Completely uncoupled and perfectly coupled gene expression in repressible systems

Evolution of a genome-encoded bias in amino acid biosynthetic pathways is a potential indicator of amino acid dynamics in the environment

Overcoming the stress of starvation is one of an organism’s most challenging phenotypic responses. Those organisms that frequently survive the challenge, by virtue of their fitness, will have evolved genomes that are shaped by their specific environments. Understanding this genotype–environment–phenotype relationship at a deep level will require quantitative predictive models of the complex molecular systems that link these aspects of an organism’s existence. Here, we treat one of the most fundamental molecular systems, protein synthesis, and the amino acid biosynthetic pathways involved in the stringent response to starvation. These systems face an inherent logical dilemma: Building an amino acid biosynthetic pathway to synthesize its product—the cognate amino acid of the pathway—may require that very amino acid when it is no longer available. To study this potential “catch-22,” we have created a generic model of amino acid biosynthesis in response to sudden starvation. Our mathematical analysis and computational results indicate that there are two distinctly different outcomes: Partial recovery to a new steady state, or full system failure. Moreover, the cell’s fate is dictated by the cognate bias, the number of cognate amino acids in the corresponding biosynthetic pathway relative to the average number of that amino acid in the proteome. We test these implications by analyzing the proteomes of over 1,800 sequenced microbes, which reveals statistically significant evidence of low cognate bias, a genetic trait that would avoid the biosynthetic quandary. Furthermore, these results suggest that the pattern of cognate bias, which is readily derived by genome sequencing, may provide evolutionary clues to an organism’s natural environment.

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.

Identifying quantitative operation principles in metabolic pathways: a systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses

BACKGROUND

Optimization methods allow designing changes in a system so that specific goals are attained. These techniques are fundamental for metabolic engineering. However, they are not directly applicable for investigating the evolution of metabolic adaptation to environmental changes. Although biological systems have evolved by natural selection and result in well-adapted systems, we can hardly expect that actual metabolic processes are at the theoretical optimum that could result from an optimization analysis. More likely, natural systems are to be found in a feasible region compatible with global physiological requirements.

RESULTS

We first present a new method for globally optimizing nonlinear models of metabolic pathways that are based on the Generalized Mass Action (GMA) representation. The optimization task is posed as a nonconvex nonlinear programming (NLP) problem that is solved by an outer-approximation algorithm. This method relies on solving iteratively reduced NLP slave subproblems and mixed-integer linear programming (MILP) master problems that provide valid upper and lower bounds, respectively, on the global solution to the original NLP. The capabilities of this method are illustrated through its application to the anaerobic fermentation pathway in Saccharomyces cerevisiae. We next introduce a method to identify the feasibility parametric regions that allow a system to meet a set of physiological constraints that can be represented in mathematical terms through algebraic equations. This technique is based on applying the outer-approximation based algorithm iteratively over a reduced search space in order to identify regions that contain feasible solutions to the problem and discard others in which no feasible solution exists. As an example, we characterize the feasible enzyme activity changes that are compatible with an appropriate adaptive response of yeast Saccharomyces cerevisiae to heat shock

CONCLUSION

Our results show the utility of the suggested approach for investigating the evolution of adaptive responses to environmental changes. The proposed method can be used in other important applications such as the evaluation of parameter changes that are compatible with health and disease states.

Recent developments in parameter estimation and structure identification of biochemical and genomic systems

The organization, regulation and dynamical responses of biological systems are in many cases too complex to allow intuitive predictions and require the support of mathematical modeling for quantitative assessments and a reliable understanding of system functioning. All steps of constructing mathematical models for biological systems are challenging, but arguably the most difficult task among them is the estimation of model parameters and the identification of the structure and regulation of the underlying biological networks. Recent advancements in modern high-throughput techniques have been allowing the generation of time series data that characterize the dynamics of genomic, proteomic, metabolic, and physiological responses and enable us, at least in principle, to tackle estimation and identification tasks using 'top-down' or 'inverse' approaches. While the rewards of a successful inverse estimation or identification are great, the process of extracting structural and regulatory information is technically difficult. The challenges can generally be categorized into four areas, namely, issues related to the data, the model, the mathematical structure of the system, and the optimization and support algorithms. Many recent articles have addressed inverse problems within the modeling framework of Biochemical Systems Theory (BST). BST was chosen for these tasks because of its unique structural flexibility and the fact that the structure and regulation of a biological system are mapped essentially one-to-one onto the parameters of the describing model. The proposed methods mainly focused on various optimization algorithms, but also on support techniques, including methods for circumventing the time consuming numerical integration of systems of differential equations, smoothing overly noisy data, estimating slopes of time series, reducing the complexity of the inference task, and constraining the parameter search space. Other methods targeted issues of data preprocessing, detection and amelioration of model redundancy, and model-free or model-based structure identification. The total number of proposed methods and their applications has by now exceeded one hundred, which makes it difficult for the newcomer, as well as the expert, to gain a comprehensive overview of available algorithmic options and limitations. To facilitate the entry into the field of inverse modeling within BST and related modeling areas, the article presented here reviews the field and proposes an operational 'work-flow' that guides the user through the estimation process, identifies possibly problematic steps, and suggests corresponding solutions based on the specific characteristics of the various available algorithms. The article concludes with a discussion of the present state of the art and with a description of open questions.

Cooperativity and saturation in biochemical networks: A saturable formalism using Taylor series approximations

Cooperative and saturable systems are common in molecular biology. Nevertheless, common canonical formalisms for kinetic modeling that are theoretically well justified do not have a saturable form. Modeling and fitting data from saturable systems are widely done using Hill-like equations. In practice, there is no theoretical justification for the generalized use of these equations, other than their ability to fit experimental data. Thus it is important to find a canonical formalism that is (a) theoretically well supported, (b) has a saturable functional form, and (c) can be justifiably applicable to any biochemical network. Here we derive such a formalism using Taylor approximations in a special transformation space defined by power-inverses and logarithms of power-inverses. This formalism is generalized for processes with n-variables, leading to a useful mathematical representation for molecular biology: the Saturable and Cooperative Formalism (SC formalism). This formalism provides an appropriate representation that can be used for modeling processes with cooperativity and saturation. We also show that the Hill equation can be seen as a special case within this formalism. Parameter estimation for the SC formalism requires information that is also necessary to build Power-Law models, Metabolic Control Analysis descriptions or (log)linear and Lin-log models. In addition, the saturation fraction of the relevant processes at the operating point needs to be considered. The practical use of the SC formalism for modeling is illustrated with a few examples. Similar models are built using different formalisms and compared to emphasize advantages and limitations of the different approaches.

Improved methods for the mathematically controlled comparison of biochemical systems

The method of mathematically controlled comparison provides a structured approach for the comparison of alternative biochemical pathways with respect to selected functional effectiveness measures. Under this approach, alternative implementations of a biochemical pathway are modeled mathematically, forced to be equivalent through the application of selected constraints, and compared with respect to selected functional effectiveness measures. While the method has been applied successfully in a variety of studies, we offer recommendations for improvements to the method that (1) relax requirements for definition of constraints sufficient to remove all degrees of freedom in forming the equivalent alternative, (2) facilitate generalization of the results thus avoiding the need to condition those findings on the selected constraints, and (3) provide additional insights into the effect of selected constraints on the functional effectiveness measures. We present improvements to the method and related statistical models, apply the method to a previously conducted comparison of network regulation in the immune system, and compare our results to those previously reported.

Identifying global regulators in transcriptional regulatory networks in bacteria

The machinery for cells to take decisions, when environmental conditions change, includes protein-DNA interactions defined by transcriptional factors and their targets around promoters. Properties of global regulators are revised attempting to reach diagnostic explicit criteria for their definition and eventual future computational identification. These include among others, the number of regulated genes, the number and type of co-regulators, the different sigma-classes of promoters and the number of transcriptional factors they regulate, the size of the evolutionary family they belong to, and the variety of conditions where they exert their control. As a consequence, global versus local regulation can be identified, as shown for Escherichia coli and eventually in other genomes.

Design principles for regulator gene expression in a repressible gene circuit

We consider the design of a type of repressible gene circuit that is common in bacteria. In this type of circuit, a regulator protein acts to coordinately repress the expression of effector genes when a signal molecule with which it interacts is present. The regulator protein can also independently influence the expression of its own gene, such that regulator gene expression is repressible (like effector genes), constitutive, or inducible. Thus, a signal-directed change in the activity of the regulator protein can result in one of three patterns of coupled regulator and effector gene expression: direct coupling, in which regulator and effector gene expression change in the same direction; uncoupling, in which regulator gene expression remains constant while effector gene expression changes; or inverse coupling, in which regulator and effector gene expression change in opposite directions. We have investigated the functional consequences of each form of coupling using a mathematical model to compare alternative circuits on the basis of engineering-inspired criteria for functional effectiveness. The results depend on whether the regulator protein acts as a repressor or activator of transcription at the promoters of effector genes. In the case of repressor control of effector gene expression, direct coupling is optimal among the three forms of coupling, whereas in the case of activator control, inverse coupling is optimal. Results also depend on the sensitivity of effector gene expression to changes in the level of a signal molecule; the optimal form of coupling can be physically realized only for circuits with sufficiently small sensitivity. These theoretical results provide a rationale for autoregulation of regulator genes in repressible gene circuits and lead to testable predictions, which we have compared with data available in the literature and electronic databases.

Comparative analysis of prototype two-component systems with either bifunctional or monofunctional sensors: differences in molecular structure and physiological function

Signal transduction by a traditional two-component system involves a sensor protein that recognizes a physiological signal, autophosphorylates and transfers its phosphate, and a response regulator protein that receives the phosphate, alters its affinity toward specific target proteins or DNA sequences and causes change in metabolic activity or gene expression. In some cases the sensor protein, when unphosphorylated, has a positive effect upon the rate of dephosphorylation of the regulator protein (bifunctional sensor), whereas in other cases it has no such effect (monofunctional sensor). In this work we identify structural and functional differences between these two designs. In the first part of the paper we use sequence data for two-component systems from several organisms and homology modelling techniques to determine structural features for response regulators and for sensors. Our results indicate that each type of reference sensor (bifunctional and monofunctional) has a distinctive structural feature, which we use to make predictions regarding the functionality of other sensors. In the second part of the paper we use mathematical models to analyse and compare the physiological function of systems that differ in the type of sensor and are otherwise equivalent. Our results show that a bifunctional sensor is better than a monofunctional sensor both at amplifying changes in the phosphorylation level of the regulator caused by signals from the sensor and at attenuating changes caused by signals from small phosphodonors. Cross-talk to or from other two-component systems is better suppressed if the transmitting sensor is monofunctional, which is the more appropriate design when such cross-talk represents pathological noise. Cross-talk to or from other two-component systems is better amplified if the transmitting sensor is bifunctional, which is the more appropriate design when such cross-talk represents a physiological signal. These results provide a functional rationale for the selection of each design that is consistent with available experimental evidence for several two-component systems.

Design principles and operating principles: the yin and yang of optimal functioning

Metabolic engineering has as a goal the improvement of yield of desired products from microorganisms and cell lines. This goal has traditionally been approached with experimental biotechnological methods, but it is becoming increasingly popular to precede the experimental phase by a mathematical modeling step that allows objective pre-screening of possible improvement strategies. The models are either linear and represent the stoichiometry and flux distribution in pathways or they are non-linear and account for the full kinetic behavior of the pathway, which is often significantly effected by regulatory signals. Linear flux analysis is simpler and requires less input information than a full kinetic analysis, and the question arises whether the consideration of non-linearities is really necessary for devising optimal strategies for yield improvements. The article analyzes this question with a generic, representative pathway. It shows that flux split ratios, which are the key criterion for linear flux analysis, are essentially sufficient for unregulated, but not for regulated branch points. The interrelationships between regulatory design on one hand and optimal patterns of operation on the other suggest the investigation of operating principles that complement design principles, like a user's manual complements the hardwiring of electronic equipment.

Evolution of enzymes in metabolism: a network perspective

Several models have been proposed to explain the origin and evolution of enzymes in metabolic pathways. Initially, the retro-evolution model proposed that, as enzymes at the end of pathways depleted their substrates in the primordial soup, there was a pressure for earlier enzymes in pathways to be created, using the later ones as initial template, in order to replenish the pools of depleted metabolites. Later, the recruitment model proposed that initial templates from other pathways could be used as long as those enzymes were similar in chemistry or substrate specificity. These two models have dominated recent studies of enzyme evolution. These studies are constrained by either the small scale of the study or the artificial restrictions imposed by pathway definitions. Here, a network approach is used to study enzyme evolution in fully sequenced genomes, thus removing both constraints. We find that homologous pairs of enzymes are roughly twice as likely to have evolved from enzymes that are less than three steps away from each other in the reaction network than pairs of non-homologous enzymes. These results, together with the conservation of the type of chemical reaction catalyzed by evolutionarily related enzymes, suggest that functional blocks of similar chemistry have evolved within metabolic networks. One possible explanation for these observations is that this local evolution phenomenon is likely to cause less global physiological disruptions in metabolism than evolution of enzymes from other enzymes that are distant from them in the metabolic network.

Control motifs for intracellular regulatory networks

A number of technological innovations are yielding unprecedented data on the networks of biochemical, genetic, and biophysical reactions that underlie cellular behavior and failure. These networks are composed of hundreds to thousands of chemical species and structures, interacting via nonlinear and possibly stochastic physical processes. A central goal of modern biology is to optimally use the data on these networks to understand how their design leads to the observed cellular behaviors and failures. Ultimately, this knowledge should enable cellular engineers to redesign cellular processes to meet industrial needs (such as optimal natural product synthesis), aid in choosing the most effective targets for pharmaceuticals, and tailor treatment for individual genotypes. The size and complexity of these networks and the inevitable lack of complete data, however, makes reaching these goals extremely difficult. If it proves possible to modularize these networks into functional subnetworks, then these smaller networks may be amenable to direct analysis and might serve as regulatory motifs. These motifs, recurring elements of control, may help to deduce the structure and function of partially known networks and form the basis for fulfilling the goals described above. A number of approaches to identifying and analyzing control motifs in intracellular networks are reviewed.

Irreversibility in unbranched pathways: preferred positions based on regulatory considerations

It has been observed experimentally that most unbranched biosynthetic pathways have irreversible reactions near their beginning, many times at the first step. If there were no functional reasons for this fact, then one would expect irreversible reactions to be equally distributed among all positions in such pathways. Since this is not the case, we have attempted to identify functional consequences of having an irreversible reaction early in the pathway. We systematically varied the position of the irreversible reaction in model pathways and compared the resulting systemic behavior according to several criteria for functional effectiveness, using the method of mathematically controlled comparisons. This technique minimizes extraneous differences in systemic behavior and identifies those that are fundamental. Our results show that a pathway with an irreversible reaction located at the first step, and with all other reactions reversible, is on average better than an otherwise equivalent pathway with all reactions reversible, which in turn is on average better than an otherwise equivalent pathway with an irreversible reaction located at any step other than the first. Pathways with an irreversible first reaction and low concentrations of intermediates (one of the primary criteria for functional effectiveness) exhibit the following profile when compared to fully reversible pathways: changes in the concentration of intermediates in response to changes in the level of initial substrate are equally low, the robustness of the intermediate concentrations and of the flux is similar, the margins of stability are similar, flux is more responsive to changes in demand for end product, intermediate concentrations are less responsive to changes in demand for end product, and transient times are shorter. These results provide a functional rationale for the positioning of irreversible reactions at the beginning of unbranched biosynthetic pathways.

Effect of overall feedback inhibition in unbranched biosynthetic pathways

We have determined the effects of control by overall feedback inhibition on the systemic behavior of unbranched metabolic pathways with an arbitrary pattern of other feedback inhibitions by using a recently developed numerical generalization of Mathematically Controlled Comparisons, a method for comparing the function of alternative molecular designs. This method allows the rigorous determination of the changes in systemic properties that can be exclusively attributed to overall feedback inhibition. Analytical results show that the unbranched pathway can achieve the same steady-state flux, concentrations, and logarithmic gains with respect to changes in substrate, with or without overall feedback inhibition. The analytical approach also shows that control by overall feedback inhibition amplifies the regulation of flux by the demand for end product while attenuating the sensitivity of the concentrations to the same demand. This approach does not provide a clear answer regarding the effect of overall feedback inhibition on the robustness, stability, and transient time of the pathway. However, the generalized numerical method we have used does clarify the answers to these questions. On average, an unbranched pathway with control by overall feedback inhibition is less sensitive to perturbations in the values of the parameters that define the system. The difference in robustness can range from a few percent to fifty percent or more, depending on the length of the pathway and on the metabolite one considers. On average, overall feedback inhibition decreases the stability margins by a minimal amount (typically less than 5%). Finally, and again on average, stable systems with overall feedback inhibition respond faster to fluctuations in the metabolite concentrations. Taken together, these results show that control by overall feedback inhibition confers several functional advantages upon unbranched pathways. These advantages provide a rationale for the prevalence of this control mechanism in unbranched metabolic pathways in vivo.

Analysis and prediction of the effect of uncertain boundary values in modeling a metabolic pathway

The integration of large quantities of biological information into mathematical models of cell metabolism provides a way for quantitatively evaluating the effect of parameter changes on simultaneous, coupled, and, often, counteracting processes. From a practical point of view, the validity of the model's predictions would critically depend on its quality. Among others, one of the critical steps that may compromise this quality is to decide which are the boundaries of the model. That is, we must decide which metabolites are assumed to be constants, and which fluxes are considered to be the inputs and outputs of the system. In this article, we analyze the effect of the experimental uncertainty on these variables on the system's characterization. Using a previously defined model of glucose fermentation in Saccharomyces cerevisiae, we characterize the effect of the uncertainty on some key variables commonly considered to be constants in many models of glucose metabolism, i.e., the intracellular pH and the pool of nucleotides. Without considering if this variability corresponds to a possible true physiological phenomenon, the goal of this article is to illustrate how this uncertainty may result in an important variability in the systemic responses predicted by the model. To characterize this variability, we analyze the utility and limitations of computing the sensitivities of logarithmic-gains (control coefficients) to the boundary parameters. With the exception of some special cases, our analysis shows that these sensitivities are good indicators of the dependence of the model systemic behavior on the parameters of interest.