Parameter estimation of dynamic biological network models using integrated fluxes

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.

Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. We present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. Our methodology has the capability to handle general nonlinear ODE systems. We demonstrate the accuracy of the proposed method through simulation studies, where the estimated parameters and recovered system trajectories are compared with other recent methods. A real data example is also provided. Supplementary materials for this article are available online.

Systems biology aims for building quantitative models to address unresolved issues in molecular biology. In order to describe the behavior of biological cells adequately, gene regulatory networks (GRNs) are intensively investigated. As the validity of models built for GRNs depends crucially on the kinetic rates, various methods have been developed to estimate these parameters from experimental data. For this purpose, it is favorable to choose the experimental conditions yielding maximal information. However, existing experimental design principles often rely on unfulfilled mathematical assumptions or become computationally demanding with growing model complexity. To solve this problem, we combined advanced methods for parameter and uncertainty estimation with experimental design considerations. As a showcase, we optimized three simulated GRNs in one of the challenges from the Dialogue for Reverse Engineering Assessment and Methods (DREAM). This article presents our approach, which was awarded the best performing procedure at the DREAM6 Estimation of Model Parameters challenge. For fast and reliable parameter estimation, local deterministic optimization of the likelihood was applied. We analyzed identifiability and precision of the estimates by calculating the profile likelihood. Furthermore, the profiles provided a way to uncover a selection of most informative experiments, from which the optimal one was chosen using additional criteria at every step of the design process. In conclusion, we provide a strategy for optimal experimental design and show its successful application on three highly nonlinear dynamic models. Although presented in the context of the GRNs to be inferred for the DREAM6 challenge, the approach is generic and applicable to most types of quantitative models in systems biology and other disciplines.

Summary: Here, we present REDEMPTION (Reduced Dimension Ensemble Modeling and Parameter estimation), a toolbox for parameter estimation and ensemble modeling of ordinary differential equations (ODEs) using time-series data. For models with more reactions than measured species, a common scenario in biological modeling, the parameter estimation is formulated as a nested optimization problem based on incremental parameter estimation strategy. REDEMPTION also includes a tool for the identification of an ensemble of parameter combinations that provide satisfactory goodness-of-fit to the data. The functionalities of REDEMPTION are accessible through a MATLAB user interface (UI), as well as through programming script. For computational speed-up, REDEMPTION provides a numerical parallelization option using MATLAB Parallel Computing toolbox.Availability and implementation: REDEMPTION can be downloaded from http://www.cabsel.ethz.ch/tools/redemption.Contact:rudi.gunawan@chem.ethz.ch

Time-series measurements of metabolite concentration have become increasingly more common, providing data for building kinetic models of metabolic networks using ordinary differential equations (ODEs). In practice, however, such time-course data are usually incomplete and noisy, and the estimation of kinetic parameters from these data is challenging. Practical limitations due to data and computational aspects, such as solving stiff ODEs and finding global optimal solution to the estimation problem, give motivations to develop a new estimation procedure that can circumvent some of these constraints. In this work, an incremental and iterative parameter estimation method is proposed that combines and iterates between two estimation phases. One phase involves a decoupling method, in which a subset of model parameters that are associated with measured metabolites, are estimated using the minimization of slope errors. Another phase follows, in which the ODE model is solved one equation at a time and the remaining model parameters are obtained by minimizing concentration errors. The performance of this two-phase method was tested on a generic branched metabolic pathway and the glycolytic pathway of Lactococcus lactis. The results showed that the method is efficient in getting accurate parameter estimates, even when some information is missing.

In this work we develop some parameter and spectral estimation methods in Magnetic Resonance Spectroscopy Signals (MRS). We do this with a dual purpose: first, to enable the full estimation process for the constituent parameters of the time-domain msignal: Amplitude, frequency, damping constant and phase. Second, to generate the parameter estimation in the frequency domain in order to evaluate the effect of the application of time domain estimation methods [2].

This paper presents focus on physical feature extraction and parameter estimation for satellites in space. In the context of increasingly prominent space security issues, it is necessary to improve the security and protection capabilities of the space system, especially the refined feature extraction and recognition technology of space targets. However, the feature extraction methods based on optical or radar imagery are limited by resolution and observation conditions, making it difficult to extract refined features. This paper is based on the attribute scattering center theory, using high-frequency electromagnetic wave detection method. Through the attribute parameter estimation result of the scattering center, the scattering structure on the space target can be identified and its distribution position can be analyzed, and the structure and size of the typical parts in the space target can be extracted. Aiming at the problem of high parameter dimension and large amount of calculation, an attributed scattering centers parameter estimation method with amplitude and phase separation is proposed, which reduces the time complexity of parameter estimation by an order of magnitude, saves system resources and increases the parameter estimation rate, and the estimation accuracy is better than 90%. In terms of the efficiency and accuracy of parameter estimation from simulation and experimental data, the effectiveness of the proposed method in this paper is verified.

A main aspect in computational modelling of biological systems is the determination of model structure and model parameters. Due to economical and technical reasons, only part of these details are well characterized, while the rest are unknown. To deal with this difficulty, many reverse engineering and parameter estimation methods have been proposed in the literature, however these methods often need an amount of experimental data not always available. In this paper we propose an alternative approach, which overcomes model indetermination solving an Optimization Problem (OP) with an objective function that, similarly to Flux Balance Analysis, is derived from an empirical biological knowledge and does not require large amounts of data. The system behaviour is described by a set of Ordinary Differential Equations (ODE). Model indetermination is resolved selecting time-varying coefficients that maximize/ minimize the objective function at each ODE integration step. Moreover, to facilitate the modelling phase we provide a graphical formalism, based on Petri Nets, which can be used to derive the corresponding ODEs and OP. Finally, the approach is illustrated on a case study focused on cancer metabolism.

Models for biological systems derived from the generalized mass action law are typically a group of nonlinear ordinary differential equations. However, parameters in such models can be separated into two groups: one group of parameters linear in the model and another group of parameters nonlinear in the model. This paper introduces a separable parameter estimation method to estimate the parameters in such models. The separable parameter estimation method has three steps: in the first step, parameters linear in a model are estimated by optimizing the objective function using linear least squares method, assuming all parameters nonlinear in the model are known. In the second step, substituting the estimated parameters in the first step into the objective function yields a new objective function which is only of parameters nonlinear in the model. Then parameters nonlinear in the model are estimated by proper nonlinear estimation methods. In the last step, the estimates of parameters linear in the model are calculated using the estimates of parameters in the second step. To investigate its performance, the separable estimation method is applied to a biological system and is compared with the conventional parameter estimation methods. Simulation results show the improvement of the separable estimation method.

The most applied principles for parameter estimation are maximum likelihood and least square error. This paper presents a new principle with regard to the parameter estimation of the three-parameter Weibull distribution. By transforming the cumulative distribution function, constructed is a mapping from the value of the random variable and its corresponding cumulative distribution probability to the scale parameter. The scale parameter estimated by such a mapping is the random variable value and the corresponding cumulative distribution probability dependent when the shape parameter and/ or location parameter applied in the mapping is subject to error. Given a set of random variable values or a set of sample values, a larger error in the shape/location parameter brings about larger differences between the scale parameter values obtained with the individual random variable values or sample values, respectively. Based on such a causal relationship between the discrepancy and the shape/ location parameter value applied in the mapping relation, a new parameter estimation method is proposed. For the Weibull distribution parameter estimation according to a set of sample values, the right shape parameter and location parameter are those minimizing the discrepancy between the scale parameter values obtained with the individual sample values, respectively. Case studies demonstrate that the proposed method outperforms the maximum likelihood method and the Weibull plot-based least squares method.

This paper is concerned with the parameter estimation and adaptive control problem for a sort of strict‐feedback nonlinear systems with unknown parameters. In order to deal with the nonlinear functions, we develop a kind of least‐squares estimator in the parameter estimation part. With such an estimator, we design the state‐feedback control law and analyze the properties of the closed‐loop system. To be concrete, the parameter estimator can guarantee the boundedness of parameter estimate error. At the same time, the adaptive control law is proposed to make the system states converge to zero and the equilibrium stable globally under some common assumptions. Finally, simulations are given to demonstrate the theoretical results.

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Ordinary differential equation (ODE)-based models of signal transduction pathways often contain parameters that are unidentifiable or unmeasurable by experimental data, and calibrating such models to data remains challenging. Here, two efficient parameter estimation methods, cluster Gauss-Newton (CGN) and CellNOpt (CNO), were applied to fit a signaling network model of U266 multiple myeloma cells to the activity dynamics of key proteins in response to vorinostat and/or bortezomib. A logic-based network model was constructed and transformed to 17 ODEs with 79 parameters estimated within broad ranges of biologically plausible values. The top 10% best-fit parameters by both methods had high uncertainties with CV > 50% for the majority of parameters. The root mean square and prediction errors were comparable without statistically significant differences between the two methods. Despite uncertain parameter estimation, protein dynamics after the sequential combination of bortezomib and vorinostat was predicted with reasonable accuracy and precision. Global sensitivity analyses of partial rank correlation coefficients and Sobol sensitivity demonstrated that apoptosis induction was most sensitive to parameters governing the activity of the proteasome-JNK-caspase-8 axis. Simulations revealed that the greatest magnitude of pharmacodynamic drug interactions between bortezomib and vorinostat occurred at caspase-9, AKT, and Bcl-2. Two sequential combinations were explored in silico, and the outcome matched qualitatively with an empirical evaluation of the pharmacodynamic interaction based on cell viability. Overall, the CGN and CNO algorithms performed similarly for this ODE-based network model calibration, and the calibrated model provided meaningful insights into cellular signaling mechanisms in response to pharmacological perturbations.

In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In addition, we provide a parameter estimation method to extract chaotic model parameters from the real data of the chaotic circuit. The parameter estimation method is based on the attractor distribution modeling in the state space, which is compatible with the chaotic system characteristics. Here, a Gaussian mixture model (GMM) is used as a main part of cost function computations in the parameter estimation method. To optimize the cost function, we also apply two recent efficient optimization methods: WOA (Whale Optimization Algorithm), and MVO (Multi-Verse Optimizer) algorithms. The results show the success of the parameter estimation procedure.

Novel recurrent neural network for modelling biological networks: Oscillatory p53 interaction dynamics