Global dynamic optimization approach to predict activation in metabolic pathways

Background

During the last decade, a number of authors have shown that the genetic regulation of metabolic networks may follow optimality principles. Optimal control theory has been succesfully used to compute optimal enzyme profiles considering simple metabolic pathways. However, applying this optimal control framework to more general networks (e.g. branched networks, or networks incorporating enzyme production dynamics) yields problems that are analytically intractable and/or numerically very challenging. Further, these previous studies have only considered a single-objective framework.

Results

In this work we consider a more general multi-objective formulation and we present solutions based on recent developments in global dynamic optimization techniques. We illustrate the performance and capabilities of these techniques considering two sets of problems. First, we consider a set of single-objective examples of increasing complexity taken from the recent literature. We analyze the multimodal character of the associated non linear optimization problems, and we also evaluate different global optimization approaches in terms of numerical robustness, efficiency and scalability. Second, we consider generalized multi-objective formulations for several examples, and we show how this framework results in more biologically meaningful results.

Conclusions

The proposed strategy was used to solve a set of single-objective case studies related to unbranched and branched metabolic networks of different levels of complexity. All problems were successfully solved in reasonable computation times with our global dynamic optimization approach, reaching solutions which were comparable or better than those reported in previous literature. Further, we considered, for the first time, multi-objective formulations, illustrating how activation in metabolic pathways can be explained in terms of the best trade-offs between conflicting objectives. This new methodology can be applied to metabolic networks with arbitrary topologies, non-linear dynamics and constraints.     

Multi-objective optimal control of dynamic bioprocesses using ACADO Toolkit

The optimal design and operation of dynamic bioprocesses gives in practice often rise to optimisation problems with multiple and conflicting objectives. As a result typically not a single optimal solution but a set of Pareto optimal solutions exist. From this set of Pareto optimal solutions, one has to be chosen by the decision maker. Hence, efficient approaches are required for a fast and accurate generation of the Pareto set such that the decision maker can easily and systematically evaluate optimal alternatives. In the current paper the multi-objective optimisation of several dynamic bioprocess examples is performed using the freely available ACADO Multi-Objective Toolkit (http://www.acadotoolkit.org). This toolkit integrates efficient multiple objective scalarisation strategies (e.g., Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint) with fast deterministic approaches for dynamic optimisation (e.g., single and multiple shooting). It has been found that the toolkit is able to efficiently and accurately produce the Pareto sets for all bioprocess examples. The resulting Pareto sets are added as supplementary material to this paper.     

Hierarchical genetic algorithm for near optimal feedforward neural network design

In this paper, we propose a genetic algorithm based design procedure for a multi layer feed forward neural network. A hierarchical genetic algorithm is used to evolve both the neural networks topology and weighting parameters. Compared with traditional genetic algorithm based designs for neural networks, the hierarchical approach addresses several deficiencies, including a feasibility check highlighted in literature. A multi objective cost function is used herein to optimize the performance and topology of the evolved neural network simultaneously. In the prediction of Mackey Glass chaotic time series, the networks designed by the proposed approach prove to be competitive, or even superior, to traditional learning algorithms for the multi layer Perceptron networks and radial basis function networks. Based upon the chosen cost function, a linear weight combination decision making approach has been applied to derive an approximated Pareto optimal solution set. Therefore, designing a set of neural networks can be considered as solving a two objective optimization problem.     

Analysis of metabolic network based on conservation of molecular structure

Recent work has revealed much about chemical reactions inside hundreds of organisms as well as universal characteristics of metabolic networks, which shed light on the evolution of the networks. However, characteristics of individual metabolites have been neglected. For example, some carbohydrates have structures that are decomposed into small molecules by metabolic reactions, but coenzymes such as ATP are mostly preserved. Such differences in metabolite characteristics are important for understanding the universal characteristics of metabolic networks. To quantify the structure conservation of metabolites, we defined the "structure conservation index" (SCI) for each metabolite as the fraction of metabolite atoms restored to their original positions through metabolic reactions. As expected, coenzymes and coenzyme-like metabolites that have reaction loops in the network show a higher SCI. Using the index, we found that the sum of metabolic fluxes is negatively correlated with the structure preservation of metabolite. Also, we found that each reaction path around high SCI metabolites changes independently, while changes in reaction paths involving low SCI metabolites coincide through evolution processes. These correlations may provide a clue to universal properties of metabolic networks.     

Inducible product gene expression technology tailored to bioprocess engineering

Bioprocess engineering has developed as a discipline to design optimal culture conditions and bioreactor operation protocols for production cell lines engineered for constitutive expression of desired protein pharmaceuticals. With the advent of heterologous gene regulation systems it has become possible to fine-tune expression of difficult-to-produce protein pharmaceuticals to optimal levels and to conditionally engineer cell metabolism for the best production performance. However, most of the small-molecules used to trigger expression of product or metabolic engineering product genes are incompatible with downstream processing regulations or process economics. Recent progress in product gene control design has resulted in the development of bioprocess-compatible regulation systems, which are responsive to physical parameters such as temperature or physiologic trigger molecules that are either an inherent part of host cell metabolism or intrinsic components of licensed protein-free cell culture media, such as redox status, vitamin H and gaseous acetaldehyde. While all of these systems have been shown to fine-tune product gene expression independent of the host cell metabolism some of them can be plugged into metabolic networks to capture critical physiologic parameters and convert them into an optimal production response. Assembly of individual product gene control modalities into synthetic networks has recently enabled construction of autonomously regulated time-delay or cell density-sensitive gene circuits, which trigger population-wide induction of product gene expression at a predefined time or culture density. We provide a comprehensive overview on the latest developments in the design of bioprocess-compatible product gene control systems.     

Novel quantitative tools for engineering analysis of hepatocyte cultures in bioartificial liver systems

Extracorporeal bioartificial liver devices (BAL) are perhaps among the most promising technologies for the treatment of liver failure, but significant technical challenges remain in order to develop systems with sufficient processing capacity and of manageable size. One key limitation is that during BAL operation, when the device is exposed to plasma from the patient, hepatocytes are prone to accumulate intracellular lipids and exhibit poor liver-specific functions. Based on hepatic intermediary metabolism, we have utilized mathematical programming techniques to optimize the biochemical environment of hepatocyte cultures towards the desired effect of increased albumin and urea synthesis. To investigate the feasible range of optimal hepatic function, we have obtained a Pareto optimal set of solutions corresponding to liver-specific functions of urea and albumin secretion in the metabolic framework using multiobjective optimization. The importance of amino acids in the supplementation and the criticality of the metabolic pathways have been investigated using logic-based programming techniques. Since the metabolite measurements are bound to be patient specific, and hence subject to variability, uncertainty has to be integrated with system analysis to improve the prediction of hepatic function. We have used the concept of two stage stochastic programming to obtain robust solutions by considering extracellular variability. The proposed analysis represents a new systematic approach to analyze behavior of hepatocyte cultures and optimize different operating parameters for an extracorporeal device based on real-time conditions.     

Automated Bayesian model development for frequency detection in biological time series

Background

The ability to perform quantitative studies using isotope tracers and metabolic flux analysis (MFA) is critical for detecting pathway bottlenecks and elucidating network regulation in biological systems, especially those that have been engineered to alter their native metabolic capacities. Mathematically, MFA models are traditionally formulated using separate state variables for reaction fluxes and isotopomer abundances. Analysis of isotope labeling experiments using this set of variables results in a non-convex optimization problem that suffers from both implementation complexity and convergence problems.

Results

This article addresses the mathematical and computational formulation of ¹³C MFA models using a new set of variables referred to as fluxomers. These composite variables combine both fluxes and isotopomer abundances, which results in a simply-posed formulation and an improved error model that is insensitive to isotopomer measurement normalization. A powerful fluxomer iterative algorithm (FIA) is developed and applied to solve the MFA optimization problem. For moderate-sized networks, the algorithm is shown to outperform the commonly used 13CFLUX cumomer-based algorithm and the more recently introduced OpenFLUX software that relies upon an elementary metabolite unit (EMU) network decomposition, both in terms of convergence time and output variability.

Conclusions

Substantial improvements in convergence time and statistical quality of results can be achieved by applying fluxomer variables and the FIA algorithm to compute best-fit solutions to MFA models. We expect that the fluxomer formulation will provide a more suitable basis for future algorithms that analyze very large scale networks and design optimal isotope labeling experiments.     

Searching for the Pareto frontier in multi-objective protein design

The goal of protein engineering and design is to identify sequences that adopt three-dimensional structures of desired function. Often, this is treated as a single-objective optimization problem, identifying the sequence–structure solution with the lowest computed free energy of folding. However, many design problems are multi-state, multi-specificity, or otherwise require concurrent optimization of multiple objectives. There may be tradeoffs among objectives, where improving one feature requires compromising another. The challenge lies in determining solutions that are part of the Pareto optimal set—designs where no further improvement can be achieved in any of the objectives without degrading one of the others. Pareto optimality problems are found in all areas of study, from economics to engineering to biology, and computational methods have been developed specifically to identify the Pareto frontier. We review progress in multi-objective protein design, the development of Pareto optimization methods, and present a specific case study using multi-objective optimization methods to model the tradeoff between three parameters, stability, specificity, and complexity, of a set of interacting synthetic collagen peptides.     

Pareto front analysis of flight time and energy use in long-distance bird migration

Optimality models are frequently used in studies of long distance bird migration to help understand and predict migration routes, stopover strategies and fuelling behaviour in a spatially varying environment. These models typically evaluate bird behaviour by focusing on a single optimization currency, such as total migration time or energy-use, without explicitly considering trade-offs between the involved objectives. In this paper, we demonstrate that this classic single-objective approach downplays the importance of variability in bird behaviour. In the light of these considerations, we therefore propose to use a full multi-criteria optimization method to isolate the set of non-dominated, efficient or Pareto optimal solutions. Unlike single-objective optimization where there is only one combination of bird behaviour maximizing fitness, the Pareto solution set represents a range of optimal solutions to conflicting objectives. Our results demonstrate that this multi-objective approach provides important new ways of analyzing how environmental factors and behavioural constraints have driven the evolution of migratory behaviour.     

Application of Response Surface Methodology to Optimise the Antioxidant Activity of a Saithe (Pollachius virens) Hydrolysate

The objective of this study was to produce, by an enzymatic hydrolysis process at a pilot scale, a saithe (Pollachius virens) hydrolysate with a high antioxidant activity. Design of experiment methodology, based on laboratory-scale experiments, was used to obtain a behavioral reduced model that allows one to determine the optimal operating conditions maximizing the antioxidant activity. Two specifications were studied: the degree of hydrolysis and the antioxidant activity. The effects of the following hydrolysis parameters (temperature, pH, enzyme concentration, and operating time) were studied and presented as response surfaces. From these results, a multifactorial optimization was performed and the Pareto optimal set of efficient solutions was evaluated. The optimal conditions were tested at laboratory scale and then validated by comparison with tests carried out on a pilot plant.     

Conservation of high-flux backbone in alternate optimal and near-optimal flux distributions of metabolic networks

Constraint-based flux balance analysis (FBA) has proven successful in predicting the flux distribution of metabolic networks in diverse environmental conditions. FBA finds one of the alternate optimal solutions that maximizes the biomass production rate. Almaas et al. have shown that the flux distribution follows a power law, and it is possible to associate with most metabolites two reactions which maximally produce and consume a given metabolite, respectively. This observation led to the concept of high-flux backbone (HFB) in metabolic networks. In previous work, the HFB has been computed using a particular optima obtained using FBA. In this paper, we investigate the conservation of HFB of a particular solution for a given medium across different alternate optima and near-optima in metabolic networks of E. coli and S. cerevisiae. Using flux variability analysis (FVA), we propose a method to determine reactions that are guaranteed to be in HFB regardless of alternate solutions. We find that the HFB of a particular optima is largely conserved across alternate optima in E. coli, while it is only moderately conserved in S. cerevisiae. However, the HFB of a particular near-optima shows a large variation across alternate near-optima in both organisms. We show that the conserved set of reactions in HFB across alternate near-optima has a large overlap with essential reactions and reactions which are both uniquely consuming (UC) and uniquely producing (UP). Our findings suggest that the structure of the metabolic network admits a high degree of redundancy and plasticity in near-optimal flow patterns enhancing system robustness for a given environmental condition.     

Calculating and controlling the error of discrete representations of Pareto surfaces in convex multi-criteria optimization

A discrete set of points and their convex combinations can serve as a sparse representation of the Pareto surface in multiple objective convex optimization. We develop a method to evaluate the quality of such a representation, and show by example that in multiple objective radiotherapy planning, the number of Pareto optimal solutions needed to represent Pareto surfaces of up to five dimensions grows at most linearly with the number of objectives. The method described is also applicable to the representation of convex sets.     

Universal Protein Families and the Functional Content of the Last Universal Common Ancestor

The phylogenetic distribution of Methanococcus jannaschii proteins can provide, for the first time, an estimate of the genome content of the last common ancestor of the three domains of life. Relying on annotation and comparison with reference to the species distribution of sequence similarities results in 324 proteins forming the universal family set. This set is very well characterized and relatively small and nonredundant, containing 301 biochemical functions, of which 246 are unique. This universal function set contains mostly genes coding for energy metabolism or information processing. It appears that the Last Universal Common Ancestor was an organism with metabolic networks and genetic machinery similar to those of extant unicellular organisms.     

An Optimized Cascaded Stochastic Resonance for the Enhancement of Brain MRI

Background

The contrast enhancement of Magnetic Resonance Imaging (MRI) data is quite challenging as the noise present in this data also get amplified in this process. Dynamic Stochastic Resonance (DSR) is the technique that utilizes the noise to enhance the contrast of MRI data.