective encoding of bimolecular CRNs into novel vertex-coloured digraphs called . The problem of enumerating non-isomorphic CRNs can then be tackled by leveraging well-established computational methods from graph theory [20]. In particular, we extend Nauty, the graph isomorphism tool suite by McKay [22]. Our method is highly parallelisable and more efficient than competing approaches, and a software package (genCRN) is freely available for reuse. Non-isomorphs are generated directly by genCRN, alleviating the need to store intermediate results. We provide the first complete count of all 2-species bimolecular CRNs and extend previous known counts for classes of CRNs of special interest, such as mass-conserving and reversible CRNs.

Chemical reaction networks are a popular formalism for modeling biological processes which supports both a deterministic and a stochastic interpretation based on ordinary differential equations and continuous-time Markov chains, respectively. In most cases, these models do not enjoy analytical solution, thus typically requiring expensive computational methods based on numerical solvers or stochastic simulations. Exact model reduction techniques can be used as an aid to lower the analysis cost by providing reduced networks that preserve the dynamics of interest to the modeler. We hereby consider a family of techniques for both deterministic and stochastic networks which are based on equivalence relations over the species in the network, leading to a coarse graining which provides the exact aggregate time-course evolution for each equivalence class. We present a large-scale empirical assessment on the BioModels repository by measuring their compression capability over 667 models. Through a number of selected case studies, we also show their ability in yielding physically interpretable reductions that can reveal dynamical patterns of the bio-molecular processes under consideration.

Algorithms based on abstract interpretation were proposed recently for predicting changes of reaction networks with partial kinetic information. Their prediction precision, however, depends heavily on which heuristics are applied in order to add linear consequences of the steady state equations of the metabolic network. In this paper we ask the question whether such heuristics can be avoided while obtaining the highest possible precision. This leads us to the first algorithm for computing the difference abstractions of a linear equation system exactly without any approximation. This algorithm relies on the usage of elementary flux modes in a nontrivial manner, first-order definitions of the abstractions, and finite domain constraint solving.

We present the BRE:IN tool, a Backend for Reasoning about Interaction Networks. Our tool supports the framework and methodology originally introduced by the RE:IN tool, where an Abstract Boolean Network (ABN) specifies partial information about the network topology, and experimental observations are used to constrain the ABN, allowing to synthesize consistent models, or prove that no consistent model exists. RE:IN has been used successfully to derive mechanistic models of biological systems allowing to gain new insights into cellular decision-making and to make predictions that were validated experimentally. BRE:IN implements translations of experimental observations to temporal logic and captures the semantics of ABNs as transition systems, enabling to use off-the-shelf model checking algorithms. We make our tool and benchmarks publicly available and demonstrate the utility of the tool, providing speed-up gains for some benchmarks, while also enabling extensions of the experimental observations specification language currently supported in RE:IN by using the rich expressive power of temporal logic.

Like during software development, interactivity is of tremendous help during model development. The more and the earlier feedback come, the more efficiently the target is reached. This is true for human as well as during mechanical model construction. If you try to mechanically learn some parameters for a model by streaming potential values for example, you would better stop as quickly as possible the simulations that behave the worst toward the goal. The Kappa simulator KaSim has been refactored to give the control to the user (human or an other program) during the simulation, allowing to pause, restart, observe, modify, prematurely stop, continue after the original end. Interventions on a simulation that can be offered as well as their consequences on the design of a stochastic simulator of rule-based models are describe here.

Snoopy is a powerful modelling and simulation tool for various types of Petri nets, which have been applied to a wide range of biochemical reaction networks. We present an extended version of Snoopy, now supporting stochastic, continuous and hybrid Petri Nets with fuzzy kinetic parameters. Fuzzy parameters are specifically useful when kinetic parameter values can not be precisely measured or estimated. By running fuzzy simulation we obtain output bands of the variables of interest induced by the effect of the fuzzy kinetic parameters.

The compartmental modeling software (CMS) is an open source computational framework that can simulate discrete, stochastic reaction models which are often utilized to describe complex systems from epidemiology and systems biology. In this article, we report the computational requirements, the novel input model language, the available numerical solvers, and the output file format for CMS. In addition, the CMS code repository also includes a library of example model files, unit and regression tests, and documentation. Two examples, one from systems biology and the other from computational epidemiology, are included that highlight the functionality of CMS. We believe the creation of computational frameworks such as CMS will advance our scientific understanding of complex systems as well as encourage collaborative efforts for code development and knowledge sharing.

This paper presents Spike - a command line tool for continuous, stochastic & hybrid simulation of biochemical reaction networks represented as (coloured) Petri nets. It supports import from and export to various Petri net data formats, and also imports SBML models. Spike’s abilities include the configuration of models by changing stoichiometries (arc weights), initial conditions (markings) and kinetic parameters. It also unfolds coloured stochastic/continuous/hybrid Petri nets. To comply with the demand for reproducible simulation experiments, Spike builds on a script language in a human-readable format. Its core features permit the design of a set of simulation experiments by a single configuration file. These simulation experiments can be executed in parallel on a multi-core machine; distributed execution is in preparation.

In this paper we present KAMIStudio, an environment for biocuration of cellular signalling knowledge based on the KAMI framework. The environment provides an interface for the aggregation of decontextualized knowledge about individual protein-protein interactions, its interactive visualization, instantiation into signalling models and the subsequent generation of Kappa scripts that can be further used to study the dynamics of the modelled systems.

Often ODE models in systems biology, medical research, epidemiology, ecology and many other areas, contain unknown parameters which need to be estimated from experimental data. Identifiability deals with the uniqueness of the relation between model parameters and ODE solution thus being a prerequisite for the well-posedness of parameter estimation. In this paper a novel extension of the software tool DAISY (Differential Algebra for Identifiability of SYstems) is presented. DAISY performs structural identifiability analysis for linear and nonlinear dynamic models described by polynomial or rational ODE’s. The major upgrades of this new version regard the ability to include in the identifiability analysis either known and unknown model initial conditions, the possibility of entering a parameter estimate to calculate all the equivalent parameter solutions, the portability to MacOS platforms and an user-friendly interface. These upgrades make DAISY surely more general and easy to use. Practical examples are presented. DAISY is available at the web site daisy.dei.unipd.it .

Chemical Reaction Networks (CRNs) are a versatile language widely used for modelling and analysis of biochemical systems [4] as well as for high-level programming of molecular devices [1, 14].

Stochasticity is a fundamental feature of biology at the single cell level. Quantitative experimental data ranging from microscopy to single-cell transcriptomic is continually expanding our understanding of the role of stochasticity in gene expression and other cellular processes.

Modern experimental methods such as flow cytometry and fluorescence in-situ hybridization (FISH) allow the measurement of cell-by-cell molecule numbers for RNA, proteins and other substances for large numbers of cells at a time, opening up new possibilities for the quantitative analysis of biological systems. Of particular interest is the study of biological reaction systems describing processes such as gene expression, cellular signalling and metabolism on a molecular level. It is well established that many of these processes are inherently stochastic [1–3] and that deterministic approaches to their study can fail to capture properties essential for our understanding of these systems [4, 5]. Despite recent technological and conceptual advances, modelling and inference for stochastic models of reaction networks remains challenging due to additional complexities not present in the deterministic case. The Chemical Master Equation (CME) [6] in particular, while frequently used to model many types of reaction networks, is difficult to solve exactly, and parameter inference in practice often relies on a variety of approximation schemes whose accuracy can vary widely and unpredictably depending on the context [6–8].

With the automation of biological experiments and the increase of quality of single cell data that can now be obtained by phosphoproteomic and time lapse videomicroscopy, automating the building of mechanistic models from these data time series becomes conceivable and a necessity for many new applications. While learning numerical parameters to fit a given model structure to observed data is now a quite well understood subject, learning the structure of the model is a more challenging problem that previous attempts failed to solve without relying quite heavily on prior knowledge about that structure. In this paper, we consider mechanistic models based on chemical reaction networks (CRN) with their continuous dynamics based on ordinary differential equations, and finite time series about the time evolution of concentration of molecular species for a given time horizon and a finite set of perturbed initial conditions. We present a greedy heuristics unsupervised statistical learning algorithm to infer reactions with a time complexity for inferring one reaction in $$\mathcal O(t.n^2)$$ O ( t . n 2 ) where n is the number of species and t the number of observed transitions in the traces. We evaluate this algorithm both on simulated data from hidden CRNs, and on real videomicroscopy single cell data about the circadian clock and cell cycle progression of NIH3T3 embryonic fibroblasts. In all cases, our algorithm is able to infer meaningful reactions, though generally not a complete set for instance in presence of multiple time scales or highly variable traces.

In our previous work, we have introduced an extension of signal temporal logic called STL* that allows expressing freezing of values referred within temporal operators. The extension is important especially to express several aspects of signals that cannot be expressed in plain STL (e.g., presence of local extremes and their mutual relationships, non-trivial oscillatory behaviour such as damped oscillations, etc.). In this short paper, we address the tool Parasim that includes an implementation of the algorithm for computing robustness with respect to an STL* specification. The tool is in its current version considered as a prototype implementation of the algorithms for STL* robust monitoring of ODE models.

This paper explains a substantial feature of symmetry breaking of dynamical systems that include bistability from the mathematical point of view to highlight important consequences of this phenomenon to biochemical and system biology studies since symmetry breaking as a bifurcation itself can serve as a source of branching. We take hematopoietic stem cells modeling as a particular case.

We summarise our recent research results on developing efficient and scalable control methods for gene regulatory networks modelled as asynchronous Boolean networks. Our methods compute a minimal subset of nodes of a given Boolean network, such that (different types of) perturbations of these nodes, in one step or a sequence of steps, can drive the network (from an initial state) to a target steady state.

Transcriptomics response of SK-N-AS cells to methamidophos (an acetylcholine esterase inhibitor) exposure was measured at 10 time points between 0.5 and 48 h. The data was analyzed using a combination of traditional statistical methods, machine learning techniques, and methods to infer causal relations between time profiles. We identified several processes that appeared to be upregulated in cells treated with methamidophos including: unfolded protein response, response to cAMP, calcium ion response, and cell-cell signaling. The data confirmed the expected consequence of acetylcholine buildup. Transcripts with potentially key roles were identified by anomaly detection using convolutional autoencoders and Generative Adversarial Networks, and causal networks relating these transcripts were inferred using Siamese convolutional networks and time warp causal inference.

Computation biology helps to understand processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a reasonable time. We propose herein a method based on algebraic separators, which are special polynomials abundantly studied in effective Galois theory. These polynomials are used in modelling discrete data related to cellular pathways affected in cancer and targeting therapies.

Chemical reaction networks describe the interaction of different molecular species in a well-stirred reactor.

Population models are widely used to model different phenomena: animal collectives such as social insects, flocking birds, schooling fish, or humans within societies, as well as molecular species inside a cell, cells forming a tissue. Animal collectives show remarkable self-organisation towards emergent behaviours without centralised control. Quantitative models of the underlying mechanisms can directly serve important societal concerns (for example, prediction of seismic activity [5]), inspire the design of distributed algorithms (for example, ant colony algorithm [1]), or aid robust design and engineering of collective, adaptive systems under given functionality and resources, which is recently gaining attention in vision of smart cities [3, 4]. Quantitative prediction of the behaviour of a population of agents over time and space, each having several behavioural modes, results in a high-dimensional, non-linear, and stochastic system [2]. Hence, computational modelling with population models is challenging, especially when the model parameters are unknown and experiments are expensive.