Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness.

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional "shooting" methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.

Liposome-based measurement of light-driven chloride transport kinetics of halorhodopsin.

We report a simple and direct fluorimetric vesicle-based method for measuring the transport rate of the light-driven ions pumps as specifically applied to the chloride pump, halorhodopsin, from Natronomonas pharaonis (pHR). Previous measurements were cell-based and methods to determine average single channel permeability challenging. We used a water-in-oil emulsion method for directional pHR reconstitution into two different types of vesicles: lipid vesicles and asymmetric lipid-block copolymer vesicles. We then used stopped-flow experiments combined with fluorescence correlation spectroscopy to determine per protein Cl- transport rates. We obtained a Cl- transport rate of 442 (±17.7) Cl-/protein/s in egg phosphatidyl choline (PC) lipid vesicles and 413 (±26) Cl-/protein/s in hybrid block copolymer/lipid (BCP/PC) vesicles with polybutadine-polyethylene oxide (PB12PEO8) on the outer leaflet and PC in the inner leaflet at a photon flux of 1450 photons/protein/s. Normalizing to a per photon basis, this corresponds to 0.30 (±0.07) Cl-/photon and 0.28 (±0.04) Cl-/photon for pure PC and BCP/PC hybrid vesicles respectively, both of which are in agreement with recently reported turnover of ~500 Cl-/protein/s from flash photolysis experiments and with voltage-clamp measurements of 0.35 (±0.16) Cl-/photon in pHR-expressing oocytes as well as with a pHR quantum efficiency of ~30%.

NetRed, an algorithm to reduce genome-scale metabolic networks and facilitate the analysis of flux predictions.

Flux balance analysis (FBA) of large, genome-scale stoichiometric models (GSMs) is a powerful and popular method to predict cell-wide metabolic activity. FBA typically generates a flux vector containing O(1,000) fluxes. The interpretation of such a flux vector is difficult, even for expert users, because of the large size and complex topology of the underlying metabolic network. This interpretation could be simplified by condensing the network to a reduced, yet fully representative version. Toward this goal we report NetRed, an algorithm that systematically reduces a stoichiometric matrix and a corresponding flux vector to a more easily interpretable form. The reduction offered by NetRed is transparent because it relies purely on matrix algebra and not on optimization. Uniquely, it involves zero information loss; therefore, the original unreduced network can be easily recovered from the reduced network. The inputs to NetRed are (i) a stoichiometric matrix, (ii) a flux vector with numerical flux values, and (iii) a list of "protected" metabolites recommended by the user to remain in the reduced network. NetRed outputs a reduced metabolic network containing a reduced number of metabolites, of which the protected metabolites are a subset. The algorithm also generates a corresponding reduced flux vector. Due to its simplified presentation and easier interpretability, the reduced network allows the user to quickly find fluxes through metabolites and reaction modes or pathways of interest. In this manuscript, we first demonstrate NetRed on a simple network consisting of glycolysis and the pentose phosphate pathway (PPP), wherein NetRed reduced the PPP to a single net reaction. We followed this with applications of NetRed to E. coli and yeast GSMs. NetRed reduced the size of an E. coli GSM by 20- to 30-fold and enabled a comprehensive comparison of aerobic and anaerobic metabolism. The application of NetRed to a yeast GSM allowed for easy mechanistic interpretation of a double-gene knockout that rerouted flux toward dihydroartemisinic acid. When applied to an E. coli strain engineered for enhanced valine production, NetRed allowed for a holistic interpretation of the metabolic rerouting resulting from multiple genetic interventions.

Light-Driven Chloride Transport Kinetics of Halorhodopsin.

Despite growing interest in light-driven ion pumps for use in optogenetics, current estimates of their transport rates span two orders of magnitude due to challenges in measuring slow transport processes and determining protein concentration and/or orientation in membranes in vitro. In this study, we report, to our knowledge, the first direct quantitative measurement of light-driven Cl- transport rates of the anion pump halorohodopsin from Natronomonas pharaonis (NpHR). We used light-interfaced voltage clamp measurements on NpHR-expressing oocytes to obtain a transport rate of 219 (± 98) Cl-/protein/s for a photon flux of 630 photons/protein/s. The measurement is consistent with the literature-reported quantum efficiency of ∼30% for NpHR, i.e., 0.3 isomerizations per photon absorbed. To reconcile our measurements with an earlier-reported 20 ms rate-limiting step, or 35 turnovers/protein/s, we conducted, to our knowledge, novel consecutive single-turnover flash experiments that demonstrate that under continuous illumination, NpHR bypasses this step in the photocycle.

Concentrating membrane proteins using ultrafiltration without concentrating detergents.

Membrane proteins (MPs) are of rapidly growing interest in the design of pharmaceutical products, novel sensors, and synthetic membranes. Ultrafiltration (UF) using commercially available centrifugal concentrators is typically employed for laboratory-scale concentration of low-yield MPs, but its use is accompanied by a concomitant increase in concentration of detergent micelles. We present a detailed analysis of the hydrodynamic processes that control detergent passage during ultrafiltration of MPs and propose methods to optimize detergent passage during protein concentration in larger-scale membrane processes. Experiments were conducted using nonionic detergents, octyl-ß-D glucoside (OG), and decyl-ß-D maltoside (DM) with the bacterial water channel protein, Aquaporin Z (AqpZ) and the light driven chloride pump, halorhodopsin (HR), respectively. The observed sieving coefficient (So ), a measure of detergent passage, was evaluated in both stirred cell and centrifugal systems. So for DM and OG increased with increasing filtrate flux and decreasing shear rates in the stirred cell, that is, with increasing concentration polarization (CP). Similar effects were observed during filtration of MP-detergent (MPD) micelles. However, lower transmission was observed in the centrifugal system for both detergent and MPD systems. This is attributed to free convection-induced shear and hence reduced CP along the membrane surface during centrifugal UF. Thus to concentrate MPs without retention of detergent, design of UF systems that promote CP is required. Biotechnol. Bioeng. 2016;113: 2122-2130. © 2016 Wiley Periodicals, Inc.