Multirate Partitioned Runge-Kutta Methods

The field of generative chemistry and automated exploration of chemical reaction space has gained much interest in recent years as it provides a feasible alternative to performing resource-intensive experiments by enabling important computational insights into new molecular systems. The results are often summarized in reaction networks, which reveal intricate relations between different key reactive events. Although various approaches to explore the available chemical space have been introduced, the information contained in the resulting reaction networks has not been fully exploited so far. We propose an automated workflow for the analysis of chemical reaction networks by applying frequent pattern mining on the corresponding directed hypergraphs to identify frequently occurring reactive patterns across a set of simulations. Furthermore, we identify reactive events that are statistically correlated with given environmental conditions by applying Fisher's exact test and controlling the family-wise error rate to ensure high statistical relevance. Minimum energy paths for frequent and statistically significant patterns are obtained with the molecular double-ended growing string method at ωB97X-3c level of theory. We showcase the pattern-mining-based analysis on the thermally controlled interstellar synthesis of carbamic acid, where we retrieve results in line with experimental data and further investigate the role of water as a protic solvent therein.

Multirate schemes for conservation laws or convection-dominated problems seem to come in two flavors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods.KeywordsMonotonicity PropertyTotal Variation DiminishMonotonicity AssumptionStrong Stability PreserveHigh Resolution SchemeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chemical organisation theory is a framework developed to simplify the analysis of long-term behaviour of chemical systems. An organisation is a set of objects which are closed and self-maintaining. In this paper, we build on these ideas to develop novel techniques for formal quantitative analysis of chemical reaction networks, using discrete stochastic models represented as continuous-time Markov chains. We propose methods to identify organisations, to study quantitative properties regarding movement between these organisations and to construct an organisation-based coarse graining of the model that can be used to approximate and predict the behaviour of the original reaction network.

Traditionally, quantitative models of reaction networks are based on classical chemical kinetics. Under the assumption of the thermodynamic limit (infinite number of molecules/volume limit), reactions are modeled as continuous, deterministic processes. It has become evident, however, that discrete fluctuations play a crucial role in cellular processes like gene expression and signal transduction, where constituents are typically present in small numbers. In this case, a modeling approach based on stochastic reaction kinetics is required, where reactions are modeled as discrete stochastic processes. The temporal evolution of the probability density function of the number of molecules is given by the chemical master equation (CME), which, however, is impractical to solve in most applications due to its high dimensionality. Instead, it is common practice to approximate an indirect solution of the CME by computing realizations of the underlying Markov jump process. A major aim is the development of such indirect approaches that enable the simulation of complex multi-scale reaction networks. This thesis deals with the promising development of hybrid methods, where fast reactions associated with large numbers of molecules are continuously and deterministically approximated, and all other reactions are modeled as discrete stochastic processes. We demonstrate the benefit of such a hybrid system description on an integrative model of the replication dynamics of the human immunodeficiency virus (HIV). Based on hybrid simulations, we are able to design and validate in silico a novel treatment strategy for HIV-infected patients that can lead to significant improvements compared to conventional treatment strategies. While current hybrid methods almost exclusively rely on indirect approximations, a novel hybrid approach is presented in this thesis that allows to solve the CME directly. Based on a multi-scale expansion, evolution equations are derived that couple a CME on a reduced state space to evolution equations of deterministically approximated variables. Thus, the impact of changes in the probability distribution of the stochastic subsystem on the dynamics of the deterministic components becomes apparent and, in contrast to indirect hybrid methods, is taken into account explicitly. We illustrate and discuss the performance of this direct hybrid approaches in application to model systems of biological interest. In the last part of this thesis, we derive effective protein synthesis rates as typically incorporated in deterministic models of biochemical systems by reduction of a detailed stochastic model of gene expression. We use this approach to derive a protein interactions model of the flagellar gene regulation cascade in Escherichia coli. Based on the deduced functional relations between transcriptional and translational processes on the one hand and the synthesis rates on the other, we find that sensitivity with respect to effective rates does not directly carry over to the aggregated subprocesses.

ABSTRACTA treatment of a chemical reaction network theory is given from the perspective of nonlinear network dynamics, in particular of consensus dynamics. By starting from the complex-balanced assumption, the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in the chemical reaction network theory, and which directly relates to the thermodynamics and port-Hamiltonian formulation of the system. Central in this formulation is the definition of a balanced Laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality. This immediately leads to the characterisation of the set of equilibria and their stability. Furthermore, the assumption of complex balancedness is revisited from the point of view of Kirchhoff's matrix tree theorem. Both the form of the dynamics and the deduced behaviour are very similar to consensus dynamics, and provide additional perspectives to the latter. Finally, using the classical idea of extending the graph of chemical complexes by a ‘zero’ complex, a complete steady-state stability analysis of mass action kinetics reaction networks with constant inflows and mass action kinetics outflows is given, and a unified framework is provided for structure-preserving model reduction of this important class of open reaction networks.

Exploration of thechemical reaction space of chemicaltransformationsin multicomponent mixtures is one of the main challenges in contemporarycomputational chemistry. To remove expert bias from mechanistic studiesand to discover new chemistries, an automated graph-theoretical methodologyis proposed, which puts forward a network formalism of homogeneouscatalysis reactions and utilizes a network analysis tool for mechanisticstudies. The method can be used for analyzing trajectories with singleand multiple catalytic species and can provide unique conformers ofcatalysts including multinuclear catalyst clusters along with othercatalytic mixture components. The presented three-step approach hasthe integrated ability to handle multicomponent catalytic systemsof arbitrary complexity (mixtures of reactants, catalyst precursors,ligands, additives, and solvents). It is not limited to predefinedchemical rules, does not require prealignment of reaction mixturecomponents consistent with a reaction coordinate, and is not agnosticto the chemical nature of transformations. Conformer exploration,reactive event identification, and reaction network analysis are themain steps taken for identifying the pathways in catalytic systemsgiven the starting precatalytic reaction mixture as the input. Sucha methodology allows us to efficiently explore catalytic systems inrealistic conditions for either previously observed or completelyunknown reactive events in the context of a network representing differentintermediates. Our workflow for the catalytic reaction space explorationexclusively focuses on the identification of thermodynamically feasibleconversion channels, representative of the (secondary) catalyst deactivationor inhibition paths, which are usually most difficult to anticipatebased solely on expert chemical knowledge. Thus, the expert bias issought to be removed at all steps, and the chemical intuition is limitedto the choice of the thermodynamic constraint imposed by the applicableexperimental conditions in terms of threshold energy values for allowedtransformations. The capabilities of the proposed methodology havebeen tested by exploring the reactivity of Mn complexes relevant forcatalytic hydrogenation chemistry to verify previously postulatedactivation mechanisms and unravel unexpected reaction channels relevantto rare deactivation events.

The example of articulated vehicles is used to introduce a new simulation concept. Modules found by functional decomposition of an engineering system can be encapsulated by an input-output representation. They are then interconnected by the system inputs and outputs, or, in the case of mechanical systems, by constraints using joints. It is shown that the resulting mechanical DAE structure can effciently be solved by a projection method. A multirate integration is proposed for efficient simulation of systems with different time scales. The simulatorNEWMOS is introduced supporting all the features described. Simulation runs show that the multirate scheme is essential to achieve a stable integration of the articulated vehicles problem.

In certain classes of chemical reaction networks (CRN), there may be two stable states. The challenge is to find a model of the CRN such that the stability properties can be predicted. In this paper we consider the problem of building a P-system designed to simulate the CRN in an attempt to determine if the CRN is stable or bistable. We found that for the networks in [2] none of the bistable CRN would have a bistable P-system by stoichiometry alone. The reaction kinetics must be included in the P-system model; the implementation of which has been considered an open problem. In this paper we conclude that a P-system for a CRN in mreactants and nproducts has at most 2(m2+ mn) membranes and 6(m2+ mn) rules. This suggests that P-system models of a chemical reaction network, including both stoichiometry and reaction kinetics can be built.

Extended Fourier Neural Operators to learn stiff chemical kinetics under unseen conditions

The input and output formats, and the structure of Rule Input Network Generator (RING), a computational tool for generation and analysis of complex reaction networks, are described with reference to the underlying algorithms from Cheminformatics and graph theory. RING consists of three modules: (a) a compiler that translates inputs written as a program in an English-like reaction language into internal representations and instructions, (b) a network generator that constructs an exhaustive reaction network from reaction rules and initial reactants specified, and (c) a post-processing module that can extract pathways, mechanisms, or lumps from the network based on user-specified instructions. RING can be used, in a rule-based manner, for constructing a large and complex reaction network from a set of elementary/overall reaction rules, and for elucidating transformations occurring in these networks through identifying pathways and mechanisms to specified products. RING is available open under GNU Lesser GPL.

The importance of stochasticity within biological systems has been shown repeatedly during the last years and has raised the need for efficient stochastic tools. We present SABRE, a tool for stochastic analysis of biochemical reaction networks. SABRE implements fast adaptive uniformization, a direct numerical approximation algorithm for computing transient solutions of biochemical reaction networks. In addition to the stochastic analysis, SABRE may also conduct deterministic analysis.

IGERS: Inferring Gibbs Energy Changes of Biochemical Reactions from Reaction Similarities

We consider problems in the functional analysis and evolution of combinatorial chemical reaction networks as rule-based, or three-level systems. The first level consists of rules, realized here as graph-grammar representations of reaction mechanisms. The second level consists of stoichiometric networks of molecules and reactions, modeled as hypergraphs. At the third level is the stochastic population process on molecule counts, solved for dynamics of population trajectories or probability distributions. Earlier levels in the hierarchy generate later levels combinatorially, and as a result constraints imposed in earlier and smaller layers can propagate to impose order in the architecture or dynamics in later and larger layers. We develop general methods to study rule algebras, emphasizing system consequences of symmetry; decomposition methods of flows on hypergraphs including the stoichiometric counterpart to Kirchhoff’s current decomposition and work/dissipation relations studied by Wachtel et al.; and the large-deviation theory for currents in a stoichiometric stochastic population process, deriving additive decompositions of the large-deviation function that relate a certain Kirchhoff flow decomposition to the extended Pythagorean theorem from information geometry. The latter result allows us to assign a natural probabilistic cost to topological changes in a reaction network of the kind produced by selection for catalyst-substrate specificity. We develop as an example a model of biological sugar-phosphate chemistry from a rule system published by Andersen et al. It is one of the most potentially combinatorial reaction systems used by biochemistry, yet one in which two ancient, widespread and nearly unique pathways have evolved in the Calvin-Benson cycle and the Pentose Phosphate pathway, which are additionally nearly reverses of one another. We propose a probabilistic accounting in which physiological costs can be traded off against the fitness advantages that select them, and which suggests criteria under which these pathways may be optimal.

We consider the problem of estimating parameter sensitivity for Markovian models of reaction networks. Sensitivity values measure the responsiveness of an output with respect to the model parameters. They help in analysing the network, understanding its robustness properties and identifying the important reactions for a specific output. Sensitivity values are commonly estimated using methods that perform finite-difference computations along with Monte Carlo simulations of the reaction dynamics. These methods are computationally efficient and easy to implement, but they produce a biased estimate which can be unreliable for certain applications. Moreover, the size of the bias is generally unknown and hence the accuracy of these methods cannot be easily determined. There also exist unbiased schemes for sensitivity estimation but these schemes can be computationally infeasible, even for very simple networks. Our goal in this paper is to present a new method for sensitivity estimation, which combines the computational efficiency of finite-difference methods with the accuracy of unbiased schemes. Our method is easy to implement and it relies on an exact representation of parameter sensitivity that we recently proved elsewhere. Through examples, we demonstrate that the proposed method can outperform the existing methods, both biased and unbiased, in many situations.

Quantitative analysis of Markov models typically proceeds through numerical methods or simulation-based evaluation. Since the state space of the models can often be large, exact or approximate state aggregation methods (such as lumping or bisimulation reduction) have been proposed to improve the scalability of the numerical schemes. However, none of the existing numerical techniques provides general, explicit bounds on the approximation error, a problem particularly relevant when the level of accuracy affects the soundness of verification results. We propose a novel numerical approach that combines the strengths of aggregation techniques (state-space reduction) with those of simulation-based approaches (automatic updates that adapt to the process dynamics). The key advantage of our scheme is that it provides rigorous precision guarantees under different measures. The new approach, which can be used in conjunction with time uniformisation techniques, is evaluated on two models of chemical reaction networks, a signalling pathway and a prokaryotic gene expression network: it demonstrates marked improvement in accuracy without performance degradation, particularly when compared to known state-space truncation techniques.KeywordsError BoundChemical Master EquationEmpirical ErrorProbabilistic Model CheckUniformisation StepThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.