The intracellular environment in which biological reactions occur is crowded with macromolecules and subdivided into microenvironments that differ in both physical properties and chemical composition. The work described here combines experimental and computational model systems to help understand the consequences of this heterogeneous reaction media on the outcome of coupled enzyme reactions. Our experimental model system for solution heterogeneity is a biphasic polyethylene glycol (PEG)/sodium citrate aqueous mixture that provides coexisting PEG-rich and citrate-rich phases. Reaction kinetics for the coupled enzyme reaction between glucose oxidase (GOX) and horseradish peroxidase (HRP) were measured in the PEG/citrate aqueous two-phase system (ATPS). Enzyme kinetics differed between the two phases, particularly for the HRP. Both enzymes, as well as the substrates glucose and H2O2, partitioned to the citrate-rich phase; however, the Amplex Red substrate necessary to complete the sequential reaction partitioned strongly to the PEG-rich phase. Reactions in ATPS were quantitatively described by a mathematical model that incorporated measured partitioning and kinetic parameters. The model was then extended to new reaction conditions, i.e., higher enzyme concentration. Both experimental and computational results suggest mass transfer across the interface is vital to maintain the observed rate of product formation, which may be a means of metabolic regulation in vivo. Although outcomes for a specific system will depend on the particulars of the enzyme reactions and the microenvironments, this work demonstrates how coupled enzymatic reactions in complex, heterogeneous media can be understood in terms of a mathematical model.

Subcellular compartmentalization of biomolecules and their reactions is common in biology and provides a general strategy for improving and/or controlling kinetics in metabolic pathways that contain multiple sequential enzymes. Enzymes can be colocalized in multiprotein complexes, on scaffolds or inside subcellular organelles. Liquid organelles formed by intracellular phase coexistence could provide an additional means of sequential enzyme colocalization. Here we use experiment and computation to explore the kinetic consequences of sequential enzyme compartmentalization into model liquid organelles in a crowded polymer solution. Two proteins of the de novo purine biosynthesis pathway, ASL (adenylosuccinate lyase, Step 8) and ATIC (5-aminoimidazole-4-carboxamide ribonucleotide transformylase/inosine monophosphate cyclohydrolase, Steps 9 and 10), were studied in a polyethylene glycol/dextran aqueous two-phase system. Dextran-rich phase droplets served as model liquid compartments for enzyme colocalization. In this system, which lacks any specific binding interactions between the phase-forming polymers and the enzymes, we did not observe significant rate enhancements from colocalization for the overall reaction under our experimental conditions. The experimental results were used to adapt a mathematical model to quantitatively describe the kinetics. The mathematical model was then used to explore additional, experimentally inaccessible conditions to predict when increased local concentrations of enzymes and substrates can (or cannot) be expected to yield increased rates of product formation. Our findings indicate that colocalization within these simplified model liquid organelles can lead to enhanced metabolic rates under some conditions, but that very strong partitioning into the phase that serves as the compartment is necessary. In vivo, this could be provided by specific binding affinities between components of the liquid compartment and the molecules to be localized within it.

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The issue of state estimation of an aggregation process through (1) using model reduction to obtain a tractable approximation of the governing dynamics and (2) designing a fast moving-horizon estimator for the reduced-order model is addressed. The method of moments is first used to reduce the governing integro-differential equation down to a nonlinear ordinary differential equation. This reduced-order model is then simulated for both batch and continuous processes and the results are shown to agree with constant Number Monte Carlo simulation results of the original model. Next, the states of the reduced order model are estimated in a moving horizon estimation approach. For this purpose, Carleman linearization is first employed and the nonlinear system is represented in a bilinear form. This representation lessens the computation burden of the estimation problem by allowing for analytical solution of the state variables as well as sensitivities with respect to decision variables. V C 2016 American Institute of Chemical Engineers AIChE J, 62: 1557-1567, 2016 Keywords: mathematical modeling, simulation, moving horizon estimation, model reduction, coagulation process IntroductionIn chemical engineering, material science, and biology, there are a multitude of processes characterized by dispersed phenomena. Systems involving such processes merit a particle population study rather than the traditional mass balance performed for continuous media. There are numerous examples of these processes ranging from industrial applications to biological reactions. Crystallization, polymerization, viral infections, and collocalization of enzymes in cells are just a few instances of these processes that occur in our everyday life and motivate the work in this article. The common characteristic of all these processes are the individual members of the population or the particles. These particles are distinguished by their type, size and/or composition. Mostly, a population balance governs the dynamic behavior of particulate systems. This results in complex mathematical models, specifically for systems in which the particles are characterized by two or more properties. The simplest system, in this area, is a twocomponent aggregation system with no chemical reactions. This system is mostly used in pharmaceutical applications where through use of a solvent called excipient the particles in a drug powder adhere together and form granules. In an ideal granulation process, the components, solvent, and solute, are well mixed and the distribution of the solute mass is uniform. The deviation from the mean of the solute mass in aggregates quantifies the granulation quality which is referred to as blending degree. However, this output property is not easily measurable during evolution of the process. This article focuses on these bicomponent granulation processes and estimation of the distribution of components in the product.The population balance equation for granulation systems determines the dynamics of a bivariate distribution function in fo...

Model Predictive Control for nonlinear systems involves a nonlinear dynamic optimization (NDO) step, which is required to be solved repeatedly. This step is computationally demanding, specially in dealing with constrained and/or nonlinear large-scale systems. This paper presents a method for accelerating the NDO in state-feedback regulation problems. Exploiting Carleman approximation, this method represents the nonlinear dynamics in a bilinear form and discretizes the resulting system in the time domain. The gradient and Hessian of the cost function with respect to the feedback gains are also analytically derived. The Carleman approximation of the nonlinear system may introduce errors in the prediction and sensitivity analysis. The manuscript derives a criterion under which the input-to-state stability of the new design is guaranteed. The proposed MPC is implemented in a chemical reactor example.