Ensemble modeling and related mathematical modeling of metabolic networks

Enhancing in silico strain design predictions through next generation metabolic modeling approaches

The reconstruction and analysis of metabolic models has garnered increasing attention due to the multitude of applications in which these have proven to be practical. The growing number of generated metabolic models has been accompanied by an exponentially expanding arsenal of tools used to analyze them. In this work, we discussed the biological relevance of a number of promising modeling frameworks, focusing on the questions and hypotheses each method is equipped to address. To this end, we critically analyzed the steady-state modeling approaches focusing on resource allocation and incorporation of thermodynamic considerations which produce promising results and aid in the generation and experimental validation of numerous predictions. For smaller networks involving more complex regulation, we addressed kinetic modeling techniques which show encouraging results in addressing questions outside the scope of steady-state modeling. Finally, we discussed the potential application of the discussed frameworks within the field of strain design. Adoption of such methodologies is believed to significantly enhance the accuracy of in silico predictions and hence decrease the number of design-build-test cycles required.

Dissecting the regulatory roles of ORM proteins in the sphingolipid pathway of plants

Sphingolipids are a vital component of plant cellular endomembranes and carry out multiple functional and regulatory roles. Different sphingolipid species confer rigidity to the membrane structure, facilitate trafficking of secretory proteins, and initiate programmed cell death. Although the regulation of the sphingolipid pathway is yet to be uncovered, increasing evidence has pointed to orosomucoid proteins (ORMs) playing a major regulatory role and potentially interacting with a number of components in the pathway, including both enzymes and sphingolipids. However, experimental exploration of new regulatory interactions is time consuming and often infeasible. In this work, a computational approach was taken to address this challenge. A metabolic network of the sphingolipid pathway in plants was reconstructed. The steady-state rates of reactions in the network were then determined through measurements of growth and cellular composition of the different sphingolipids in Arabidopsis seedlings. The Ensemble modeling framework was modified to accurately account for activation mechanisms and subsequently used to generate sets of kinetic parameters that converge to the measured steady-state fluxes in a thermodynamically consistent manner. In addition, the framework was appended with an additional module to automate screening the parameters and to output models consistent with previously reported network responses to different perturbations. By analyzing the network's response in the presence of different combinations of regulatory mechanisms, the model captured the experimentally observed repressive effect of ORMs on serine palmitoyltransferase (SPT). Furthermore, predictions point to a second regulatory role of ORM proteins, namely as an activator of class II (or LOH1 and LOH3) ceramide synthases. This activating role was found to be modulated by the concentration of free ceramides, where an accumulation of these sphingolipid species dampened the activating effect of ORMs on ceramide synthase. The predictions pave the way for future guided experiments and have implications in engineering crops with higher biotic stress tolerance.

A simplified modelling framework facilitates more complex representations of plant circadian clocks

The circadian clock orchestrates biological processes so that they occur at specific times of the day, thereby facilitating adaptation to diurnal and seasonal environmental changes. In plants, mathematical modelling has been comprehensively integrated with experimental studies to gain a better mechanistic understanding of the complex genetic regulatory network comprising the clock. However, with an increasing number of circadian genes being discovered, there is a pressing need for methods facilitating the expansion of computational models to incorporate these newly-discovered components. Conventionally, plant clock models have comprised differential equation systems based on Michaelis-Menten kinetics. However, the difficulties associated with modifying interactions using this approach-and the concomitant problem of robustly identifying regulation types-has contributed to a complexity bottleneck, with quantitative fits to experimental data rapidly becoming computationally intractable for models possessing more than ≈50 parameters. Here, we address these issues by constructing the first plant clock models based on the S-System formalism originally developed by Savageau for analysing biochemical networks. We show that despite its relative simplicity, this approach yields clock models with comparable accuracy to the conventional Michaelis-Menten formalism. The S-System formulation also confers several key advantages in terms of model construction and expansion. In particular, it simplifies the inclusion of new interactions, whilst also facilitating the modification of regulation types, thereby making it well-suited to network inference. Furthermore, S-System models mitigate the issue of parameter identifiability. Finally, by applying linear systems theory to the models considered, we provide some justification for the increased use of aggregated protein equations in recent plant clock modelling, replacing the separate cytoplasmic/nuclear protein compartments that were characteristic of the earlier models. We conclude that as well as providing a simplified framework for model development, the S-System formalism also possesses significant potential as a robust modelling method for designing synthetic gene circuits.

Acceleration Strategies to Enhance Metabolic Ensemble Modeling Performance

Developing reliable, predictive kinetic models of metabolism is a difficult, yet necessary, priority toward understanding and deliberately altering cellular behavior. Constraint-based modeling has enabled the fields of metabolic engineering and systems biology to make great strides in interrogating cellular metabolism but does not provide sufficient insight into regulation or kinetic limitations of metabolic pathways. Moreover, the growth-optimized assumptions that constraint-based models often rely on do not hold when studying stationary or persistor cell populations. However, developing kinetic models provides many unique challenges, as many of the kinetic parameters and rate laws governing individual enzymes are unknown. Ensemble modeling (EM) was developed to circumnavigate this challenge and effectively sample the large kinetic parameter solution space using consistent experimental datasets. Unfortunately, EM, in its base form, requires long solve times to complete and often leads to unstable kinetic model predictions. Furthermore, these limitations scale prohibitively with increasing model size. As larger metabolic models are developed with increasing genetic information and experimental validation, the demand to incorporate kinetic information increases. Therefore, in this work, we have begun to tackle the challenges of EM by introducing additional steps to the existing method framework specifically through reducing computation time and optimizing parameter sampling. We first reduce the structural complexity of the network by removing dependent species, and second, we sample locally stable parameter sets to reflect realistic biological states of cells. Lastly, we presort the screening data to eliminate the most incorrect predictions in the earliest screening stages, saving further calculations in later stages. Our complementary improvements to this EM framework are easily incorporated into concurrent EM efforts and broaden the application opportunities and accessibility of kinetic modeling across the field.

A system of recurrent neural networks for modularising, parameterising and dynamic analysis of cell signalling networks

In this paper, we show how to extend our previously proposed novel continuous time Recurrent Neural Networks (RNN) approach that retains the advantage of continuous dynamics offered by Ordinary Differential Equations (ODE) while enabling parameter estimation through adaptation, to larger signalling networks using a modular approach. Specifically, the signalling network is decomposed into several sub-models based on important temporal events in the network. Each sub-model is represented by the proposed RNN and trained using data generated from the corresponding ODE model. Trained sub-models are assembled into a whole system RNN which is then subjected to systems dynamics and sensitivity analyses. The concept is illustrated by application to G1/S transition in cell cycle using Iwamoto et al. (2008) ODE model. We decomposed the G1/S network into 3 sub-models: (i) E2F transcription factor release; (ii) E2F and CycE positive feedback loop for elevating cyclin levels; and (iii) E2F and CycA negative feedback to degrade E2F. The trained sub-models accurately represented system dynamics and parameters were in good agreement with the ODE model. The whole system RNN however revealed couple of parameters contributing to compounding errors due to feedback and required refinement to sub-model 2. These related to the reversible reaction between CycE/CDK2 and p27, its inhibitor. The revised whole system RNN model very accurately matched dynamics of the ODE system. Local sensitivity analysis of the whole system model further revealed the most dominant influence of the above two parameters in perturbing G1/S transition, giving support to a recent hypothesis that the release of inhibitor p27 from Cyc/CDK complex triggers cell cycle stage transition. To make the model useful in a practical setting, we modified each RNN sub-model with a time relay switch to facilitate larger interval input data (≈20min) (original model used data for 30s or less) and retrained them that produced parameters and protein concentrations similar to the original RNN system. Results thus demonstrated the reliability of the proposed RNN method for modelling relatively large networks by modularisation for practical settings. Advantages of the method are its ability to represent accurate continuous system dynamics and ease of: parameter estimation through training with data from a practical setting, model analysis (40% faster than ODE), fine tuning parameters when more data are available, sub-model extension when new elements and/or interactions come to light and model expansion with addition of sub-models.

Constructing kinetic models of metabolism at genome-scales: A review

Constraint-based modeling of biological networks (metabolism, transcription and signal transduction), although used successfully in many applications, suffer from specific limitations such as the lack of representation of metabolite concentrations and enzymatic regulation, which are necessary for a complete physiologically relevant model. Kinetic models conversely overcome these shortcomings and enable dynamic analysis of biological systems for enhanced in silico hypothesis generation. Nonetheless, kinetic models also have limitations for modeling at genome-scales chiefly due to: (i) model non-linearity; (ii) computational tractability; (iii) parameter identifiability; (iv) estimability; and (v) uncertainty. In order to support further development of kinetic models as viable alternatives to constraint-based models, this review presents a brief description of the existing obstacles towards building genome-scale kinetic models. Specific kinetic modeling frameworks capable of overcoming these obstacles are covered in this review. The tractability and physiological feasibility of these models are discussed with the objective of using available in vivo experimental observations to define the model parameter space. Among the different methods discussed, Monte Carlo kinetic models of metabolism stand out as potentially tractable methods to model genome scale networks while also addressing in vivo parameter uncertainty.

Explicit consideration of topological and parameter uncertainty gives new insights into a well-established model of glycolysis

Previous models of glycolysis in the sleeping sickness parasite Trypanosoma brucei assumed that the core part of glycolysis in this unicellular parasite is tightly compartimentalized within an organelle, the glycosome, which had previously been shown to contain most of the glycolytic enzymes. The glycosomes were assumed to be largely impermeable, and exchange of metabolites between the cytosol and the glycosome was assumed to be regulated by specific transporters in the glycosomal membrane. This tight compartmentalization was considered to be essential for parasite viability. Recently, size-specific metabolite pores were discovered in the membrane of glycosomes. These channels are proposed to allow smaller metabolites to diffuse across the membrane but not larger ones. In light of this new finding, we re-analyzed the model taking into account uncertainty about the topology of the metabolic system in T. brucei, as well as uncertainty about the values of all parameters of individual enzymatic reactions. Our analysis shows that these newly-discovered nonspecific pores are not necessarily incompatible with our current knowledge of the glycosomal metabolic system, provided that the known cytosolic activities of the glycosomal enzymes play an important role in the regulation of glycolytic fluxes and the concentration of metabolic intermediates of the pathway.

Dynamic modelling under uncertainty: the case of Trypanosoma brucei energy metabolism

Kinetic models of metabolism require detailed knowledge of kinetic parameters. However, due to measurement errors or lack of data this knowledge is often uncertain. The model of glycolysis in the parasitic protozoan Trypanosoma brucei is a particularly well analysed example of a quantitative metabolic model, but so far it has been studied with a fixed set of parameters only. Here we evaluate the effect of parameter uncertainty. In order to define probability distributions for each parameter, information about the experimental sources and confidence intervals for all parameters were collected. We created a wiki-based website dedicated to the detailed documentation of this information: the SilicoTryp wiki (http://silicotryp.ibls.gla.ac.uk/wiki/Glycolysis). Using information collected in the wiki, we then assigned probability distributions to all parameters of the model. This allowed us to sample sets of alternative models, accurately representing our degree of uncertainty. Some properties of the model, such as the repartition of the glycolytic flux between the glycerol and pyruvate producing branches, are robust to these uncertainties. However, our analysis also allowed us to identify fragilities of the model leading to the accumulation of 3-phosphoglycerate and/or pyruvate. The analysis of the control coefficients revealed the importance of taking into account the uncertainties about the parameters, as the ranking of the reactions can be greatly affected. This work will now form the basis for a comprehensive Bayesian analysis and extension of the model considering alternative topologies.