Optimization-based models have been used to predict cellular behavior for over 25 years. The constraints in these models are derived from genome annotations, measured macro-molecular composition of cells, and by measuring the cell's growth rate and metabolism in different conditions. The cellular goal (the optimization problem that the cell is trying to solve) can be challenging to derive experimentally for many organisms, including human or mammalian cells, which have complex metabolic capabilities and are not well understood. Existing approaches to learning goals from data include (a) estimating a linear objective function, or (b) estimating linear constraints that model complex biochemical reactions and constrain the cell's operation. The latter approach is important because often the known observed biochemical reactions are not enough to explain observations, and hence there is a need to extend automatically the model complexity by learning new chemical reactions. However, this leads to nonconvex optimization problems, and existing tools cannot scale to realistically large metabolic models. Hence, constraint estimation is still used sparingly despite its benefits for modeling cell metabolism, which is important for developing novel antimicrobials against pathogens, discovering cancer drug targets, and producing value-added chemicals. Here, we develop the first approach to estimating constraint reactions from data that can scale to realistically large metabolic models. Previous tools have been used on problems having less than 75 biochemical reactions and 60 metabolites, which limits real-life-size applications. We perform extensive experiments using 75 large-scale metabolic network models for different organisms (including bacteria, yeasts, and mammals) and show that our algorithm can recover cellular constraint reactions, even when some measurements are missing.
While the aforementioned studies have focused on using pre-defined optimization problems, a number of studies have investigated data-driven estimation of cellular goals. Two types of method exist. Methods of the first type estimate how important the different chemical reactions are for the cell's operation, i.e., estimate @math in . For example, ObjFind @cite_1 does this through a nonconvex optimization formulation, and the more recent invFBA @cite_41 solves a linear program. The second type, and the focus of our work, estimate the stoichiometric coefficients of a new chemical reaction, i.e., estimate a new column for matrix @math in . This approach is important because often the known biochemical reactions are not enough to explain observed data. We want to extend the model complexity automatically by learning new chemical reactions, i.e., new columns for @math .
@cite_41 @cite_1
2517280042 2745064379 1579213377 2059145118
Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the Chemical Master Equation. Despite its simple structure, no analytic solutions to the Chemical Master Equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient approximation and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several approximation methods, including the chemical Langevin equation, the system size expansion, moment closure approximations, time-scale separation approximations and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, approximations and inference methods for stochastic chemical kinetics. The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non-unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schl "ogl's model for a chemical bifurcation, a two-dimensional toggle switch, a model for bursty gene expression, and a dimerisation model with multiple stationary distributions. The optimal selection of experimental conditions is essential in maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. A general Bayesian framework for optimal experimental design with nonlinear simulation-based models is proposed. The formulation accounts for uncertainty in model parameters, observables, and experimental conditions. Straightforward Monte Carlo evaluation of the objective function - which reflects expected information gain (Kullback-Leibler divergence) from prior to posterior - is intractable when the likelihood is computationally intensive. Instead, polynomial chaos expansions are introduced to capture the dependence of observables on model parameters and on design conditions. Under suitable regularity conditions, these expansions converge exponentially fast. Since both the parameter space and the design space can be high-dimensional, dimension-adaptive sparse quadrature is used to construct the polynomial expansions. Stochastic optimization methods will be used in the future to maximize the expected utility. While this approach is broadly applicable, it is demonstrated on a chemical kinetic system with strong nonlinearities. In particular, the Arrhenius rate parameters in a combustion reaction mechanism are estimated from observations of autoignition. Results show multiple order-of-magnitude speedups in both experimental design and parameter inference. Detailed modeling of complex reaction systems is becoming increasingly important in the development, analysis, design, and control of chemical reaction processes. For industrial processes, complete incorporation of the chemistry into process models facilitates the minimization of byproduct and pollutant formation, increased efficiency, and improved product quality. Processes that involve complex reaction networks include a variety of noncatalytic and homogeneous or heterogeneous catalytic processes (such as fluid catalytic cracking, combustion, chemical vapor deposition, and alkylation). For some systems, large sets of relevant reactions have been identified for use in simulations.1-3 For others, the availability of advanced computing environments has enabled the automated generation of reaction networks and their models, based on computational descriptions of the reaction types occurring in the system.4-6 The use of such complex models is hindered by two obstacles. First, because of their sheer size and the presence of multiple time scales, these models are difficult to solve. Second, the models contain large numbers of uncertain (and sometimes unknown) kinetic parameters; regression to determine the parameters of complex nonlinear models is both difficult and unreliable, and the sensitivity of simulations to parameter uncertainties cannot be easily ascertained. Furthermore, for the purpose of gaining insights into the reaction system’s behavior, it is usually preferable to obtain simpler models that bring out the key features and components of the system. For these reasons, model simplification and order reduction are becoming central problems in the study of complex reaction systems. The simulation, monitoring, and control of a complex chemical process benefit from the derivation of accurate and reliable reduced models tailored to particular process modeling tasks. Model simplification is directly linked to identification of key reactions and sets of species that give valuable insights into the behavior of the network and how it may be influenced. Advanced control schemes such as model predictive control7 or multiple model adaptive control8 must be based on selecting appropriate reduced models and tracking key sets of species. Ideally, a model order reduction algorithm should have broad applicability, enable analysis at several levels of detail, and provide an assessment of the modeling error.