Computational Methods in Systems Biology

The Turing completeness of continuous chemical reaction networks (CRNs) states that any computable real function can be computed by a continuous CRN on a finite set of molecular species, possibly restricted to elementary reactions, i.e. with at most two reactants and mass action law kinetics. In this paper, we introduce a notion of online analog computation for the CRNs that stabilize the concentration of their output species to the result of some function of the concentration values of their input species, whatever changes are operated on the inputs during the computation. We prove that the set of real functions stabilized by a CRN with mass action law kinetics is precisely the set of real algebraic functions.

We propose to simulate chemical reaction networks with the deterministic semantics abstractly, without any precise knowledge on the initial concentrations. For this, the concentrations of species are abstracted to Booleans stating whether the species is present or absent, and the derivatives of the concentrations are abstracted to signs saying whether the concentration is increasing, decreasing, or unchanged. We use abstract interpretation over the structure of signs for mapping the ODEs of a reaction network to a Boolean network with nondeterministic updates. The abstract state transition graph of such Boolean networks can be computed by finite domain constraint programming over the finite structure of signs. Constraints on the abstraction of the initial concentrations can be added naturally, leading to an abstract simulation algorithm that produces only the part of the abstract state transition graph that is reachable from the abstraction of the initial state. We prove the soundness of our abstract simulation algorithm, and show its applicability to reaction networks in the SBML format from the BioModels database.

Simulating chemical reaction networks is often computationally demanding, in particular due to stiffness. We propose a novel simulation scheme where long runs are not simulated as a whole but assembled from shorter precomputed segments of simulation runs. On the one hand, this speeds up the simulation process to obtain multiple runs since we can reuse the segments. On the other hand, questions on diversity and genuineness of our runs arise. However, we ensure that we generate runs close to their true distribution by generating an appropriate abstraction of the original system and utilizing it in the simulation process. Interestingly, as a by-product, we also obtain a yet more efficient simulation scheme, yielding runs over the system’s abstraction. These provide a very faithful approximation of concrete runs on the desired level of granularity, at a low cost. Our experiments demonstrate the speedups in the simulations while preserving key dynamical as well as quantitative properties.

In systems biology, Boolean networks (BNs) aim at modeling the qualitative dynamics of quantitative biological systems. Contrary to their (a) synchronous interpretations, the Most Permissive (MP) interpretation guarantees capturing all the trajectories of any quantitative system compatible with the BN, without additional parameters. Notably, the MP mode has the ability to capture transitions related to the heterogeneity of time scales and concentration scales in the abstracted quantitative system and which are not captured by asynchronous modes. So far, the analysis of MPBNs has focused on Boolean dynamical properties, such as the existence of particular trajectories or attractors.

Boolean modelling of gene regulation but also of post-transcriptomic systems has proven over the years that it can bring powerful analyses and corresponding insight to the many cases where precise biological data is not sufficiently available to build a detailed quantitative model. This is even more true for very large models where such data is frequently missing and led to a constant increase in size of logical models à la Thomas. Besides simulation, the analysis of such models is mostly based on attractor computation, since those correspond roughly to observable biological phenotypes. The recent use of trap spaces made a real breakthrough in that field allowing to consider medium-sized models that used to be out of reach. However, with the continuing increase in model-size, the state-of-the-art computation of minimal trap spaces based on prime-implicants shows its limits as there can be a huge number of implicants.

In this article we present an alternative method to compute minimal trap spaces, and hence complex attractors, of a Boolean model. It replaces the need for prime-implicants by a completely different technique, namely the enumeration of maximal siphons in the Petri net encoding of the original model. After some technical preliminaries, we expose the concrete need for such a method and detail its implementation using Answer Set Programming. We then demonstrate its efficiency and compare it to implicant-based methods on some large Boolean models from the literature.

The skin microbiome plays an important role in the maintenance of a healthy skin. It is an ecosystem, composed of several species, competing for resources and interacting with the skin cells. Imbalance in the cutaneous microbiome, also called dysbiosis, has been correlated with several skin conditions, including acne and atopic dermatitis. Generally, dysbiosis is linked to colonization of the skin by a population of opportunistic pathogenic bacteria (for example C. acnes in acne or S. aureus in atopic dermatitis). Treatments consisting in non-specific elimination of cutaneous microflora have shown conflicting results. It is therefore necessary to understand the factors influencing shifts of the skin microbiome composition. In this work, we introduce a mathematical model based on ordinary differential equations, with 2 types of bacteria populations (skin commensals and opportunistic pathogens) to study the mechanisms driving the dominance of one population over the other. By using published experimental data, assumed to correspond to the observation of stable states in our model, we derive constraints that allow us to reduce the number of parameters of the model from 13 to 5. Interestingly, a meta-stable state settled at around 2 d following the introduction of bacteria in the model, is followed by a reversed stable state after 300 h. On the time scale of the experiments, we show that certain changes of the environment, like the elevation of skin surface pH, create favorable conditions for the emergence and colonization of the skin by the opportunistic pathogen population. Such predictions help identifying potential therapeutic targets for the treatment of skin conditions involving dysbiosis of the microbiome, and question the importance of meta-stable states in mathematical models of biological processes.

Detailed dynamical systems models used in life sciences may include dozens or even hundreds of state variables. Models of large dimension are not only harder from the numerical perspective (e.g., for parameter estimation or simulation), but it is also becoming challenging to derive mechanistic insights from such models. Exact model reduction is a way to address this issue by finding a self-consistent lower-dimensional projection of the corresponding dynamical system. A recent algorithm CLUE allows one to construct an exact linear reduction of the smallest possible dimension such that the fixed variables of interest are preserved. However, CLUE is restricted to systems with polynomial dynamics. Since rational dynamics occurs frequently in the life sciences (e.g., Michaelis-Menten or Hill kinetics), it is desirable to extend CLUE to the models with rational dynamics. In this paper, we present an extension of CLUE to the case of rational dynamics and demonstrate its applicability on examples from literature. Our implementation is available in version 1.5 of CLUE (https://​github.​com/​pogudingleb/​CLUE).

Many gene regulatory networks have periodic behavior, for instance the cell cycle or the circadian clock. Therefore, the study of formal methods to analyze limit cycles in mathematical models of gene regulatory networks is of interest. In this work, we study a pre-existing hybrid modeling framework (HGRN) which extends René Thomas’ widespread discrete modeling. We propose a new formal method to find all limit cycles that are simple and deterministic, and analyze their stability, that is, the ability of the model to converge back to the cycle after a small perturbation. Up to now, only limit cycles in two dimensions (with two genes) have been studied; our work fills this gap by proposing a generic approach applicable in higher dimensions. For this, the hybrid states are abstracted to consider only their borders, in order to enumerate all simple abstract cycles containing possible concrete trajectories. Then, a Poincaré map is used, based on the notion of transition matrix of the concrete continuous dynamics inside these abstract paths. We successfully applied this method on existing models: three HGRNs of negative feedback loops with 3 components, and a HGRN of the cell cycle with 5 components.

Biochemical reactions inside living cells often occur in the presence of crowders - molecules that do not participate in the reactions but influence the reaction rates through excluded volume effects. However the standard approach to modelling stochastic intracellular reaction kinetics is based on the chemical master equation (CME) whose propensities are derived assuming no crowding effects. Here, we propose a machine learning strategy based on Bayesian Optimisation utilising synthetic data obtained from spatial cellular automata (CA) simulations (that explicitly model volume-exclusion effects) to learn effective propensity functions for CMEs. The predictions from a small CA training data set can then be extended to the whole range of parameter space describing physiologically relevant levels of crowding by means of Gaussian Process regression. We demonstrate the method on an enzyme-catalyzed reaction and a genetic feedback loop, showing good agreement between the time-dependent distributions of molecule numbers predicted by the effective CME and CA simulations.

A broad range of natural and social systems from human microbiome to financial markets can go through critical transitions, where the system suddenly collapses to another stable configuration. Anticipating such transition early and accurately can facilitate controlled system manipulation and mitigation of undesired outcomes. Generic data-driven indicators, such as autocorrelation and variance, have been shown to increase in the vicinity of an approaching tipping point, and statistical early warning signals have been reported across a range of systems. In practice, obtaining reliable predictions has proven to challenging, as the available methods deal with simplified one-dimensional representations of complex systems, and rely on the availability of large amounts of data. Here, we demonstrate that a probabilistic data aggregation strategy can provide new ways to improve early warning detection by more efficiently utilizing the available information from multivariate time series. In particular, we consider a probabilistic variant of a vector autoregression model as a novel early warning indicator and argue that it has certain advantages in model regularization, treatment of uncertainties, and parameter interpretation. We evaluate the performance against alternatives in a simulation benchmark and show improved sensitivity in warning signal detection in a common ecological model encompassing multiple interacting species.

Chemical reaction networks are widely used to model biochemical systems. However, when the complexity of these systems increases, the chemical reaction networks are prone to errors in the initial modeling and subsequent updates of the model.

The dynamics of biochemical reaction networks require a stochastic description when copy number fluctuations become significant. Such description is provided by moment equations that capture the statistical properties of the involved molecular components such as their average abundance and variability. Certain applications require a special form of moment equations, where the statistics of some components are described conditionally on complete trajectories of other components. Typical examples include information theoretical analyses of biochemical networks, model reduction and subnetwork simulation, or statistical inference where time-varying molecular signals are inferred from counting observations. These conditional moment equations have so far been limited to relatively simple reaction systems as their manual derivation becomes difficult for systems involving many components and interactions. Here, we present a Python tool for the automated derivation of moment equations conditional on complete time trajectories for arbitrary user-defined reaction systems and showcase its utility in the context of subnetwork simulation. With this automated tool, conditional moment equations become applicable to a broad class of biochemical systems.

Boolean Networks (BN) are established tools for modelling biological systems. However, their analysis is hindered by the state space explosion: the exponentially many states on the variables of a BN. We present an extension of the tool for model reduction ERODE with support for BNs and their reduction with a recent method called Backward Boolean Equivalence (BBE). BBE identifies maximal sets of variables that retain the same value whenever initialized equally. ERODE has been also extended to support importing and exporting between different formats and model repositories, enhancing interoperability with other tools.

eBCSgen is a software tool for developing and analysing models written in Biochemical Space Language (BCSL). BCSL is a rule-based language designed for the description of biological systems with rewriting rules in the form of behavioural patterns. This tool paper describes a new version of the tool, implementing the support for regulations, a mechanism suitable for reducing the branching behaviour of concurrent systems. Additionally, the presented version provides export to SBML, and support for CTL model checking. The paper artefact is available via https://​doi.​org/​10.​5281/​zenodo.​6644973.