A Spatially Continuous Model of Carbohydrate Digestion and Transport Processes in the Colon

A spatially continuous mathematical model of transport processes, anaerobic digestion and microbial complexity as would be expected in the human colon is presented. The model is a system of first-order partial differential equations with context determined number of dependent variables, and stiff, non-linear source terms. Numerical simulation of the model is used to elucidate information about the colon-microbiota complex. It is found that the composition of materials on outflow of the model does not well-describe the composition of material in other model locations, and inferences using outflow data varies according to model reactor representation. Additionally, increased microbial complexity allows the total microbial community to withstand major system perturbations in diet and community structure. However, distribution of strains and functional groups within the microbial community can be modified depending on perturbation length and microbial kinetic parameters. Preliminary model extensions and potential investigative opportunities using the computational model are discussed.     

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.     

Analysis of Chinese hamster ovary cell metabolism through a combined computational and experimental approach

Optimization of cell culture processes can benefit from the systematic analysis of experimental data and their organization in mathematical models, which can be used to decipher the effect of individual process variables on multiple outputs of interest. Towards this goal, a kinetic model of cytosolic glucose metabolism coupled with a population-level model of Chinese hamster ovary cells was used to analyse metabolic behavior under batch and fed-batch cell culture conditions. The model was parameterized using experimental data for cell growth dynamics, extracellular and intracellular metabolite profiles. The results highlight significant differences between the two culture conditions in terms of metabolic efficiency and motivate the exploration of lactate as a secondary carbon source. Finally, the application of global sensitivity analysis to the model parameters highlights the need for additional experimental information on cell cycle distribution to complement metabolomic analyses with a view to parameterize kinetic models.     

Extracting Information from the Power Spectrum of Synaptic Noise

In cortical neurons, synaptic "noise" is caused by the nearly random release of thousands of synapses. Few methods are presently available to analyze synaptic noise and deduce properties of the underlying synaptic inputs. We focus here on the power spectral density (PSD) of several models of synaptic noise. We examine different classes of analytically solvable kinetic models for synaptic currents, such as the "delta kinetic models," which use Dirac delta functions to represent the activation of the ion channel. We first show that, for this class of kinetic models, one can obtain an analytic expression for the PSD of the total synaptic conductance and derive equivalent stochastic models with only a few variables. This yields a method for constraining models of synaptic currents by analyzing voltage-clamp recordings of synaptic noise. Second, we show that a similar approach can be followed for the PSD of the the membrane potential (Vm) through an effective-leak approximation. Third, we show that this approach is also valid for inputs distributed in dendrites. In this case, the frequency scaling of the Vm PSD is preserved, suggesting that this approach may be applied to intracellular recordings of real neurons. In conclusion, using simple mathematical tools, we show that Vm recordings can be used to constrain kinetic models of synaptic currents, as well as to estimate equivalent stochastic models. This approach, therefore, provides a direct link between intracellular recordings in vivo and the design of models consistent with the dynamics and spectral structure of synaptic noise.     

WebCell: a web-based environment for kinetic modeling and dynamic simulation of cellular networks

SUMMARY: WebCell is a web-based environment for managing quantitative and qualitative information on cellular networks and for interactively exploring their steady-state and dynamic behaviors in response to systemic perturbations. It is designed as a user-friendly web interface, allowing users to efficiently construct, visualize, analyze and store reaction network models, thereby facilitating kinetic modeling and in silico simulation of biological systems of interest. A collected model library is also available to provide comprehensive implications for cellular dynamics of the published models.     

Transport and kinetic processes underlying biomolecular interactions in the BIACORE optical biosensor

The transport and kinetic processes describing biomolecular interactions in the BIACORE optical biosensor have been studied with the help of a mathematical model. In comparison to previous models, the model presented here couples, for the first time, transport phenomena in the flow channel with hindered diffusive transport and reactions inside the hydrogel. Simulated experiments based on this model, and two simpler models extant in the literature, are used to identify cases under which the detailed model is essential for accurate prediction of kinetic parameters. It is shown that this model can substantially improve the accuracy of kinetic parameter estimation when transport limitations in the flow channel and/or the hydrogel significantly influence the observed instrument response curves. The model can extend the range of the instrument's applicability to higher concentrations of immobilized species within the hydrogel. It can also be used for accurate design of experiments with the purpose of minimizing errors in the estimation of the kinetic parameters.     

Symbolic-numeric estimation of parameters in biochemical models by quantifier elimination

The sequencing of complete genomes allows analyses of the interactions between various biological molecules on a genomic scale, which prompted us to simulate the global behaviors of biological phenomena on the molecular level. One of the basic mathematical problems in the simulation is the parameter optimization in the kinetic model for complex dynamics, and many estimation methods have been designed. We introduce a new approach to estimate the parameters in biological kinetic models by quantifier elimination (QE), in combination with numerical simulation methods. The estimation method was applied to a model for the inhibition kinetics of HIV proteinase with ten parameters and nine variables, and attained the goodness of fit to 300 points of observed data with the same magnitude as that obtained by the previous estimation methods, remarkably by using only one or two points of data. Furthermore, the utilization of QE demonstrated the feasibility of the present method for elucidating the behavior of the parameters and the variables in the analyzed model. Therefore, the present symbolic-numeric method is a powerful approach to reveal the fundamental mechanisms of kinetic models, in addition to being a computational engine.     

Prediction in the face of uncertainty: a Monte Carlo-based approach for systems biology of cancer treatment

Cancer is known to be a complex disease and its therapy is difficult. Much information is available on molecules and pathways involved in cancer onset and progression and this data provides a valuable resource for the development of predictive computer models that can help to identify new potential drug targets or to improve therapies. Modeling cancer treatment has to take into account many cellular pathways usually leading to the construction of large mathematical models. The development of such models is complicated by the fact that relevant parameters are either completely unknown, or can at best be measured under highly artificial conditions. Here we propose an approach for constructing predictive models of such complex biological networks in the absence of accurate knowledge on parameter values, and apply this strategy to predict the effects of perturbations induced by anti-cancer drug target inhibitions on an epidermal growth factor (EGF) signaling network. The strategy is based on a Monte Carlo approach, in which the kinetic parameters are repeatedly sampled from specific probability distributions and used for multiple parallel simulations. Simulation results from different forms of the model (e.g., a model that expresses a certain mutation or mutation pattern or the treatment by a certain drug or drug combination) can be compared with the unperturbed control model and used for the prediction of the perturbation effects. This framework opens the way to experiment with complex biological networks in the computer, likely to save costs in drug development and to improve patient therapy.     

Making sense of parameter estimation and model simulation in bioprocesses

Most articles that report fitted parameters for kinetic models do not include meaningful statistical information. This study demonstrates the importance of reporting a complete statistical analysis and shows a methodology to perform it, using functionalities implemented in computational tools. As an example, alginate production is studied in a batch stirred-tank fermenter and modeled using the kinetic model proposed by Klimek and Ollis (1980). The model parameters and their 95% confidence intervals are estimated by nonlinear regression. The significance of the parameters value is checked using a hypothesis test. The uncertainty of the parameters is propagated to the output model variables through prediction intervals, showing that the kinetic model of Klimek and Ollis (1980) can simulate with high certainty the dynamic of the alginate production process. Finally, the results obtained in other studies are compared to show how the lack of statistical analysis can hold back a deeper understanding about bioprocesses.     

Incorporating qualitative knowledge in enzyme kinetic models using fuzzy logic

Modeling of metabolic pathway dynamics requires detailed kinetic equations at the enzyme level. In particular, the kinetic equations must account for metabolite effectors that contribute significantly to the pathway regulation in vivo. Unfortunately, most kinetic rate laws available in the literature do not consider all the effectors simultaneously, and much kinetic information exists in a qualitative or semiquantitative form. In this article, we present a strategy to incorporate such information into the kinetic equation. This strategy uses fuzzy logic‐based factors to modify algebraic rate laws that account for partial kinetic characteristics. The parameters introduced by the fuzzy factors are then optimized by use of a hybrid of simplex and genetic algorithms. The resulting model provides a flexible form that can simulate various kinetic behaviors. Such kinetic models are suitable for pathway modeling without complete enzyme mechanisms. Three enzymes in Escherichia coli central metabolism are used as examples: phosphoenolpyruvate carboxylase; phosphoenolpyruvate carboxykinase; and pyruvate kinase I. Results show that, with fuzzy logic‐augmented models, the kinetic data can be much better described. In particular, complex behavior, such as allosteric inhibition, can be captured using fuzzy rules. The resulting models, even though they do not provide additional physical meaning in enzyme mechanisms, allow the model to incorporate semiquantitative information in metabolic pathway models. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 62: 722–729, 1999.     

A comparison of mathematical model predictions to experimental measurements for growth and recombinant protein production in induced cultures of Escherichia coli

Recombinant cell growth and protein synthesis by a recombinant Escherichia coli under various inducing conditions are compared to the predictions of a mathematical model. The mathematical model used was a combination of two literature models: (1) an empirical kinetic model for recombinant growth and product formation and (2) a genetically structured model of the lac promoter-operator on a multicopy plasmid. The experimental system utilized was recombinant E. coli CSH22 bearing the temperature-sensitive plasmid pVH106/172, which codes for the synthesis of beta-galactosidase and the other lac operon genes under the control of a lac promoter. Mathematical model predictions for recombinant beta-galactosidase yield and specific growth rate were compared with fermentation measurements of these same quantities for conditions of chemical induction with cyclic AMP and IPTG, copy number amplification (by shifting culture temperature), and combined chemical induction and copy number amplification. The model successfully predicted experimental product yields for most cases of chemical induction even though the product yields varied from 0.34 x 10(3) to 1500 x 10(3) units/g cell mass. The kinetic model also correctly predicted a decline in the specific growth rate with increasing levels of plasmid and recombinant protein. The model was less successful at predicting product amplification at high copy numbers. A comparison of model predictions and experimental results was also used to investigate some of the assumptions used in constructing the mathematical models.     

Mathematical modeling and optimization of cellulase protein production using Trichoderma reesei RL-P37

The enzyme cellulase, a multienzyme complex made up of several proteins, catalyzes the conversion of cellulose to glucose in an enzymatic hydrolysis-based biomass-to-ethanol process. Production of cellulase enzyme proteins in large quantities using the fungus Trichoderma reesei requires understanding the dynamics of growth and enzyme production. The method of neural network parameter function modeling, which combines the approximation capabilities of neural networks with fundamental process knowledge, is utilized to develop a mathematical model of this dynamic system. In addition, kinetic models are also developed. Laboratory data from bench-scale fermentations involving growth and protein production by T. reesei on lactose and xylose are used to estimate the parameters in these models. The relative performances of the various models and the results of optimizing these models on two different performance measures are presented. An approximately 33% lower root-mean-squared error (RMSE) in protein predictions and about 40% lower total RMSE is obtained with the neural network-based model as opposed to kinetic models. Using the neural network-based model, the RMSE in predicting optimal conditions for two performance indices, is about 67% and 40% lower, respectively, when compared with the kinetic models. Thus, both model predictions and optimization results from the neural network-based model are found to be closer to the experimental data than the kinetic models developed in this work. It is shown that the neural network parameter function modeling method can be useful as a "macromodeling" technique to rapidly develop dynamic models of a process.     

Kinetic isotope effects significantly influence intracellular metabolite 13C labeling patterns and flux determination

Rigorous mathematical modeling of carbon-labeling experiments allows estimation of fluxes through the pathways of central carbon metabolism, yielding powerful information for basic scientific studies as well as for a wide range of applications. However, the mathematical models that have been developed for flux determination from ¹³C labeling data have commonly neglected the influence of kinetic isotope effects on the distribution of ¹³C label in intracellular metabolites, as these effects have often been assumed to be inconsequential. We have used measurements of the ¹³C isotope effects on the pyruvate dehydrogenase enzyme from the literature to model isotopic fractionation at the pyruvate node and quantify the modeling errors expected to result from the assumption that isotope effects are negligible. We show that under some conditions kinetic isotope effects have a significant impact on the ¹³C labeling patterns of intracellular metabolites, and the errors associated with neglecting isotope effects in ¹³C-metabolic flux analysis models can be comparable in size to measurement errors associated with GC–MS. Thus, kinetic isotope effects must be considered in any rigorous assessment of errors in ¹³C labeling data, goodness-of-fit between model and data, confidence intervals of estimated metabolic fluxes, and statistical significance of differences between estimated metabolic flux distributions.     

Reduction of stochastic conductance-based neuron models with time-scales separation

We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.     

Process performance models in the optimization of multiproduct protein production plants

In this work we propose a model that simultaneously optimizes the process variables and the structure of a multiproduct batch plant for the production of recombinant proteins. The complete model includes process performance models for the unit stages and a posynomial representation for the multiproduct batch plant. Although the constant time and size factor models are the most commonly used to model multiproduct batch processes, process performance models describe these time and size factors as functions of the process variables selected for optimization. These process performance models are expressed as algebraic equations obtained from the analytical integration of simplified mass balances and kinetic expressions that describe each unit operation. They are kept as simple as possible while retaining the influence of the process variables selected to optimize the plant. The resulting mixed-integer nonlinear program simultaneously calculates the plant structure (parallel units in or out of phase, and allocation of intermediate storage tanks), the batch plant decision variables (equipment sizes, batch sizes, and operating times of semicontinuous items), and the process decision variables (e.g., final concentration at selected stages, volumetric ratio of phases in the liquid-liquid extraction). A noteworthy feature of the proposed approach is that the mathematical model for the plant is the same as that used in the constant factor model. The process performance models are handled as extra constraints. A plant consisting of eight stages operating in the single product campaign mode (one fermentation, two microfiltrations, two ultrafiltrations, one homogenization, one liquid-liquid extraction, and one chromatography) for producing four different recombinant proteins by the genetically engineered yeast Saccharomyces cerevisiae was modeled and optimized. Using this example, it is shown that the presence of additional degrees of freedom introduced by the process performance models, with respect to a fixed size and time factor model, represents an important development in improving plant design.     

Structure-preserving model reduction of passive and quasi-active neurons

The spatial component of input signals often carries information crucial to a neuron’s function, but models mapping synaptic inputs to the transmembrane potential can be computationally expensive. Existing reduced models of the neuron either merge compartments, thereby sacrificing the spatial specificity of inputs, or apply model reduction techniques that sacrifice the underlying electrophysiology of the model. We use Krylov subspace projection methods to construct reduced models of passive and quasi-active neurons that preserve both the spatial specificity of inputs and the electrophysiological interpretation as an RC and RLC circuit, respectively. Each reduced model accurately computes the potential at the spike initiation zone (SIZ) given a much smaller dimension and simulation time, as we show numerically and theoretically. The structure is preserved through the similarity in the circuit representations, for which we provide circuit diagrams and mathematical expressions for the circuit elements. Furthermore, the transformation from the full to the reduced system is straightforward and depends on intrinsic properties of the dendrite. As each reduced model is accurate and has a clear electrophysiological interpretation, the reduced models can be used not only to simulate morphologically accurate neurons but also to examine computations performed in dendrites.     

Uncovering biases in high throughput screens of G-protein coupled receptors

The ability of high throughput membrane binding assays to detect ligands for G-protein coupled receptors was examined using mathematical models. Membrane assay models were developed using the extended ternary complex model (Samama et al., 1993) as a basis. Ligand binding to whole cells was modeled by adding a G-protein activation step. Results show that inverse agonists bind more slowly and with a lower affinity to receptors in the membrane binding assay than to receptors in whole cells, causing the membrane assay to miss pharmaceutically important inverse agonists. Assay modifications to allow detection of inverse agonists are discussed. Finally, kinetic binding data are shown to provide information about ligand efficacy. This work demonstrates the utility of mathematical modeling in detecting biases in drug-screening assay, and also in suggesting techniques to correct those biases.