A robust methodology for kinetic model parameter estimation for biocatalytic reactions

Effective estimation of parameters in biocatalytic reaction kinetic expressions are very important when building process models to enable evaluation of process technology options and alternative biocatalysts. The kinetic models used to describe enzyme‐catalyzed reactions generally include several parameters, which are strongly correlated with each other. State‐of‐the‐art methodologies such as nonlinear regression (using progress curves) or graphical analysis (using initial rate data, for example, the Lineweaver‐Burke plot, Hanes plot or Dixon plot) often incorporate errors in the estimates and rarely lead to globally optimized parameter values. In this article, a robust methodology to estimate parameters for biocatalytic reaction kinetic expressions is proposed. The methodology determines the parameters in a systematic manner by exploiting the best features of several of the current approaches. The parameter estimation problem is decomposed into five hierarchical steps, where the solution of each of the steps becomes the input for the subsequent step to achieve the final model with the corresponding regressed parameters. The model is further used for validating its performance and determining the correlation of the parameters. The final model with the fitted parameters is able to describe both initial rate and dynamic experiments. Application of the methodology is illustrated with a case study using the ω‐transaminase catalyzed synthesis of 1‐phenylethylamine from acetophenone and 2‐propylamine. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Parameter estimation in a special reaction-diffusion system modelling man-environment diseases

An approximation scheme for a reaction-diffusion system with distributed feedback through the boundary is developed. It is used to estimate the strength of the feedback mechanisms from measurements of the states. The results are illustrated by numerical examples.     

A different approach to generating the data for parameter estimation with the heterotrophic activity method

This paper outlines a different approach to generating the data for V_max and T_t estimation with the Wright-Hobbie [1] method of measuring heterotrophic activities in aquatic environments. To be certain that the incubation times chosen are appropriate for all concentrations of substrate tested, and to increase the precision of the kinetic parameter estimates, we have adopted the approach of using kinetic plots derived from independent time-course studies performed at each concentration of substrate and analyzed by non-linear regression analysis. In keeping with our interest in the impact of acidification on aquatic microbial activities, we have applied this approach to the sediments and water column of the acid-stressed Silver Lake.     

Statistical methods for model comparison in parameter estimation problems for distributed systems

In this note we outline some recent results on the development of a statistical testing methodology for inverse problems involving partial differential equation models. Applications to several problems from biology are presented. The statistical tests, which are in the spirit of analysis of variance (ANOVA), are based on asymptotic distributional results for estimators and residuals in a least squares approach.Research supported in part under grants NSF MCS 8504316, NASA NAG-1-517, and AFOSRF-49620-86-C-0111. Part of this research was carried out while the first author was a visiting scientist at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA, which is operated under NASA contracts NASI-18107 and NASI-18605     

Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems

Background

Parameter estimation for differential equation models of intracellular processes is a highly relevant bu challenging task. The available experimental data do not usually contain enough information to identify all parameters uniquely, resulting in ill-posed estimation problems with often highly correlated parameters. Sampling-based Bayesian statistical approaches are appropriate for tackling this problem. The samples are typically generated via Markov chain Monte Carlo, however such methods are computationally expensive and their convergence may be slow, especially if there are strong correlations between parameters. Monte Carlo methods based on Euclidean or Riemannian Hamiltonian dynamics have been shown to outperform other samplers by making proposal moves that take the local sensitivities of the system’s states into account and accepting these moves with high probability. However, the high computational cost involved with calculating the Hamiltonian trajectories prevents their widespread use for all but the smallest differential equation models. The further development of efficient sampling algorithms is therefore an important step towards improving the statistical analysis of predictive models of intracellular processes.

Results

We show how state of the art Hamiltonian Monte Carlo methods may be significantly improved for steady state dynamical models. We present a novel approach for efficiently calculating the required geometric quantities by tracking steady states across the Hamiltonian trajectories using a Newton-Raphson method and employing local sensitivity information. Using our approach, we compare both Euclidean and Riemannian versions of Hamiltonian Monte Carlo on three models for intracellular processes with real data and demonstrate at least an order of magnitude improvement in the effective sampling speed. We further demonstrate the wider applicability of our approach to other gradient based MCMC methods, such as those based on Langevin diffusions.

Conclusion

Our approach is strictly benefitial in all test cases. The Matlab sources implementing our MCMC methodology is available from https://github.com/a-kramer/ode_rmhmc.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-253) contains supplementary material, which is available to authorized users.     

A parameter sensitivity methodology in the context of HIV delay equation models

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.     

Robustness of post-reconstruction and direct kinetic parameter estimates under rigid head motion in dynamic brain PET imaging

ObjectiveDynamic PET imaging is extensively used in brain imaging to estimate parametric maps. Inter-frame motion can substantially disrupt the voxel-wise time-activity curves (TACs), leading to erroneous maps during kinetic modelling. Therefore, it is important to characterize the robustness of kinetic parameters under various motion and kinetic model related factors.MethodsFully 4D brain simulations ([¹⁵O]H₂O and [¹⁸F]FDG dynamic datasets) were performed using a variety of clinically observed motion patterns. Increasing levels of head motion were investigated as well as varying temporal frames of motion initiation. Kinetic parameter estimation was performed using both post-reconstruction kinetic analysis and direct 4D image reconstruction to assess bias from inter-frame emission blurring and emission/attenuation mismatch.ResultsKinetic parameter bias heavily depends on the time point of motion initiation. Motion initiated towards the end of the scan results in the most biased parameters. For the [¹⁸F]FDG data, k₄ is the more sensitive parameter to positional changes, while K₁ and blood volume were proven to be relatively robust to motion. Direct 4D image reconstruction appeared more sensitive to changes in TACs due to motion, with parameter bias spatially propagating and depending on the level of motion.ConclusionKinetic parameter bias highly depends upon the time frame at which motion occurred, with late frame motion-induced TAC discontinuities resulting in the least accurate parameters. This is of importance during prolonged data acquisition as is often the case in neuro-receptor imaging studies. In the absence of a motion correction, use of TOF information within 4D image reconstruction could limit the error propagation.     

A simple calibrated model of Amazon rainforest productivity based on leaf biochemical properties

A simple ‘big leaf’ ecosystem gas exchange model was developed, using eddy covariance data collected at an undisturbed tropical rainforest in south-western Amazonia (Brazil). The model used mechanistic equations of canopy biochemistry combined with an empirical stomatal model describing responses to light, temperature and humidity. After calibration, the model was driven using hourly data from a weather station at the top of the tower at the measurement site, yielding an estimate of gross primary productivity (annual photosynthesis) in 1992/1993 of about 200 mol C m^?2 year ^?. Although incoming photon flux density emerged as the major control on photosynthesis in this forest, at a given PAR CO₂ assimilation rates were higher in the mornings than in the afternoons. This was attributable to stomatal closure in the afternoon in response to increasing canopy-to-air vapour pressure differences. Although most morning gas exchange was clearly limited by the rate of electron transport, afternoon gas exchange was generally observed to be very nearly co-limited by both Rubisco activity (V_max) and electron transport rate. The sensitivity of the model to changes in nitrogen allocation showed that the modelled ratio of V_max to electron transport (J_max) served nearly to maximize the annual carbon gain, and indeed, would have resulted in almost maximum annual carbon gain at the pre-industrial revolution atmospheric CO₂ concentration of 27 Pa. Modelled gross primary productivity (GPP) was somewhat lower at 27 Pa, being about 160 mol C m^?2 year^?1. The model suggests that, in the absence of any negative feedbacks on GPP, future higher concentrations of atmospheric CO₂ will continue to increase the GPP of this rainforest, up to about 230 mol C m^?2 year^?1 at 70 Pa.     

Hodgkin-Huxley type ion channel characterization: an improved method of voltage clamp experiment parameter estimation

The Hodgkin-Huxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.     

采用遥感手段估算海洋初级生产力研究进展

海洋初级生产力的精确估算对渔业资源评估与管理、海洋生态系统和全球变化等研究具有重要意义.传统的现场测量与估算方法必须依赖于随船采样数据.卫星遥感具有能够获取实时的、大尺度的、动态的海洋环境参数的优点,因此卫星遥感日益成为大尺度海洋初级生产力估算的重要手段.本文从海洋水色传感器的发展历程出发,着重归纳了以叶绿素、浮游植物碳和浮游植物吸收系数为参量的海洋初级生产力的遥感估算方法,并就这3类模型的适应性和复杂程度进行了讨论.在此基础上,进一步分析评价了全球海洋初级生产力遥感估算的研究现状.鉴于当前海洋初级生产力遥感估算研究中存在的问题,今后的研究需要在4个方面进一步加强:1)对全球海洋初级生产力估算进行分区域研究;2)加深对浮游植物吸收系数的研究;3)提高海洋遥感技术水平;4)加强实地测量技术的研究.     

Experimental design for optimal parameter estimation of an enzyme kinetic process based on the analysis of the Fisher information matrix

The investigation of enzyme kinetics is increasingly important, especially for finding active substances and understanding intracellular behaviors. Therefore, the determination of an enzyme's kinetic parameters is crucial. For this a systematic experimental design procedure is necessary to avoid wasting time and resources. The parameter estimation error of a Michaelis-Menten enzyme kinetic process is analysed analytically to reduce the search area as well as numerically to specify the optimum for parameter estimation. From analytical analysis of the Fisher information matrix the fact is obtained, that an enzyme feed will not improve the estimation process, but substrate feeding is favorable with small volume flow. Unconstrained and constrained process conditions are considered. If substrate fed-batch process design is used instead of pure batch experiments the improvements of the Cramer-Rao lower bound of the variance of parameter estimation error reduces to 82% for mu(max) and to 60% for K(m) of the batch values in average.     

Regularized EM algorithm for sparse parameter estimation in nonlinear dynamic systems with application to gene regulatory network inference

Parameter estimation in dynamic systems finds applications in various disciplines, including system biology. The well-known expectation-maximization (EM) algorithm is a popular method and has been widely used to solve system identification and parameter estimation problems. However, the conventional EM algorithm cannot exploit the sparsity. On the other hand, in gene regulatory network inference problems, the parameters to be estimated often exhibit sparse structure. In this paper, a regularized expectation-maximization (rEM) algorithm for sparse parameter estimation in nonlinear dynamic systems is proposed that is based on the maximum a posteriori (MAP) estimation and can incorporate the sparse prior. The expectation step involves the forward Gaussian approximation filtering and the backward Gaussian approximation smoothing. The maximization step employs a re-weighted iterative thresholding method. The proposed algorithm is then applied to gene regulatory network inference. Results based on both synthetic and real data show the effectiveness of the proposed algorithm.     

Parameter estimation in cardiac ionic models

We examine the problem of parameter estimation in mathematical models of excitable cell cardiac electrical activity using the well-known Beeler–Reuter (1977) ionic equations for the ventricular action potential. The estimation problem can be regarded as equivalent to the accurate reconstruction of ionic current kinetics and amplitudes in an excitable cell model, given only action potential experimental data. We show that in the Beeler–Reuter case, all ionic currents may be reasonably reconstructed using an experimental design consisting of action potential recordings perturbed by pseudo-random injection currents.

The Beeler–Reuter model was parameterised into 63 parameters completely defining all membrane current amplitudes and kinetics. Total membrane current was fitted to model-generated experimental data using a ‘data-clamp’ protocol. The experimental data consisted of a default action-potential waveform and an optional series of perturbed waveforms generated by current injections. Local parameter identifiability was ascertained from the reciprocal condition value (1/λ) of the Hessian at the known solution. When fitting to a single action potential waveform, the model was found to be over-determined, having a 1/λ value of 3.6e−14. This value improved slightly to 1.4e−10 when an additional 2 perturbed waveforms were included in the fitting process, suggesting that the additional data did not overly improve the identifiability problem. The additional data, however, did allow the accurate reconstruction of all ionic currents. This indicates that by appropriate experimental design, it may be possible to infer the properties of underlying membrane currents from observation of transmembrane potential waveforms perturbed by pseudo-random currents.     

Parameter estimation for models of chaotic time series

We illustrate the inadequacy for chaotic time series data of widely-used dynamical parameter-estimation procedures based on whole-trajectory comparisons (observation-error fitting), using as case studies the record of measles outbreaks in New York City and simulated data generated by the logistic equation. We explore and reject alternative estimation methods based on matching emergent features of strange attractors; specifically, the Lyapunov exponents and the Hausdorff dimension. We show that partial-trajectory comparison methods (process-error fitting) work well when the system state is completely known through the course of the experiment.     

A general model of reaction kinetics in biological systems

Dynamic mathematical models in biotechnology require, besides the information about the stoichiometry of the biological reaction system, knowledge about the reaction kinetics. Modulation phenomena like limitation, inhibition and activation occur in different forms of competition with the key enzymes responsible for the respective metabolic reaction steps. The identification of a priori unknown reaction kinetics is often a critical task due to the non-linearity and (over-) parameterization of the model equations introduced to account for all the possible modulation phenomena. The contribution of this paper is to propose a general formulation of reaction kinetics, as an extension of the Michaelis-Menten kinetics, which allows limitation/activation and inhibition effects to be described with a reduced number of parameters. The versatility of the new model structure is demonstrated with application examples.     

ECOBAS — A tool to develop ecosystem models exemplified by the shallow lake model EMMO

The modelling and simulation tool ECOBAS was extended in order to include special features supporting the development of ecological models. The «Graphical Model Editor» allows the connection of at least 2 modules in order to build a whole model to run simulations. With the ECOBAS simulation system the model can be tested extensively in order to find appropriate parameter sets («Parameter analysis» and «Parameter estimation») and to identify critical parameters («Sensitivity analysis»). The «Interaction Analysis» shows the internal dependencies of a model. ECOBAS integrates the steps of ecological modelling and creates well readable and complete documentations within one working step, supports modularization of models and the user is rid of the technical and numerical aspects of modelling. Hence ECOBAS is setting up complete, consistent and syntactical correct models.All new features of the ECOBAS-system will be introduced by applying it on the existing ecosystem model EMMO.