Michael A. Henson

Optimization of fed-batch Saccharomyces cerevisiae fermentation using dynamic flux balance models.

We developed a dynamic flux balance model for fed‐batch Saccharomyces cerevisiae fermentation that couples a detailed steady‐state description of primary carbon metabolism with dynamic mass balances on key extracellular species. Model‐based dynamic optimization is performed to determine fed‐batch operating policies that maximize ethanol productivity and/or ethanol yield on glucose. The initial volume and glucose concentrations, the feed flow rate and dissolved oxygen concentration profiles, and the final batch time are treated as decision variables in the dynamic optimization problem. Optimal solutions are generated to analyze the tradeoff between maximal productivity and yield objectives. We find that for both cases the prediction of a microaerobic region is significant. The optimization results are sensitive to network model parameters for the growth associated maintenance and P/O ratio. The results of our computational study motivate continued development of dynamic flux balance models and further exploration of their application to productivity optimization in biochemical reactors.

Selection of model parameters for off-line parameter estimation

Mechanistic dynamic models often contain unknown parameters whose values are difficult to determine even with highly specialized laboratory experiments. A practical approach is to estimate such parameters from available process data. Typically only a subset of the parameters can be estimated due to restrictions imposed by the model structure, lack of measurements, and limited data. We present a simple parameter selection method which accounts for the first two factors independent of the data available for parameter estimation. The magnitude of each parameter effect on the measured variables is quantified by applying principal-component analysis to the steady-state parameter-output sensitivity matrix. The uniqueness of each parameter effect is determined by computing the minimum distance between the sensitivity vector of the particular parameter and the vector spaces spanned by sensitivity vectors of the parameters already selected for estimation. A recursive algorithm that provides a tradeoff between the magnitude and linear independence of parameter effects yields a ranking of the parameters according to their inherent ease of estimation. The parameter-selection procedure is applied to the problem of kinetic parameter estimation for an industrial model of a polymerization reactor. For this specific example, the proposed method yields superior estimation results than those obtained with a parameter-selection technique based on the Fisher information matrix (FIM).

Steady-state and dynamic flux balance analysis of ethanol production by Saccharomyces cerevisiae

Steady-state and dynamic flux balance analysis (DFBA) was used to investigate the effects of metabolic model complexity and parameters on ethanol production predictions for wild-type and engineered Saccharomyces cerevisiae. Three metabolic network models ranging from a single compartment representation of metabolism to a genome-scale reconstruction with seven compartments and detailed charge balancing were studied. Steady-state analysis showed that the models generated similar wild-type predictions for the biomass and ethanol yields, but for ten engineered strains the seven compartment model produced smaller ethanol yield enhancements. Simplification of the seven compartment model to two intracellular compartments produced increased ethanol yields, suggesting that reaction localisation had an impact on mutant phenotype predictions. Further analysis with the seven compartment model demonstrated that steady-state predictions can be sensitive to intracellular model parameters, with the biomass yield exhibiting high sensitivity to ATP utilisation parameters and the biomass composition. The incorporation of gene expression data through the zeroing of metabolic reactions associated with unexpressed genes was shown to produce negligible changes in steady-state predictions when the oxygen uptake rate was suitably constrained. Dynamic extensions of the single and seven compartment models were developed through the addition of glucose and oxygen uptake expressions and transient extracellular balances. While the dynamic models produced similar predictions of the optimal batch ethanol productivity for the wild type, the single compartment model produced significantly different predictions for four implementable gene insertions. A combined deletion/overexpression/insertion mutant with improved ethanol productivity capabilities was computationally identified by dynamically screening multiple combinations of the ten metabolic engineering strategies. The authors concluded that extensive compartmentalisation and detailed charge balancing can be important for reliably screening metabolic engineering strategies that rely on modification of the global redox balance and that DFBA offers the potential to identify novel mutants for enhanced metabolite production in batch and fed-batch cultures.

Small-World Network Models of Intercellular Coupling Predict Enhanced Synchronization in the Suprachiasmatic Nucleus

The suprachiasmatic nucleus (SCN) of the hypothalamus is a multioscillator system that drives daily rhythms in mammalian behavior and physiology. Based on recent data implicating vasoactive intestinal polypeptide (VIP) as the key intercellular synchronizing agent, we developed a multicellular SCN model to investigate the effects of cellular heterogeneity and intercellular connectivity on circadian behavior. A 2-dimensional grid was populated with 400 model cells that were heterogeneous with respect to their uncoupled rhythmic behavior (intrinsic and damped pacemakers with a range of oscillation periods) and VIP release characteristics (VIP producers and nonproducers). We constructed small-world network architectures in which local connections between VIP producing cells and their 4 nearest neighbors were augmented with random connections, resulting in long-range coupling across the grid. With only 10% of the total possible connections, the small-world network model was able to produce similar phase synchronization indices as a mean-field model with VIP producing cells connected to all other cells. Partial removal of random connections decreased the synchrony among neurons, the amplitude of VIP and cAMP response element binding protein oscillations, the mean period of intrinsic periods across the population, and the percentage of oscillating cells. These results indicate that small-world connectivity provides the optimal compromise between the number of connections and control of circadian amplitude and synchrony. This model predicts that small decreases in long-range VIP connections in the SCN could have dramatic effects on period and amplitude of daily rhythms, features commonly described with aging.

Cell population modelling of yeast glycolytic oscillations

We investigated a cell-population modelling technique in which the population is constructed from an ensemble of individual cell models. The average value or the number distribution of any intracellular property captured by the individual cell model can be calculated by simulation of a sufficient number of individual cells. The proposed method is applied to a simple model of yeast glycolytic oscillations where synchronization of the cell population is mediated by the action of an excreted metabolite. We show that smooth one-dimensional distributions can be obtained with ensembles comprising 1000 individual cells. Random variations in the state and/or structure of individual cells are shown to produce complex dynamic behaviours which cannot be adequately captured by small ensembles.

Integrating Cell Cycle Progression, Drug Penetration and Energy Metabolism to Identify Improved Cancer Therapeutic Strategies

The effectiveness of chemotherapeutic drugs in tumors is reduced by multiple effects including drug diffusion and variable susceptibility of local cell populations. We hypothesized that quantifying the interactions between drugs and tumor microenvironments could be used to identify more effective anti-cancer strategies. To test this hypothesis we created a mathematical model that integrated intracellular metabolism, nutrient and drug diffusion, cell-cycle progression, cellular drug effects, and drug pharmacokinetics. To our knowledge, this is the first model that combines these elements and has coupled them to experimentally derived parameters. Drug cytotoxicity was assumed to be cell-cycle phase specific, and progression through the cell cycle was assumed to be dependent on ATP generation. The model consisted of a coupled set of nonlinear partial differential, ordinary differential and algebraic equations with an outer free boundary, which was solved using orthogonal collocation on a moving grid of finite elements. Model simulations showed the existence of an optimum drug diffusion coefficient: a low diffusivity prevents effective penetration before the drug is cleared from the blood and a high diffusivity limits drug retention. This result suggests that increasing the molecular weight of the anti-cancer drug paclitaxel from 854 to approximately 20,000 by nanoparticle conjugation would improve its efficacy. The simulations also showed that fast growing tumors are less responsive to therapy than are slower tumors with more quiescent cells, demonstrating the competing effects of regrowth and cytotoxicity. The therapeutic implications of the simulation results are that (1) monolayer cultures are inadequate for accurately determining therapeutic effects in vitro, (2) decreasing the diffusivity of paclitaxel could increase its efficacy, and (3) measuring the proliferation fraction in tumors could enhance the prediction of therapeutic efficacy.

Direct adaptive control of partially known nonlinear systems

A direct adaptive control strategy for a class of single-input/single-output nonlinear systems is presented. The major advantage of the proposed method is that a detailed dynamic nonlinear model is not required for controller design. The only information required about the plant is measurements of the state variables, the relative degree, and the sign of a Lie derivative which appears in the associated input-output linearizing control law. Unknown controller functions are approximated using locally supported radial basis functions that are introduced only in regions of the state space where the closed-loop system actually evolves. Lyapunov stability analysis is used to derive parameter update laws which ensure (under certain assumptions) the state vector remains bounded and the plant output asymptotically tracks the output of a linear reference model. The technique is successfully applied to a nonlinear biochemical reactor model.

A Multiscale Model to Investigate Circadian Rhythmicity of Pacemaker Neurons in the Suprachiasmatic Nucleus

The suprachiasmatic nucleus (SCN) of the hypothalamus is a multicellular system that drives daily rhythms in mammalian behavior and physiology. Although the gene regulatory network that produces daily oscillations within individual neurons is well characterized, less is known about the electrophysiology of the SCN cells and how firing rate correlates with circadian gene expression. We developed a firing rate code model to incorporate known electrophysiological properties of SCN pacemaker cells, including circadian dependent changes in membrane voltage and ion conductances. Calcium dynamics were included in the model as the putative link between electrical firing and gene expression. Individual ion currents exhibited oscillatory patterns matching experimental data both in current levels and phase relationships. VIP and GABA neurotransmitters, which encode synaptic signals across the SCN, were found to play critical roles in daily oscillations of membrane excitability and gene expression. Blocking various mechanisms of intracellular calcium accumulation by simulated pharmacological agents (nimodipine, IP3- and ryanodine-blockers) reproduced experimentally observed trends in firing rate dynamics and core-clock gene transcription. The intracellular calcium concentration was shown to regulate diverse circadian processes such as firing frequency, gene expression and system periodicity. The model predicted a direct relationship between firing frequency and gene expression amplitudes, demonstrated the importance of intracellular pathways for single cell behavior and provided a novel multiscale framework which captured characteristics of the SCN at both the electrophysiological and gene regulatory levels.

Constrained output feedback control of a multivariable polymerization reactor

A multivariable nonlinear control strategy which accounts for unmeasured state variables and input constraints is developed for the free-radical polymerization of methyl methacrylate in a continuous stirred tank reactor. Monomer concentration and reactor temperature are controlled using a technique which combines nonlinear input-output decoupling and linear model predictive control. Input constraints are handled explicitly by applying linear model predictive control to the constrained linear system obtained after feedback linearization and constraint transformation. Unmeasured initiator and solvent concentrations are accounted for by treating the live polymer concentration as an unknown parameter which is estimated online. The performance of the control strategy is compared to other nonlinear control techniques through closed-loop simulations.

Feedback linearizing control of discrete-time nonlinear systems with input constraints

A feedback linearizing control strategy for discrete-time nonlinear systems subject to input constraints is proposed. The control system comprises: (i) a feedback linearizing controller; (ii) a constraint mapping algorithm that transforms the actual input constraints into constraints on the feedback linearized system; and (iii) a linear model predictive controller that regulates the resulting constrained linear system. Closed-loop stability analysis is a challenging problem because the transformed constraints are state dependent. Sufficient conditions for asymptotic stability are presented for fully and partially feedback linearizable systems. As part of the analysis, a new stability result for unconstrained discrete-time nonlinear systems which parallels a well-known continuous-time result is derived.