Computational Methods in Systems Biology

The adaptation of microorganisms to their environment is controlled at the molecular level by large and complex networks of biochemical reactions involving genes, RNAs, proteins, metabolites, and small signalling molecules. In theory, it is possible to write down mathematical models of these networks, and study these by means of classical analysis and simulation tools. In practice, this is not easy to achieve though, as quantitative data on kinetic parameters are usually absent for most systems of biological interest. Moreover, the models consist of a large number of variables, are strongly nonlinear and include different time-scales, which make them difficult to handle both mathematically and computationally.

Bio-PEPA is a novel stochastic process algebra which has been recently developed for modelling biochemical pathways [5,6]. In Bio-PEPA a reagent-centric style of modelling is adopted, and a variety of analysis techniques can be applied to a single model expression. Such an approach facilitates easy validation of analysis results when the analyses address the same issues [3] and enhanced insight when the analyses are complementary [4]. Currently supported analysis techniques include stochastic simulation at the molecular level, ordinary di..erential equations, probabilistic model checking and numerical analysis of a continuous time Markov chain.

As we obtain better abilities to observe cellular biochemistry at the single cell / molecular levels, such as through fluorescent correlation spectroscopy and single particle tracking, evidences are accumulating that the cells may be taking advantage of intracellular spatial features to realize and optimize their functions. Computer simulation is a useful means to bridge the gap between the microscopic, physico-chemical picture of how macro-molecules diffuse and react, and the scales of time and space where biochemistry and physiology take place.

Transduction and transmission of an input signal by a neuronal dendrite involves generation, integration and propagation of at least four kinds of information: Chemical concentration such as calcium ions, chemical modification such as phosphorylation cascades, conformational information such as allosteric modulations, and electrical signals such as membrane depolarisation. One cannot claim to understand neuronal function when focussing on a single aspect. However, developing models of the four requires using different formalisms. Furthermore running simulations implies widely different requirements in terms of compute power, storage or results and duration. I will present a few results we obtained about the synaptic function and plasticity in the striatal medium-spiny neuron, using models of signalling networks, allosteric regulations, single particle diffusion and multi-compartment electrical models. I will then discuss how we can sometimes encapsulate the results obtained at a certain level of resolution in order to increase the realism of more abstract models. I will end by outlining how one could envision to build a model striatal neuron that embodies chemical, biochemical and electrical signalling.

Systems biology is spread over all branches of life science and attracts biologists, mathematicians, physicists, computer scientists, and engineers equally. Full of promises and visions it often signalizes that the in silico eucaryotic cell is close to realization and experimental work will be needed in the future only for confirmation. At this point science becomes fiction and destroys the great potential of interdisciplinary research aiming for added value in describing a living system or its composing modules by theoretical/simulation approaches on the basis of experimental facts. As a reliable working definition of molecular systems biology the following is useful: Modelling of cells or a modules of cells with an incomplete data set. The model (ensemble of models) must have predictive value to induce experiments which lead to falsification (verification) of subsets of models until, on the basis of available data, optimally only one model is left. The approach can be either “bottom up” or “top down”. We use halophilic archaea, especially the model organism Halobacterium salinarum for systems biological experiments. These procaryotes living in concentrated brines offer biochemical features which make them very suitable for systematic analysis. A first module is signal transduction where photon absorption via two photoreceptors causes three different reactions of the target, which is the flagellar motor. The system guarantees a balanced response of the cell to light for active search of the optimal conditions for photosynthesis. Experimentally, quantitative data can be collected, which link the size of stimulus to the reaction time of the flagellar motor. Further, genome wide data on members of the network, their molecular properties and protein protein interactions were made available. Altogether a model was developed, which allows to simulate all experimental results reported so far. Bioenergetics are a second module, which is ready for modelling with a bottom up approach and the central metabolism of the cell presents an example of top down modelling with about 800 reactions in the cell. Experimental data on the course of sixteen amino acids added to the growth medium as carbon source and on the rate growth were collected and a model created which is able to quantitatively predict growth curve and carbon source usage. The lecture will give account on the details of the experimental methods used, describe the modelling approaches and summarize the results, we so far obtained.

Background: Inference and understanding of gene networks from experimental data is an important but complex problem in molecular biology. Mapping of gene pathways typically involves inferences arising from various studies performed on individual pathway components. Although pathways are often conceptualized as distinct entities, it is often understood that inter-pathway cross-talk and other properties of networks reflect underlying complexities that cannot by explained by consideration of individual pathways in isolation. In order to consider interaction between individual paths, a global multivariate approach is required. In this paper, we propose an extended form of Granger causality can be used to infer interactions between sets of time series data.

Results: We successfully tested our method on several artificial datasets, each one depicting various possibilities of connections among the participating entities. We also demonstrate the ability of our method to deal with latent and exogenous variables present in the system. We then applied this method to a highly replicated gene expression microarray time series data to infer causal influences between gene expression events involved in activation of human T-cells. The application of our method to the T-cell dataset revealed a set of strong causal links between the participating genes, with many links already experimentally verified and reported in the biological literature.

Conclusions: We have proposed a novel form of Granger causality to reverse-engineer a causal network structure from a time series dataset involving multiple entities. We have extensively and successfully tested our method on synthesized as well as real time series microarray data.

The origin and evolution of metabolism is an interesting field of research with many unsolved questions. Simulation approaches, even though mostly very abstract and specific, have proven to be helpful in explaining properties and behavior observed in real world metabolic reaction networks, such as the occurrence of hub-metabolites. We propose here a more complex and intuitive graph-based model combined with an artificial chemistry. Instead of differential equations, enzymes are represented as graph rewriting rules and reaction rates are derived from energy calculations of the involved metabolite graphs. The generated networks were shown to possess the typical properties and further studied using our metabolic pathway analysis tool implemented for the observation of system properties such as robustness and modularity. The analysis of our simulations also leads to hypotheses about the evolution of catalytic molecules and its effect on the emergence of the properties mentioned above.

RNA structure is fundamentally important for many biological processes. In the past decades, diverse structure prediction algorithms and tools were developed but due to missing descriptions in clearly defined modelling formalisms it’s difficult or even impossible to integrate them into larger system models. We present an RNA secondary structure folding model described in ml-Devs, a variant of the Devs formalism, which enables the hierarchical combination with other model components like RNA binding proteins. An example of transcriptional attenuation will be given where model components of RNA polymerase, the folding RNA molecule, and the translating ribosome play together in a composed dynamic model.

CBS is a Calculus of Biochemical Systems intended to allow the modelling of metabolic, signalling and regulatory networks in a natural and modular manner. In this paper we extend CBS with features directed towards practical, large-scale applications, thus yielding LBS: a Language for Biochemical Systems. The two main extensions are expressions for modifying large complexes in a step-wise manner and parameterised modules with a notion of subtyping; LBS also has nested declarations of species and compartments. The extensions are demonstrated with examples from the yeast pheromone pathway. A formal specification of LBS is then given through an abstract syntax, static semantics and a translation to a variant of coloured Petri nets. Translation to other formalisms such as ordinary differential equations and continuous time Markov chains is also possible.

The attributed pi calculus \(({\phi({\mathcal L})})\) forms an extension of the pi calculus with attributed processes and attribute dependent synchronization. To ensure flexibility, the calculus is parametrized with the language \({\mathcal L}\) which defines possible values of attributes. \({\phi({\mathcal L})}\) can express polyadic synchronization as in pi@ and thus diverse compartment organizations. A non-deterministic and a stochastic semantics, where rates may depend on attribute values, is introduced. The stochastic semantics is based on continuous time Markov chains. A simulation algorithm is developed which is firmly rooted in this stochastic semantics. Two examples underline the applicability of \({\phi({\mathcal L})}\) to systems biology: Euglena’s movement in phototaxis, and cooperative protein binding in gene regulation of bacteriophage lambda.

We introduce the continuous π -calculus, a process algebra for modelling behaviour and variation in molecular systems. Key features of the language are: its expressive succinctness; support for diverse interaction between agents via a flexible network of molecular affinities; and operational semantics for a continuous space of processes. This compositional semantics also gives a modular way to generate conventional differential equations for system behaviour over time. We illustrate these features with a model of an existing biological system, a simple oscillatory pathway in cyanobacteria. We then discuss future research directions, in particular routes to applying the calculus in the study of evolutionary properties of biochemical pathways.

Kinetic models for biochemical systems often comprise a large amount of coupled differential equations with species concentrations varying on different time scales. In this paper we present and apply two novel methods aimed at automatic complexity and model reduction by numerical algorithms. The first method combines dynamic sensitivity analysis with singular value decomposition. The aim is to determine the minimal dimension of the kinetic model necessary to describe the active dynamics of the system accurately enough within a user-defined error tolerance for particular species concentrations and to determine each species’ contribution to the active dynamics. The second method treats the explicit numerical reduction of the model to a lower dimension according to the results of the first method and allows any species combination to be chosen as a parameterization of the reduced model which may either be tabulated in the form of look-up tables or computed in situ during numerical simulations. A reduced representation of a multiple time scale system is particularly beneficial in the context of spatiotemporal simulations which require high computational efforts. Both the complexity analysis and model reduction method operate in a fully automatic and numerically highly efficient way and have been implemented in a software package. The methods are applied to a biochemical example model describing the ERK signaling pathway. With this example, we demonstrate the value of the methods for various applications in systems biology.

We present the Piecewise Linear Approximation Model of Ion Channel contribution (PLAMIC) to cardiac excitation. We use the PLAMIC model to conduct formal analysis of cardiac arrhythmic events, namely Early Afterdepolarizations (EADs). The goal is to quantify (for the first time) the contribution of the overall sodium (Na⁺), potassium (K⁺) and calcium (Ca^2 +) currents to the occurrence of EADs during the plateau phase of the cardiac action potential (AP). Our analysis yields exact mathematical criteria for the separation of the parameter space for normal and EAD-producing APs, which is validated by simulations with classical AP models based on complex systems of nonlinear differential equations. Our approach offers a simple formal technique for the prediction of conditions leading to arrhythmias (EADs) from a limited set of experimental measurements, and can be invaluable for devising new anti-arrhythmic strategies.

The distribution of mutational effects on fitness (DME ^F ) is of fundamental importance for many questions in biology. Previously, wet-lab experiments and population genetic methods have been used to infer the sizes of effects of mutations. Both approaches have important limitations. Here we propose a new framework for estimating the DME ^F by constructing fitness correlates in molecular systems biology models. This new framework can complement the other approaches in estimating small effects on fitness. We present a notation for the various DMEs that can be present in a molecular systems biology model. Then we apply this new framework to a simple circadian clock model and estimate various DMEs in that system. Circadian clocks are responsible for the daily rhythms of activity in a wide range of organisms. Mutations in the corresponding genes can have large effects on fitness by changing survival or fecundity. We define potential fitness correlates, describe methods for automatically measuring them from simulations and implement a simple clock using the Gillespie stochastic simulation algorithm within StochKit. We determine what fraction of examined mutations with small effects on the rates of the reactions involved in this system are advantageous or deleterious for emerging features of the system like a fitness correlate, cycle length and cycle amplitude. We find that the DME can depend on the wild type reference used in its construction. Analyzing many models with our new approach will open up a third source of information about the distribution of mutational effects, one of the fundamental quantities that shape life.

Share and reuse of biochemical models have become two of the main issues in the field of Computational Systems Biology. There already exist widely-accepted formats to encode the structure of models. However, the problem of describing the simulations to be run using those models has not yet been tackled in a satisfactory way. The community believes that providing detailed information about simulation recipes will highly improve the efficient use of existing models. Accordingly a set of guidelines called the Minimum Information About a Simulation Experiment (MIASE) is currently under development. It covers information about the simulation settings, including information about the models, changes on them, simulation settings applied to the models and output definitions. Here we present the Simulation Experiment Description Markup Language (SED-ML), an XML format that enables the storage and exchange of part of the information required to implement the MIASE guidelines. SED-ML is independent of the formats used to encode the models – as long as they are expressed in XML –, and it is independent of the software tools used to run the simulations. Several test implementations are being developed to benchmark SED-ML on simple cases, and pave the way to a more complete support of MIASE.

The parallel simulation of biochemical reactions is a very interesting problem: biochemical systems are inherently parallel, yet the majority of the algorithms to simulate them, including the well-known and widespread Gillespie SSA, are strictly sequential. Here we investigate, in a general way, how to characterize the simulation of biochemical systems in terms of Discrete Event Simulation. We dissect their inherent parallelism in order both to exploit the work done in this area and to speed-up their simulation. We study the peculiar characteristics of discrete biological simulations in order to select the parallelization technique which provides the greater benefits, as well as to touch its limits. We then focus on reaction-diffusion systems: we design and implement an efficient parallel algorithm for simulating such systems that include both reactions between entities and movements throughout the space.

BioLab models are encoded in the BioNetGen language, and properties are expressed as formulae in probabilistic bounded linear temporal logic. Temporal logic is a formalism for representing and reasoning about propositions qualified in terms of time. Properties are then verified using sequential hypothesis testing on executions generated using stochastic simulation. BioLab is optimal, in the sense that it generates the minimum number of executions necessary to verify the given property. BioLab also provides guarantees on the probability of it generating Type-I (i.e., false-positive) and Type-II (i.e., false-negative) errors. Moreover, these error bounds are pre-specified by the user. We demonstrate BioLab by verifying stochastic effects and bistability in the dynamics of the T-cell receptor signaling network.

Finding mathematical models satisfying a specification built from the formalization of biological experiments, is a common task of the modeller that techniques like model-checking help solving, in the qualitative but also in the quantitative case. In this article we propose to go one step further by defining a continuous degree of satisfaction of a temporal logic formula with constraints. We show how such a satisfaction measure can be used as a fitness function with state-of-the-art search methods in order to find biochemical kinetic parameter values satisfying a set of biological properties formalized in temporal logic. We also show how it can be used to define a measure of robustness of a biological model with respect to some specification. These methods are evaluated on models of the cell cycle and of the MAPK signalling cascade.

Model checking has historically been an important tool to verify models of a wide variety of systems. Typically a model has to exhibit certain properties to be classed ‘acceptable’. In this work we use model checking in a new setting; parameter estimation. We characterise the desired behaviour of a model in a temporal logic property and alter the model to make it conform to the property (determined through model checking). We have implemented a computational system called MC2(GA) which pairs a model checker with a genetic algorithm. To drive parameter estimation, the fitness of set of parameters in a model is the inverse of the distance between its actual behaviour and the desired behaviour. The model checker used is the simulation-based Monte Carlo Model Checker for Probabilistic Linear-time Temporal Logic with numerical constraints, MC2(PLTLc). Numerical constraints as well as the overall probability of the behaviour expressed in temporal logic are used to minimise the behavioural distance. We define the theory underlying our parameter estimation approach in both the stochastic and continuous worlds. We apply our approach to biochemical systems and present an illustrative example where we estimate the kinetic rate constants in a continuous model of a signalling pathway.

Constraint-based approaches have proved successful in analyzing complex metabolic networks. They restrict the range of all possible behaviors that a metabolic system can display under governing constraints. The set of all possible flux distributions over a metabolic network at steady state defines a polyhedral cone, the steady-state flux cone. This cone can be analyzed using an inner description based on sets of generating vectors such as elementary flux modes or extreme pathways. Another possibility is the use of an outer description based on sets of non-negativity constraints. In this paper, we study the relationship between inner and outer descriptions of the cone. We give a generic procedure to show how inner descriptions can be computed from the outer one. Then we use this procedure to explain why, for large-scale metabolic networks, the size of the inner descriptions may be several orders of magnitude larger than that of the outer description.

For many aspects of health and disease, it is important to understand different phenomena in biology and medicine. To gain the required insight, experimental data are provided and need to be interpreted, thus the challenging task is to generate all models that explain the observed phenomena. In systems biology the framework of Petri nets is often used to describe models for the regulatory mechanisms of biological systems. The aim of this paper is to present an exact combinatorial approach for the reconstruction of such models from experimental data.

As we obtain better abilities to observe cellular biochemistry at the single cell / molecular levels, such as through fluorescent correlation spectroscopy and single particle tracking, evidences are accumulating that the cells may be taking advantage of intracellular spatial features to realize and optimize their functions. Computer simulation is a useful means to bridge the gap between the microscopic, physico-chemical picture of how macro-molecules diffuse and react, and the scales of time and space where biochemistry and physiology take place.

We present a formal analysis of amino acid replacement during mRNA translation. Building on an abstract stochastic model of arrival of tRNAs and their processing at the ribosome, we compute probabilities of the insertion of amino acids into the nascent polypeptide chain. To this end, we integrate the probabilistic model checker Prism in the Matlab environment. We construct the substitution matrix containing the probabilities of an amino acid replacing another. The resulting matrix depends on various parameters, including availability and concentration of tRNA species, as well as their assignment to individual codons. We draw a parallel with the standard mutation matrices like Dayhoff and PET91, and analyze the mutual replacement of biologically similar amino acids.

The therapeutic potential of stem cells due to their ability to build and maintain tissues and organs is widely recognised. Much can be learned by studying stem cell turnover dynamics and Bromodeoxyuridine (BrdU) is often used for this purpose. Good computational models are however needed for a full understanding of BrdU data and in this paper we present such a model. Our approach is to model single cells as well as their chromosomes as agents which make probabilistic decisions over fixed intervals of time. We demonstrate the power of our model by comparing its performance to a deterministic BrdU model used in a recently published study on asymmetric chromosome segregation in Hematopoietic stem cells.

We present a discrete formal model of the central pattern generator (CPG) located in the buccal ganglia of Aplysia that is responsible for mediating the rhythmic movements of its foregut during feeding. Our starting point is the continuous dynamical model for pattern generation underlying fictive feeding in Aplysia proposed by Baxter et. al. [1]. The discrete model is obtained as a composition of discrete models of ten individual neurons in the CPG. The individual neurons are interconnected through excitatory and inhibitory synaptic connections and electric connections. We used Symbolic Analysis Laboratory (SAL) to formally build the model and analyzed it using the SAL model checkers. Using abstract discrete models of the individual neurons helps in understanding the buccal motor programs generated by the network in terms of the network connection topology. It also eliminates the need for detailed knowledge of the unknown parameters in the continuous model of Baxter et. al. [1].