Complex coordination of multi-scale cellular responses to environmental stress

Model Parameterization with Quantitative Proteomics: Case Study with Trehalose Metabolism in Saccharomyces cerevisiae

When Saccharomyces cerevisiae undergoes heat stress it stimulates several changes that are necessary for its survival, notably in carbon metabolism. Notable changes include increase in trehalose production and glycolytic flux. The increase in glycolytic flux has been postulated to be due to the regulatory effects in upper glycolysis, but this has not been confirmed. Additionally, trehalose is a useful industrial compound for its protective properties. A model of trehalose metabolism in S. cerevisiae was constructed using Convenient Modeller, a software that uses a combination of convenience kinetics and a genetic algorithm. The model was parameterized with quantitative omics under standard conditions and validated using data collected under heat stress conditions. The completed model was used to show that feedforward activation of pyruvate kinase by fructose 1,6-bisphosphate during heat stress contributes to the increase in metabolic flux. We were also able to demonstrate in silico that overexpression of enzymes involved in production and degradation of trehalose can lead to higher trehalose yield in the cell. By integrating quantitative proteomics with metabolic modelling, we were able to confirm that the flux increase in trehalose metabolic pathways during heat stress is due to regulatory effects and not purely changes in enzyme expression. The overexpression of enzymes involved in trehalose metabolism is a potential approach to be exploited for trehalose production without need for increasing temperature.

The three-legged stool of understanding metabolism: integrating metabolomics with biochemical genetics and computational modeling

Traditional biochemical research has resulted in a good understanding of many aspects of metabolism. However, this reductionist approach is time consuming and requires substantial resources, thus raising the question whether modern metabolomics and genomics should take over and replace the targeted experiments of old. We proffer that such a replacement is neither feasible not desirable and propose instead the tight integration of modern, system-wide omics with traditional experimental bench science and dedicated computational approaches. This integration is an important prerequisite toward the optimal acquisition of knowledge regarding metabolism and physiology in health and disease. The commentary describes advantages and drawbacks of current approaches to assessing metabolism and highlights the challenges to be overcome as we strive to achieve a deeper level of metabolic understanding in the future.

Stepwise inference of likely dynamic flux distributions from metabolic time series data

MOTIVATION

Most metabolic pathways contain more reactions than metabolites and therefore have a wide stoichiometric matrix that corresponds to infinitely many possible flux distributions that are perfectly compatible with the dynamics of the metabolites in a given dataset. This under-determinedness poses a challenge for the quantitative characterization of flux distributions from time series data and thus for the design of adequate, predictive models. Here we propose a method that reduces the degrees of freedom in a stepwise manner and leads to a dynamic flux distribution that is, in a statistical sense, likely to be close to the true distribution.

RESULTS

We applied the proposed method to the lignin biosynthesis pathway in switchgrass. The system consists of 16 metabolites and 23 enzymatic reactions. It has seven degrees of freedom and therefore admits a large space of dynamic flux distributions that all fit a set of metabolic time series data equally well. The proposed method reduces this space in a systematic and biologically reasonable manner and converges to a likely dynamic flux distribution in just a few iterations. The estimated solution and the true flux distribution, which is known in this case, show excellent agreement and thereby lend support to the method.

AVAILABILITY AND IMPLEMENTATION

The computational model was implemented in MATLAB (version R2014a, The MathWorks, Natick, MA). The source code is available at https://github.gatech.edu/VoitLab/Stepwise-Inference-of-Likely-Dynamic-Flux-Distributions and www.bst.bme.gatech.edu/research.php .

CONTACT

mojdeh@gatech.edu or eberhard.voit@bme.gatech.edu.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Nonparametric dynamic modeling

Challenging as it typically is, the estimation of parameter values seems to be an unavoidable step in the design and implementation of any dynamic model. Here, we demonstrate that it is possible to set up, diagnose, and simulate dynamic models without the need to estimate parameter values, if the situation is favorable. Specifically, it is possible to establish nonparametric models for nonlinear compartment models, including metabolic pathway models, if sufficiently many high-quality time series data are available that describe the biological phenomenon under investigation in an appropriate and representative manner. The proposed nonparametric strategy is a variant of the method of Dynamic Flux Estimation (DFE), which permits the estimation of numerical flux profiles from metabolic time series data. However, instead of attempting to formulate these numerical profiles as explicit functions and to optimize their parameter values, as it is done in DFE, the metabolite and flux profiles are used here directly as a scaffold for a library from which values are interpolated and retrieved for the simulation of the differential equations describing the model. Beyond simulations, the proposed methods render it possible to determine steady states from non-steady state data, perform sensitivity analyses, and estimate the Jacobian of the system at a steady state.

Dynamics of the Heat Stress Response of Ceramides with Different Fatty-Acyl Chain Lengths in Baker's Yeast

The article demonstrates that computational modeling has the capacity to convert metabolic snapshots, taken sequentially over time, into a description of cellular, dynamic strategies. The specific application is a detailed analysis of a set of actions with which Saccharomyces cerevisiae responds to heat stress. Using time dependent metabolic concentration data, we use a combination of mathematical modeling, reverse engineering, and optimization to infer dynamic changes in enzyme activities within the sphingolipid pathway. The details of the sphingolipid responses to heat stress are important, because they guide some of the longer-term alterations in gene expression, with which the cells adapt to the increased temperature. The analysis indicates that all enzyme activities in the system are affected and that the shapes of the time trends in activities depend on the fatty-acyl CoA chain lengths of the different ceramide species in the system.

Characterizability of metabolic pathway systems from time series data

Over the past decade, the biomathematical community has devoted substantial effort to the complicated challenge of estimating parameter values for biological systems models. An even more difficult issue is the characterization of functional forms for the processes that govern these systems. Most parameter estimation approaches tacitly assume that these forms are known or can be assumed with some validity. However, this assumption is not always true. The recently proposed method of Dynamic Flux Estimation (DFE) addresses this problem in a genuinely novel fashion for metabolic pathway systems. Specifically, DFE allows the characterization of fluxes within such systems through an analysis of metabolic time series data. Its main drawback is the fact that DFE can only directly be applied if the pathway system contains as many metabolites as unknown fluxes. This situation is unfortunately rare. To overcome this roadblock, earlier work in this field had proposed strategies for augmenting the set of unknown fluxes with independent kinetic information, which however is not always available. Employing Moore-Penrose pseudo-inverse methods of linear algebra, the present article discusses an approach for characterizing fluxes from metabolic time series data that is applicable even if the pathway system is underdetermined and contains more fluxes than metabolites. Intriguingly, this approach is independent of a specific modeling framework and unaffected by noise in the experimental time series data. The results reveal whether any fluxes may be characterized and, if so, which subset is characterizable. They also help with the identification of fluxes that, if they could be determined independently, would allow the application of DFE.

Transcriptional regulation is insufficient to explain substrate-induced flux changes in Bacillus subtilis

Regulation of enzyme expression is one key mechanism by which cells control their metabolic programs. In this work, a quantitative analysis of metabolism in a model bacterium under different conditions shows that expression alone cannot explain the majority of the observed metabolic changes.

Most enzymes are indeed highly expressed in conditions where they are more active.

Quantitatively, however, the observed changes in expression between conditions do not match the changes in activity for most enzymes.

A good quantitative match is only observed for enzymes involved in the TCA cycle.

Metabolomics reveals that increased substrate availability explains only a few instances of changes in activity.

One of the key ways in which microbes are thought to regulate their metabolism is by modulating the availability of enzymes through transcriptional regulation. However, the limited success of efforts to manipulate metabolic fluxes by rewiring the transcriptional network has cast doubt on the idea that transcript abundance controls metabolic fluxes. In this study, we investigate control of metabolic flux in the model bacterium Bacillus subtilis by quantifying fluxes, transcripts, and metabolites in eight metabolic states enforced by different environmental conditions. We find that most enzymes whose flux switches between on and off states, such as those involved in substrate uptake, exhibit large corresponding transcriptional changes. However, for the majority of enzymes in central metabolism, enzyme concentrations were insufficient to explain the observed fluxes—only for a number of reactions in the tricarboxylic acid cycle were enzyme changes approximately proportional to flux changes. Surprisingly, substrate changes revealed by metabolomics were also insufficient to explain observed fluxes, leaving a large role for allosteric regulation and enzyme modification in the control of metabolic fluxes.

Coordination of rapid sphingolipid responses to heat stress in yeast

The regulatory roles of sphingolipids in diverse cell functions have been characterized extensively. However, the dynamics and interactions among the different sphingolipid species are difficult to assess, because de novo biosynthesis, metabolic inter-conversions, and the retrieval of sphingolipids from membranes form a complex, highly regulated pathway system. Here we analyze the heat stress response of this system in the yeast Saccharomyces cerevisiae and demonstrate how the cell dynamically adjusts its enzyme profile so that it is appropriate for operation under stress conditions before changes in gene expression become effective. The analysis uses metabolic time series data, a complex mathematical model, and a custom-tailored optimization strategy. The results demonstrate that all enzyme activities rapidly increase in an immediate response to the elevated temperature. After just a few minutes, different functional clusters of enzymes follow distinct activity patterns. Interestingly, starting after about six minutes, both de novo biosynthesis and all exit routes from central sphingolipid metabolism become blocked, and the remaining metabolic activity consists entirely of an internal redistribution among different sphingoid base and ceramide pools. After about 30 minutes, heat stress is still in effect and the enzyme activity profile is still significantly changed. Importantly, however, the metabolites have regained concentrations that are essentially the same as those under optimal conditions.

Sphingolipids play dual roles by serving as components of membrane rafts and by regulating numerous key cell functions. Although sphingolipids have been studied extensively, the details of their functioning are difficult to understand, because their synthesis, pathways of inter-conversion, and utilization constitute a complex, dynamically changing system. We analyze the role of yeast sphingolipids in the response to heat stress. Data show that the profile of these lipids changes almost immediately with the initiation of the stress, but it is a priori unclear how this response is organized. Using experimental data, a sophisticated dynamic model, and a novel optimization strategy, we show how changes in enzyme activities are temporally organized. Intriguingly, the results show that the cells take up as much material as possible in the first few minutes of heat stress and then shut down entry and exit routes of the biosynthetic pathway system. After about 30 minutes, when heat stress is still in effect, the enzyme activity profile is still significantly changed, but the metabolites have regained concentrations that are essentially the same as those under optimal conditions. The results demonstrate how novel insights are achievable with an effective combination of experimental and theoretical research.

Biochemical Systems Theory: A Review

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

Ensemble kinetic modeling of metabolic networks from dynamic metabolic profiles

Kinetic modeling of metabolic pathways has important applications in metabolic engineering, but significant challenges still remain. The difficulties faced vary from finding best-fit parameters in a highly multidimensional search space to incomplete parameter identifiability. To meet some of these challenges, an ensemble modeling method is developed for characterizing a subset of kinetic parameters that give statistically equivalent goodness-of-fit to time series concentration data. The method is based on the incremental identification approach, where the parameter estimation is done in a step-wise manner. Numerical efficacy is achieved by reducing the dimensionality of parameter space and using efficient random parameter exploration algorithms. The shift toward using model ensembles, instead of the traditional "best-fit" models, is necessary to directly account for model uncertainty during the application of such models. The performance of the ensemble modeling approach has been demonstrated in the modeling of a generic branched pathway and the trehalose pathway in Saccharomyces cerevisiae using generalized mass action (GMA) kinetics.

Estimation of dynamic flux profiles from metabolic time series data

Background

Advances in modern high-throughput techniques of molecular biology have enabled top-down approaches for the estimation of parameter values in metabolic systems, based on time series data. Special among them is the recent method of dynamic flux estimation (DFE), which uses such data not only for parameter estimation but also for the identification of functional forms of the processes governing a metabolic system. DFE furthermore provides diagnostic tools for the evaluation of model validity and of the quality of a model fit beyond residual errors. Unfortunately, DFE works only when the data are more or less complete and the system contains as many independent fluxes as metabolites. These drawbacks may be ameliorated with other types of estimation and information. However, such supplementations incur their own limitations. In particular, assumptions must be made regarding the functional forms of some processes and detailed kinetic information must be available, in addition to the time series data.

Results

The authors propose here a systematic approach that supplements DFE and overcomes some of its shortcomings. Like DFE, the approach is model-free and requires only minimal assumptions. If sufficient time series data are available, the approach allows the determination of a subset of fluxes that enables the subsequent applicability of DFE to the rest of the flux system. The authors demonstrate the procedure with three artificial pathway systems exhibiting distinct characteristics and with actual data of the trehalose pathway in Saccharomyces cerevisiae.

Conclusions

The results demonstrate that the proposed method successfully complements DFE under various situations and without a priori assumptions regarding the model representation. The proposed method also permits an examination of whether at all, to what degree, or within what range the available time series data can be validly represented in a particular functional format of a flux within a pathway system. Based on these results, further experiments may be designed to generate data points that genuinely add new information to the structure identification and parameter estimation tasks at hand.

Canonical modeling of the multi-scale regulation of the heat stress response in yeast

Heat is one of the most fundamental and ancient environmental stresses, and response mechanisms are found in prokaryotes and shared among most eukaryotes. In the budding yeast Saccharomyces cerevisiae, the heat stress response involves coordinated changes at all biological levels, from gene expression to protein and metabolite abundances, and to temporary adjustments in physiology. Due to its integrative multi-level-multi-scale nature, heat adaptation constitutes a complex dynamic process, which has forced most experimental and modeling analyses in the past to focus on just one or a few of its aspects. Here we review the basic components of the heat stress response in yeast and outline what has been done, and what needs to be done, to merge the available information into computational structures that permit comprehensive diagnostics, interrogation, and interpretation. We illustrate the process in particular with the coordination of two metabolic responses, namely the dramatic accumulation of the protective disaccharide trehalose and the substantial change in the profile of sphingolipids, which in turn affect gene expression. The proposed methods primarily use differential equations in the canonical modeling framework of Biochemical Systems Theory (BST), which permits the relatively easy construction of coarse, initial models even in systems that are incompletely characterized.

Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models

Background

Design of newly engineered microbial strains for biotechnological purposes would greatly benefit from the development of realistic mathematical models for the processes to be optimized. Such models can then be analyzed and, with the development and application of appropriate optimization techniques, one could identify the modifications that need to be made to the organism in order to achieve the desired biotechnological goal. As appropriate models to perform such an analysis are necessarily non-linear and typically non-convex, finding their global optimum is a challenging task. Canonical modeling techniques, such as Generalized Mass Action (GMA) models based on the power-law formalism, offer a possible solution to this problem because they have a mathematical structure that enables the development of specific algorithms for global optimization.

Results

Based on the GMA canonical representation, we have developed in previous works a highly efficient optimization algorithm and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolism. Here, we explore the possibility of recasting kinetic non-linear models into an equivalent GMA model, so that global optimization on the recast GMA model can be performed. With this technique, optimization is greatly facilitated and the results are transposable to the original non-linear problem. This procedure is straightforward for a particular class of non-linear models known as Saturable and Cooperative (SC) models that extend the power-law formalism to deal with saturation and cooperativity.

Conclusions

Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.

Statistical inference methods for sparse biological time series data

Background

Comparing metabolic profiles under different biological perturbations has become a powerful approach to investigating the functioning of cells. The profiles can be taken as single snapshots of a system, but more information is gained if they are measured longitudinally over time. The results are short time series consisting of relatively sparse data that cannot be analyzed effectively with standard time series techniques, such as autocorrelation and frequency domain methods. In this work, we study longitudinal time series profiles of glucose consumption in the yeast Saccharomyces cerevisiae under different temperatures and preconditioning regimens, which we obtained with methods of in vivo nuclear magnetic resonance (NMR) spectroscopy. For the statistical analysis we first fit several nonlinear mixed effect regression models to the longitudinal profiles and then used an ANOVA likelihood ratio method in order to test for significant differences between the profiles.

Results

The proposed methods are capable of distinguishing metabolic time trends resulting from different treatments and associate significance levels to these differences. Among several nonlinear mixed-effects regression models tested, a three-parameter logistic function represents the data with highest accuracy. ANOVA and likelihood ratio tests suggest that there are significant differences between the glucose consumption rate profiles for cells that had been--or had not been--preconditioned by heat during growth. Furthermore, pair-wise t-tests reveal significant differences in the longitudinal profiles for glucose consumption rates between optimal conditions and heat stress, optimal and recovery conditions, and heat stress and recovery conditions (p-values <0.0001).

Conclusion

We have developed a nonlinear mixed effects model that is appropriate for the analysis of sparse metabolic and physiological time profiles. The model permits sound statistical inference procedures, based on ANOVA likelihood ratio tests, for testing the significance of differences between short time course data under different biological perturbations.

Analysis of operating principles with S-system models

Operating principles address general questions regarding the response dynamics of biological systems as we observe or hypothesize them, in comparison to a priori equally valid alternatives. In analogy to design principles, the question arises: Why are some operating strategies encountered more frequently than others and in what sense might they be superior? It is at this point impossible to study operation principles in complete generality, but the work here discusses the important situation where a biological system must shift operation from its normal steady state to a new steady state. This situation is quite common and includes many stress responses. We present two distinct methods for determining different solutions to this task of achieving a new target steady state. Both methods utilize the property of S-system models within Biochemical Systems Theory (BST) that steady states can be explicitly represented as systems of linear algebraic equations. The first method uses matrix inversion, a pseudo-inverse, or regression to characterize the entire admissible solution space. Operations on the basis of the solution space permit modest alterations of the transients toward the target steady state. The second method uses standard or mixed integer linear programming to determine admissible solutions that satisfy criteria of functional effectiveness, which are specified beforehand. As an illustration, we use both methods to characterize alternative response patterns of yeast subjected to heat stress, and compare them with observations from the literature.