Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control

Cybernetic modeling strives to uncover the inbuilt regulatory programs of biological systems and leverage them toward computational prediction of metabolic dynamics. Because of its focus on incorporating the global aims of metabolism, cybernetic modeling provides a systems-oriented approach for describing regulatory inputs and inferring the impact of regulation within biochemical networks. Combining cybernetic control laws with concepts from metabolic pathway analysis has culminated in a systematic strategy for constructing cybernetic models, which was previously lacking. The newly devised framework relies upon the simultaneous application of local controls that maximize the net flux through each elementary flux mode and global controls that modulate the activities of these modes to optimize the overall nutritional state of the cell. The modeling concepts are illustrated using a simple linear pathway and a larger network representing anaerobic E. coli central metabolism. The E. coli model successfully describes the metabolic shift that occurs upon deleting the pta-ackA operon that is responsible for fermentative acetate production. The model also furnishes predictions that are consistent with experimental results obtained from additional knockout strains as well as strains expressing heterologous genes. Because of the stabilizing influence of the included control variables, the resulting cybernetic models are more robust and reliable than their predecessors in simulating the network response to imposed genetic and environmental perturbations.     

Prediction of dynamic behavior of mutant strains from limited wild-type data

Metabolic engineering is the field of introducing genetic changes in organisms so as to modify their function towards synthesizing new products of high impact to society. However, engineered cells frequently have impaired growth rates thus seriously limiting the rate at which such products are made. The problem is attributable to inadequate understanding of how a metabolic network functions in a dynamic sense. Predictions of mutant strain behavior in the past have been based on steady state theories such as flux balance analysis (FBA), minimization of metabolic adjustment (MOMA), and regulatory on/off minimization (ROOM). Such predictions are restricted to product yields and cannot address productivity, which is of focal interest to applications. We demonstrate that our framework ( [Song and Ramkrishna, 2010] and [Song and Ramkrishna, 2011]), based on a “cybernetic” view of metabolic systems, makes predictions of the dynamic behavior of mutant strains of Escherichia coli from a limited amount of data obtained from the wild-type. Dynamic frameworks must necessarily address the issue of metabolic regulation, which the cybernetic approach does by postulating that metabolism is an optimal dynamic response of the organism to the environment in driving reactions towards ensuring survival. The predictions made in this paper are without parallel in the literature and lay the foundation for rational metabolic engineering.     

Growth of microbial populations in nonminimal media: some considerations for modeling

Ramkrishna and his co-workers have developed so-called cybernetic models which purport to describe, among other things, how microorganisms make choices when presented with two or more functionally equivalent, or substitutable, nutrients that are sources of carbon and available energy. In general, however, organisms are presented with choices not just between nutrients that are substitutable for one another, but also between sets of nutrients some of which are by no means substitutable for one another. It is postulated herein that the main ideas of cybernetic modeling apply to these more general choices as they seem to apply to the choices considered by Ramkrishna and his co-workers. Some consequences of the postulate are worked out for steady-state growth situations where two, or in one case three, nutrients limit or potentially limit growth rate. If predicted phenomena are observed experimentally so as to verify the postulate, a significantly improved basis for understanding growth of microorganisms in practical fermentation media as well as in natural situations will be provided by this application of cybernetic modeling ideas.     

Cybernetic modeling of bacteriol cultures at low growth rates: single-substrate systems

The cybernetic framework developed by Ramkrishna and co-workers has been expanded to include the effects of cellular maintenance energy requirements on biomass levels in slow-growing, carbon-substrate-limited cultures. A simple structured model, based on the existence of distinct key enzymes for growth and maintenance functions, is presented. Comparisons of the model with experimental data for the growth of Klebsiella oxytoca in constant fed-batch culture on glucose, fructose, arabinose, and xylose show good agreement. In addition, perturbed fed-batch culture experiments indicate that slow-growing cultures respond less rapidly to a removal of the growth limitation than do faster-growing ones. The possibility of a growth-rate dependent "critical resource" is discussed.     

Allele-Specific Behavior of Molecular Networks: Understanding Small-Molecule Drug Response in Yeast

The study of systems genetics is changing the way the genetic and molecular basis of phenotypic variation, such as disease susceptibility and drug response, is being analyzed. Moreover, systems genetics aids in the translation of insights from systems biology into genetics. The use of systems genetics enables greater attention to be focused on the potential impact of genetic perturbations on the molecular states of networks that in turn affects complex traits. In this study, we developed models to detect allele-specific perturbations on interactions, in which a genetic locus with alternative alleles exerted a differing influence on an interaction. We utilized the models to investigate the dynamic behavior of an integrated molecular network undergoing genetic perturbations in yeast. Our results revealed the complexity of regulatory relationships between genetic loci and networks, in which different genetic loci perturb specific network modules. In addition, significant within-module functional coherence was found. We then used the network perturbation model to elucidate the underlying molecular mechanisms of individual differences in response to 100 diverse small molecule drugs. As a result, we identified sub-networks in the integrated network that responded to variations in DNA associated with response to diverse compounds and were significantly enriched for known drug targets. Literature mining results provided strong independent evidence for the effectiveness of these genetic perturbing networks in the elucidation of small-molecule responses in yeast.     

Learning Gene Networks under SNP Perturbations Using eQTL Datasets

The standard approach for identifying gene networks is based on experimental perturbations of gene regulatory systems such as gene knock-out experiments, followed by a genome-wide profiling of differential gene expressions. However, this approach is significantly limited in that it is not possible to perturb more than one or two genes simultaneously to discover complex gene interactions or to distinguish between direct and indirect downstream regulations of the differentially-expressed genes. As an alternative, genetical genomics study has been proposed to treat naturally-occurring genetic variants as potential perturbants of gene regulatory system and to recover gene networks via analysis of population gene-expression and genotype data. Despite many advantages of genetical genomics data analysis, the computational challenge that the effects of multifactorial genetic perturbations should be decoded simultaneously from data has prevented a widespread application of genetical genomics analysis. In this article, we propose a statistical framework for learning gene networks that overcomes the limitations of experimental perturbation methods and addresses the challenges of genetical genomics analysis. We introduce a new statistical model, called a sparse conditional Gaussian graphical model, and describe an efficient learning algorithm that simultaneously decodes the perturbations of gene regulatory system by a large number of SNPs to identify a gene network along with expression quantitative trait loci (eQTLs) that perturb this network. While our statistical model captures direct genetic perturbations of gene network, by performing inference on the probabilistic graphical model, we obtain detailed characterizations of how the direct SNP perturbation effects propagate through the gene network to perturb other genes indirectly. We demonstrate our statistical method using HapMap-simulated and yeast eQTL datasets. In particular, the yeast gene network identified computationally by our method under SNP perturbations is well supported by the results from experimental perturbation studies related to DNA replication stress response.     

Application of hybrid cybernetic model in simulating myeloma cell culture co-consuming glucose and glutamine with mixed consumption patterns

A dynamic model called hybrid cybernetic model (HCM) based on structured metabolic network is established for simulating mammalian cell metabolism featured with partially substitutable and partially complementary consumption patterns of two substrates, glucose and glutamine. Benefiting from the application of elementary mode analysis (EMA), the complicated metabolic network is decomposed into elementary modes (EMs) facilitating the employment of the hybrid cybernetic framework to investigate the external and internal flux distribution and the regulation mechanism among them. According to different substrate combination, two groups of EMs are obtained, i.e., EMs associated with glucose uptake and simultaneous uptake of glucose and glutamine. Uptake fluxes through various EMs are coupled together via cybernetic variables to maximize substrate uptake. External fluxes and internal fluxes could be calculated and estimated respectively, by the combination of the stoichiometrics of metabolic networks and fluxes through regulated EMs. The model performance is well validated via three sets of experimental data. Through parameter identification of limited number of experimental data, other external metabolites are precisely predicted. The obtained kinetic parameters of three experimental cultures have similar values, which indicates the robustness of the model. Furthermore, the prediction performance of the model is successfully validated based on identified parameters.     

On the matching and proportional laws of cybernetic models

The Matching and Proportional Laws are heuristic control policies that have found widespread use in cybernetic models of biological systems. Within this context, the laws serve as optimization surrogates for predicting the response of metabolic control circuits that modulate enzyme levels and activities. The key result of the current contribution is to demonstrate clearly the optimality properties of these laws and the assumptions that underlie their development. In doing so, we arrive at generalized versions of the Matching and Proportional Laws that are shown to collapse to the forms originally derived by Kompala et al. (Biotechnol. Bioeng. 1986, 28, 1044-1055) when certain simplifications are applied. As a further line of investigation, we show how Kompala et al.'s cybernetic laws compare with alternative control policies in their ability to describe diauxic growth behavior of microbial cultures. We find that Kompala et al.'s model describes the experimental observations more accurately than other limiting-case models that are either too aggressive or too passive in capturing the mixed-substrate growth rates and intermediate lag periods. Monte Carlo analysis of computational growth experiments in which strains obeying different regulatory policies directly compete for available nutrients reveals that the Matching and Proportional Law policy does not maximize the average growth rate of the culture. However, it allocates metabolic resources more frugally than other policies that outperform it and may be more realistic in reflecting the cell's true fitness-to-cost tradeoff as judged by its agreement with experimental growth data.     

Reconstructing the regulatory network controlling commitment and sporulation in Physarum polycephalum based on hierarchical Petri Net modelling and simulation

We reconstruct the regulatory network controlling commitment and sporulation of Physarum polycephalum from experimental results using a hierarchical Petri Net-based modelling and simulation framework. The stochastic Petri Net consistently describes the structure and simulates the dynamics of the molecular network as analysed by genetic, biochemical and physiological experiments within a single coherent model. The Petri Net then is extended to simulate time-resolved somatic complementation experiments performed by mixing the cytoplasms of mutants altered in the sporulation response, to systematically explore the network structure and to probe its dynamics. This reverse engineering approach presumably can be employed to explore other molecular or genetic signalling systems where the activity of genes or their products can be experimentally controlled in a time-resolved manner.     

A cybernetic view of microbial growth: modeling of cells as optimal strategists

A cybernetic framework is presented which views microbial response in multiple substrate environments as a judicious investment of cellular resources in synthesizing different key proteins according to an optimal regulatory strategy. A mathematical model is developed within the cybernetic framework for the diauxic growth of Klebsiella pneumoniae on a mixture of D-glucose and D-xylose. The "bang-bang" optimal policy describes well the experimental observations obtained using a fermenter coupled to an Apple II microcomputer. Striking variations in respiratory levels are observed experimentally during the switching of the cell's adaptive machinery for the utilization of the less preferred substrate.     

A Bayesian framework for inference of the genotype-phenotype map for segregating populations

Complex genetic interactions lie at the foundation of many diseases. Understanding the nature of these interactions is critical to developing rational intervention strategies. In mammalian systems hypothesis testing in vivo is expensive, time consuming, and often restricted to a few physiological endpoints. Thus, computational methods that generate causal hypotheses can help to prioritize targets for experimental intervention. We propose a Bayesian statistical method to infer networks of causal relationships among genotypes and phenotypes using expression quantitative trait loci (eQTL) data from genetically randomized populations. Causal relationships between network variables are described with hierarchical regression models. Prior distributions on the network structure enforce graph sparsity and have the potential to encode prior biological knowledge about the network. An efficient Monte Carlo method is used to search across the model space and sample highly probable networks. The result is an ensemble of networks that provide a measure of confidence in the estimated network topology. These networks can be used to make predictions of system-wide response to perturbations. We applied our method to kidney gene expression data from an MRL/MpJ × SM/J intercross population and predicted a previously uncharacterized feedback loop in the local renin-angiotensin system.     

Complex growth dynamics in batch cultures: experiments and cybernetic models

The cybernetic framework developed by Ramkrishna and co-workers is shown to encompass the regulation of nutrient transport processes as well as the effect of nutrient transport on the biotic phase. A structured model which accounts for both an abiotic or environmental phase and a biotic or cellular phase is proposed to describe bacterial growth on lactose as the limiting carbon and energy source. In the presence of lactose, competing uptake mechanisms are proposed. At low lactose concentrations, an energy-requiring transport process is the preferred uptake mechanism. The coupling between cellular energetics and nutrient uptake results in an interesting intermittent growth phenomenon. As the concentration of lactose increases, a nonenergetic transport process is preferred and cellular growth ceases to be intermittent. Model simulations are compared with previously reported experimental results and exhibit good agreement over the entire range of initial lactose concentrations.     

Transient sensitivity analysis of a cybernetic model of microbial growth on two substrates

When a microbe has a choice of two substrates for its growth in a fermentation medium, its preference varies with the substrates and with time. This "informed" choice is conveniently expressed by cybernetic models. For the growth of Klebsiella oxytoca in a medium of glucose and lactose, a one-parameter cybernetic model of growth has been employed in a batch bioreactor to analyse sensitivity of the fermentation to perturbations in the parameter, ! (0 h ! h 1). The sensitivity surfaces in the (!-time) space show interesting variations which are discussed. An important observation is that while growth is best promoted with !=1, i.e. sequential consumption of the substrates [8], low sensitivities require smaller values of !, i.e. simultaneous utilisation. Thus, in a realistic operation it may be necessary to compromise between high growth (with subsequent high productivity) and low sensitivity.     

Modeling beta-adrenergic control of cardiac myocyte contractility in silico

The beta-adrenergic signaling pathway regulates cardiac myocyte contractility through a combination of feedforward and feedback mechanisms. We used systems analysis to investigate how the components and topology of this signaling network permit neurohormonal control of excitation-contraction coupling in the rat ventricular myocyte. A kinetic model integrating beta-adrenergic signaling with excitation-contraction coupling was formulated, and each subsystem was validated with independent biochemical and physiological measurements. Model analysis was used to investigate quantitatively the effects of specific molecular perturbations. 3-Fold overexpression of adenylyl cyclase in the model allowed an 85% higher rate of cyclic AMP synthesis than an equivalent overexpression of beta 1-adrenergic receptor, and manipulating the affinity of Gs alpha for adenylyl cyclase was a more potent regulator of cyclic AMP production. The model predicted that less than 40% of adenylyl cyclase molecules may be stimulated under maximal receptor activation, and an experimental protocol is suggested for validating this prediction. The model also predicted that the endogenous heat-stable protein kinase inhibitor may enhance basal cyclic AMP buffering by 68% and increasing the apparent Hill coefficient of protein kinase A activation from 1.0 to 2.0. Finally, phosphorylation of the L-type calcium channel and phospholamban were found sufficient to predict the dominant changes in myocyte contractility, including a 2.6x increase in systolic calcium (inotropy) and a 28% decrease in calcium half-relaxation time (lusitropy). By performing systems analysis, the consequences of molecular perturbations in the beta-adrenergic signaling network may be understood within the context of integrative cellular physiology.     

Cybernetic models based on lumped elementary modes accurately predict strain‐specific metabolic function

In a recent article, Song and Ramkrishna (Song and Ramkrishna [2010]. Biotechnol Bioeng 106(2):271–284) proposed a lumped hybrid cybernetic model (L‐HCM) towards extracting maximum information about metabolic function from a minimum of data. This approach views the total uptake flux as distributed among lumped elementary modes (L‐EMs) so as to maximize a prescribed metabolic objective such as growth or uptake rate. L‐EM is computed as a weighted average of EMs where the weights are related to the yields of vital products (i.e., biomass and ATP). In this article, we further enhance the predictive power of L‐HCMs through modifications in lumping weights with additional parameters that can be tuned with data viewed to be critical. The resulting model is able to make predictions of diverse metabolic behaviors varying greatly with strain types as evidenced from case studies of anaerobic growth of various Escherichia coli strains. Incorporation of the new lumping formula into L‐HCM remarkably improves model predictions with a few critical data, thus presenting L‐HCM as a dynamic tool as being not only qualitatively correct but also quantitatively accurate. Biotechnol. Bioeng. 2011; 108:127–140. © 2010 Wiley Periodicals, Inc.     

Exacting predictions by cybernetic model confirmed experimentally: Steady state multiplicity in the chemostat

We demonstrate strong experimental support for the cybernetic model based on maximizing carbon uptake rate in describing the microorganism's regulatory behavior by verifying exacting predictions of steady state multiplicity in a chemostat. Experiments with a feed mixture of glucose and pyruvate show multiple steady state behavior as predicted by the cybernetic model. When multiplicity occurs at a dilution (growth) rate, it results in hysteretic behavior following switches in dilution rate from above and below. This phenomenon is caused by transient paths leading to different steady states through dynamic maximization of the carbon uptake rate. Thus steady state multiplicity is a manifestation of the nonlinearity arising from cybernetic mechanisms rather than of the nonlinear kinetics. The predicted metabolic multiplicity would extend to intracellular states such as enzyme levels and fluxes to be verified in future experiments. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Possible functional roles of phase resetting during walking

The walking rhythm is known to show phase shift or "reset" in response to external impulsive perturbations. We tried to elucidate functional roles of the phase reset possibly used for the neural control of locomotion. To this end, a system with a double pendulum as a simplified model of the locomotor control and a model of bipedal locomotion were employed and analyzed in detail. In these models, a movement corresponding to the normal steady-state walking was realized as a stable limit cycle solution of the system. Unexpected external perturbations applied to the system can push the state point of the system away from its limit cycle, either outside or inside the basin of attraction of the limit cycle. Our mathematical analyses of the models suggested functional roles of the phase reset during walking as follows. Function 1: an appropriate amount of the phase reset for a given perturbation can contribute to relocating the system's state point outside the basin of attraction of the limit cycle back to the inside. Function 2: it can also be useful to reduce the convergence time (the time necessary for the state point to return to the limit cycle). In experimental studies during walking of animals and humans, the reset of walking rhythm induced by perturbations was investigated using the phase transition curve (PTC) or the phase resetting curve (PRC) representing phase-dependent responses of the walking. We showed, for the simple double-pendulum model, the existence of the optimal phase control and the corresponding PTC that could optimally realize the aforementioned functions in response to impulsive force perturbations. Moreover, possible forms of PRC that can avoid falling against the force perturbations were predicted by the biped model, and they were compared with the experimentally observed PRC during human walking. Finally, physiological implications of the results were discussed.     

Cybernetic modeling based on pathway analysis for Penicillium chrysogenum fed-batch fermentation

A macrokinetic model employing cybernetic methodology is proposed to describe mycelium growth and penicillin production. Based on the primordial and complete metabolic network of Penicillium chrysogenum found in the literature, the modeling procedure is guided by metabolic flux analysis and cybernetic modeling framework. The abstracted cybernetic model describes the transients of the consumption rates of the substrates, the assimilation rates of intermediates, the biomass growth rate, as well as the penicillin formation rate. Combined with the bioreactor model, these reaction rates are linked with the most important state variables, i.e., mycelium, substrate and product concentrations. Simplex method is used to estimate the sensitive parameters of the model. Finally, validation of the model is carried out with 20 batches of industrial-scale penicillin cultivation.     

Cybernetic modeling of iron-limited growth and siderophore production

The cybernetic modeling framework developed by Ramkrishna and co-workers has been applied to a case of bacterial metabolite production, namely the production of siderophores (iron-chelating agents) associated with iron-limiting fermentation conditions. Experimental growth data showed that, even though final biomass levels were controlled by exhaustion of the carbon source, iron-limiting conditions also affected the biomass yield. A structured model which includes the process of an iron-limiting energy resource production was able to quantitatively account for this apparent dual-substrate limitation over a wide range of batch and continuous operating conditions. The experiments data also showed quite large difference in iron uptake over the wide range of operating condition and iron levels investigated. The inclusion in the model of the processes of low and high (siderophore-mediated) affinity iron transport, and siderophore production led to simulation results that were in good quantitative agreement with the siderophore, medium and cell iron levels, in both batch and steady-state continuous culture operating conditions.     

Genetics of gene expression responses to temperature stress in a sea urchin gene network

Stress responses play an important role in shaping species distributions and robustness to climate change. We investigated how stress responses alter the contribution of additive genetic variation to gene expression during development of the purple sea urchin, Strongylocentrotus purpuratus, under increased temperatures that model realistic climate change scenarios. We first measured gene expression responses in the embryos by RNA‐seq to characterize molecular signatures of mild, chronic temperature stress in an unbiased manner. We found that an increase from 12 to 18 °C caused widespread alterations in gene expression including in genes involved in protein folding, RNA processing and development. To understand the quantitative genetic architecture of this response, we then focused on a well‐characterized gene network involved in endomesoderm and ectoderm specification. Using a breeding design with wild‐caught individuals, we measured genetic and gene–environment interaction effects on 72 genes within this network. We found genetic or maternal effects in 33 of these genes and that the genetic effects were correlated in the network. Fourteen network genes also responded to higher temperatures, but we found no significant genotype–environment interactions in any of the genes. This absence may be owing to an effective buffering of the temperature perturbations within the network. In support of this hypothesis, perturbations to regulatory genes did not affect the expression of the genes that they regulate. Together, these results provide novel insights into the relationship between environmental change and developmental evolution and suggest that climate change may not expose large amounts of cryptic genetic variation to selection in this species.