State selection in coupled identical biochemical systems with coexisting stable states

This work examines state selection for coupled biochemical systems with coexisting stable states. For biochemically identical biochemical systems, different coupled systems are examined for the coexistence of (a) one steady state and one oscillatory state or (b) two oscillatory states. For case (a), it is revealed that state selection is always governed by two key factors: the values of kinetic parameters and the coupling strength. When the coupling strength is small, the coupled systems remain in the basin of attraction of their original states. When it is sufficiently large, all coupled systems are always entrained, independently of their original states. Furthermore, for the entrainment, which of the two coexisting states is selected depends sensitively on the activity of recycling enzyme (one of kinetic parameters). It is shown that this is because changing the activity of recycling enzyme alters the size of basin of attraction of each state. When both systems in the same oscillatory state are coupled, an additional factor, namely phase shift between two oscillations, may also affect state selection, and coupling may cause the systems to select either the original oscillatory state or the coexisting steady state. In addition to the features of case (a), case (b) also supports quasiperiodic oscillations and synchronisation of two periodic oscillations. Implications of the results for understanding state selection during the evolution of coupled biochemical systems with coexisting stable states are discussed.     

Identification in an anaerobic batch system: global sensitivity analysis, multi-start strategy and optimization criterion selection

Several mathematical models have been developed in anaerobic digestion systems and a variety of methods have been used for parameter estimation and model validation. However, structural and parametric identifiability questions are relatively seldom addressed in the reported AD modeling studies. This paper presents a 3-step procedure for the reliable estimation of a set of kinetic and stoichiometric parameters in a simplified model of the anaerobic digestion process. This procedure includes the application of global sensitivity analysis, which allows to evaluate the interaction among the identified parameters, multi-start strategy that gives a picture of the possible local minima and the selection of optimization criteria or cost functions. This procedure is applied to the experimental data collected from a lab-scale sequencing batch reactor. Two kinetic parameters and two stoichiometric coefficients are estimated and their accuracy was also determined. The classical least-squares cost function appears to be the best choice in this case study.     

Concerted simulations reveal how peroxidase compound III formation results in cellular oscillations

A major problem in mathematical modeling of the dynamics of complex biological systems is the frequent lack of knowledge of kinetic parameters. Here, we apply Brownian dynamics simulations, based on protein three-dimensional structures, to estimate a previously undetermined kinetic parameter, which is then used in biochemical network simulations. The peroxidase-oxidase reaction involves many elementary steps and displays oscillatory dynamics important for immune response. Brownian dynamics simulations were performed for three different peroxidases to estimate the rate constant for one of the elementary steps crucial for oscillations in the peroxidase-oxidase reaction, the association of superoxide with peroxidase. Computed second-order rate constants agree well with available experimental data and permit prediction of rate constants at physiological conditions. The simulations show that electrostatic interactions depress the rate of superoxide association with myeloperoxidase, bringing it into the range necessary for oscillatory behavior in activated neutrophils. Such negative electrostatic steering of enzyme-substrate association presents a novel control mechanism and lies in sharp contrast to the electrostatically-steered fast association of superoxide and Cu/Zn superoxide dismutase, which is also simulated here. The results demonstrate the potential of an integrated and concerted application of structure-based simulations and biochemical network simulations in cellular systems biology.     

Complex,non-monotonic dose-response curves with multiple maxima: Do we (ever) sample densely enough?

We usually expect the dose-response curves of biological responses to quantifiable stimuli to be simple, either monotonic or exhibiting a single maximum or minimum. Deviations are often viewed as experimental noise. However, detailed measurements in plant primary tissue cultures (stem pith explants of kale and tobacco) exposed to varying doses of sucrose, cytokinins (BA or kinetin) or auxins (IAA or NAA) revealed that growth and several biochemical parameters exhibit multiple reproducible, statistically significant maxima over a wide range of exogenous substance concentrations. This results in complex, non-monotonic dose-response curves, reminiscent of previous reports of analogous observations in both metazoan and plant systems responding to diverse pharmacological treatments. These findings suggest the existence of a hitherto neglected class of biological phenomena resulting in dose-response curves exhibiting periodic patterns of maxima and minima, whose causes remain so far uncharacterized, partly due to insufficient sampling frequency used in many studies.     

Protein dynamics and 1/f noise.

It has long been recognized that protein dynamical processes occur over a wide temporal range. However, the functionality of this spectrum of events remains unclear. In this work, a generalized noise function analysis is applied to a collection of diverse protein dynamical systems. It is shown that a power law model with an oscillatory component can adequately describe the time course of a variety of processes. These results suggest that under the appropriate conditions, proteins are in a metastable state. A microscopic, chemical kinetic model based on a Poisson distribution of activation energies is presented. From this model specific functional forms for the parameters of the generalized noise model can be derived. Additionally, a model is presented to described kinetic hole burning effects observed at low temperatures. Scaling laws are derived for these models that provide a connection with the generalized noise analysis.     

RankAggreg, an R package for weighted rank aggregation

Background

Determining the parameters of a mathematical model from quantitative measurements is the main bottleneck of modelling biological systems. Parameter values can be estimated from steady-state data or from dynamic data. The nature of suitable data for these two types of estimation is rather different. For instance, estimations of parameter values in pathway models, such as kinetic orders, rate constants, flux control coefficients or elasticities, from steady-state data are generally based on experiments that measure how a biochemical system responds to small perturbations around the steady state. In contrast, parameter estimation from dynamic data requires time series measurements for all dependent variables. Almost no literature has so far discussed the combined use of both steady-state and transient data for estimating parameter values of biochemical systems.

Results

In this study we introduce a constrained optimization method for estimating parameter values of biochemical pathway models using steady-state information and transient measurements. The constraints are derived from the flux connectivity relationships of the system at the steady state. Two case studies demonstrate the estimation results with and without flux connectivity constraints. The unconstrained optimal estimates from dynamic data may fit the experiments well, but they do not necessarily maintain the connectivity relationships. As a consequence, individual fluxes may be misrepresented, which may cause problems in later extrapolations. By contrast, the constrained estimation accounting for flux connectivity information reduces this misrepresentation and thereby yields improved model parameters.

Conclusion

The method combines transient metabolic profiles and steady-state information and leads to the formulation of an inverse parameter estimation task as a constrained optimization problem. Parameter estimation and model selection are simultaneously carried out on the constrained optimization problem and yield realistic model parameters that are more likely to hold up in extrapolations with the model.     

A novel method for the observation of membrane transporter dynamics.

A new method is proposed for measuring the dynamic properties of a membrane transporter by means of steady-state fluxes. Any voltage-sensitive transporter will give a flow of substrate in the presence of a steady-state periodic membrane potential. The periodic steady-state flow, averaged over one period, is a flux that can be measured by traditional steady-state techniques, such as the radioactive tracer method. The average flux, solely due to the periodic field, is described by a set of Lorentzian functions that depend on the applied periodic field amplitude and frequency. The normal mode amplitudes and frequencies of these Lorentzians are model-independent parameters of the transport mechanism. Measurement of the average flux as a function of the applied periodic frequency permits determination of system relaxation times as the reciprocals of the midpoints of the Lorentzian curves, which in turn can be used to estimate individual rate constants of specific models. It was found by simulation of a six-state model of the electrogenic Na+/glucose cotransporter, using published estimates of the model rate constants, that the periodic field effects can be large and rich with measurable details that can be used to study the mechanism thoroughly. The new method serves in this case to complement and expand on the information obtainable by means of the voltage clamp method. It was also found by means of simulations of a nonelectrogenic six-state cotransporter model that experimentally measurable effects are expected and that results can be used to distinguished among alternative kinetic models as well as to estimate individual rate constants.(ABSTRACT TRUNCATED AT 250 WORDS)     

Frequency encoded biochemical regulation is more accurate than amplitude dependent control

It is generally recognised in engineering that encoding information in a frequency provides resistance to degradation by noise and an enhanced precision of control. This paper demonstrates how the same arguments can be applied to biochemical control networks. It shows that the conversion of an analogue demand signal to an oscillation is stable against corruption by noise in the input and even against corruption by certain internal chaotic motions. The paper also argues that intracellular transmission of frequency encoded information is robust against noise. These arguments are proposed as a partial explanation of why so many biological regulatory systems are periodic.     

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.     

Optimization of bioreactor using metabolic control analysis approach

A new methodology based on a metabolic control analysis (MCA) approach is developed for the optimization of continuous cascade bioreactor system. A general framework for representation of a cascade bioreactor system consisting of a large number of reactors as a single network is proposed. The kinetic and transport processes occurring in the system are represented as a reaction network with appropriate stoichiometry. Such representation of the bioreactor systems makes it amenable to the direct application of the MCA approach. The process sensitivity information is extracted using MCA methodology in the form of flux and concentration control coefficients. The process sensitivity information is shown to be a useful guide for determining the choice of decision variables for the purpose of optimization. A generalized problem of optimization of the bioreactor is formulated in which the decision variables are the operating conditions and kinetic parameters. The gradient of the objective function to be maximized with respect to all decision variables is obtained in the form of response coefficients. This gradient information can be used in any gradient-based optimization algorithm. The efficiency of the proposed technique is demonstrated with two examples taken from literature: biotransformation of crotonobetaine and alcohol fermentation in cascade bioreactor system.     

Bridging from molecular simulation to biochemical networks

How can we make the connection between the three-dimensional structures of individual proteins and understanding how complex biological systems involving many proteins work? The modelling and simulation of protein structures can help to answer this question for systems ranging from multimacromolecular complexes to organelles and cells. On one hand, multiscale modelling and simulation techniques are advancing to permit the spatial and temporal properties of large systems to be simulated using atomic-detail structures. On the other hand, the estimation of kinetic parameters for the mathematical modelling of biochemical pathways using protein structure information provides a basis for iterative manipulation of biochemical pathways guided by protein structure. Recent advances include the structural modelling of protein complexes on the genomic level, novel coarse-graining strategies to increase the size of the system and the time span that can be simulated, and comparative molecular field analyses to estimate enzyme kinetic parameters.     

When the optimal is not the best: parameter estimation in complex biological models

Background

The vast computational resources that became available during the past decade enabled the development and simulation of increasingly complex mathematical models of cancer growth. These models typically involve many free parameters whose determination is a substantial obstacle to model development. Direct measurement of biochemical parameters in vivo is often difficult and sometimes impracticable, while fitting them under data-poor conditions may result in biologically implausible values.

Results

We discuss different methodological approaches to estimate parameters in complex biological models. We make use of the high computational power of the Blue Gene technology to perform an extensive study of the parameter space in a model of avascular tumor growth. We explicitly show that the landscape of the cost function used to optimize the model to the data has a very rugged surface in parameter space. This cost function has many local minima with unrealistic solutions, including the global minimum corresponding to the best fit.

Conclusions

The case studied in this paper shows one example in which model parameters that optimally fit the data are not necessarily the best ones from a biological point of view. To avoid force-fitting a model to a dataset, we propose that the best model parameters should be found by choosing, among suboptimal parameters, those that match criteria other than the ones used to fit the model. We also conclude that the model, data and optimization approach form a new complex system and point to the need of a theory that addresses this problem more generally.     

Dynamics of granzyme B-induced apoptosis: mathematical modeling

Apoptosis is mediated by an intracellular biochemical system that mainly includes proteins (procaspases, caspases, inhibitors, Bcl-2 protein family as well as substances released from mitochondrial intermembrane space). The dynamics of caspase activation and target cleavage in apoptosis induced by granzyme B in a single K562 cell was studied using a mathematical model of the dynamics of granzyme B-induced apoptosis developed in this work. Also the first application of optimization approach to determination of unknown kinetic constants of biochemical apoptotic reactions was presented. The optimization approach involves solving of two problems: direct and inverse. Solving the direct optimization problem, we obtain the initial (baseline) concentrations of procaspases for known kinetic constants through conditional minimization of a cost function based on the principle of minimum protein consumption by the apoptosis system. The inverse optimization problem is aimed at determination of unknown kinetic constants of apoptotic biochemical reactions proceeding from the condition that the optimal concentrations of procaspases resulting from the solution of the direct optimization problem coincide with the observed ones, that is, those determined by biochemical methods. The Multidimensional Index Method was used to perform numerical solution of the inverse optimization problem.     

Error analysis in the determination of the electron microscopical contrast transfer function parameters from experimental power Spectra

Background  

The transmission electron microscope is used to acquire structural information of macromolecular complexes. However, as any other imaging device, it introduces optical aberrations that must be corrected if high-resolution structural information is to be obtained. The set of all aberrations are usually modeled in Fourier space by the so-called Contrast Transfer Function (CTF). Before correcting for the CTF, we must first estimate it from the electron micrographs. This is usually done by estimating a number of parameters specifying a theoretical model of the CTF. This estimation is performed by minimizing some error measure between the theoretical Power Spectrum Density (PSD) and the experimentally observed PSD. The high noise present in the micrographs, the possible local minima of the error function for estimating the CTF parameters, and the cross-talking between CTF parameters may cause errors in the estimated CTF parameters.     

Instrumental noise estimates stabilize and quantify endothelial cell micro-impedance barrier function parameter estimates

A novel transcellular micro-impedance biosensor, referred to as the electric cell-substrate impedance sensor or ECIS, has become increasingly applied to the study and quantification of endothelial cell physiology. In principle, frequency dependent impedance measurements obtained from this sensor can be used to estimate the cell–cell and cell–matrix impedance components of endothelial cell barrier function based on simple geometric models. Few studies, however, have examined the numerical optimization of these barrier function parameters and established their error bounds. This study, therefore, illustrates the implementation of a multi-response Levenberg–Marquardt algorithm that includes instrumental noise estimates and applies it to frequency dependent porcine pulmonary artery endothelial cell impedance measurements. The stability of cell–cell, cell–matrix and membrane impedance parameter estimates based on this approach is carefully examined, and several forms of parameter instability and refinement illustrated. Including frequency dependent noise variance estimates in the numerical optimization reduced the parameter value dependence on the frequency range of measured impedances. The increased stability provided by a multi-response non-linear fit over one-dimensional algorithms indicated that both real and imaginary data should be used in the parameter optimization. Error estimates based on single fits and Monte Carlo simulations showed that the model barrier parameters were often highly correlated with each other. Independently resolving the different parameters can, therefore, present a challenge to the experimentalist and demand the use of non-linear multivariate statistical methods when comparing different sets of parameters.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

Noise propagation in two-step series MAPK cascade

Series MAPK enzymatic cascades, ubiquitously found in signaling networks, act as signal amplifiers and play a key role in processing information during signal transduction in cells. In activated cascades, cell-to-cell variability or noise is bound to occur and thereby strongly affects the cellular response. Commonly used linearization method (LM) applied to Langevin type stochastic model of the MAPK cascade fails to accurately predict intrinsic noise propagation in the cascade. We prove this by using extensive stochastic simulations for various ranges of biochemical parameters. This failure is due to the fact that the LM ignores the nonlinear effects on the noise. However, LM provides a good estimate of the extrinsic noise propagation. We show that the correct estimate of intrinsic noise propagation in signaling networks that contain at least one enzymatic step can be obtained only through stochastic simulations. Noise propagation in the cascade depends on the underlying biochemical parameters which are often unavailable. Based on a combination of global sensitivity analysis (GSA) and stochastic simulations, we developed a systematic methodology to characterize noise propagation in the cascade. GSA predicts that noise propagation in MAPK cascade is sensitive to the total number of upstream enzyme molecules and the total number of molecules of the two substrates involved in the cascade. We argue that the general systematic approach proposed and demonstrated on MAPK cascade must accompany noise propagation studies in biological networks.     

A class of parametrically excited calcium oscillation detectors.

Intracellular Ca2+ oscillations are often a response to external signals such as hormones. Changes in the external signal can alter the frequency, amplitude, or form of the oscillations suggesting that information is encoded in the pattern of Ca2+ oscillations. How might a cell decode this signal? We show that an excitable system whose kinetic parameters are modulated by the Ca2+ concentration can function as a Ca2+ oscillation detector. Such systems have the following properties: (1) They are more sensitive to an oscillatory than to a steady Ca2+ signal. (2) Their response is largely independent of the signal amplitude. (3) They can extract information from a noisy signal. (4) Unlike other frequency sensitive detectors, they have a flat frequency response. These properties make a Ca(2+)-sensitive excitable system nearly ideal for detecting and decoding Ca2+ oscillations. We suggest that Ca2+ oscillations, in concert with these detectors, can act as cellular timekeepers to coordinate related biochemical reactions and enhance their overall efficiency.