Abstract

To synthesize natural or artificial life, it is critically important to understand the design principles of how biochemical networks generate particular cellular functions and evolve complex systems in comparison with engineering systems. Cellular systems maintain their robustness in the face of perturbations arising from environmental and genetic variations. In analogy to control engineering architectures, the complexity of modular structures within a cell can be attributed to the necessity of achieving robustness. To reveal such biological design, the E. coli ammonia assimilation system is analyzed, which consists of complex but highly structured modules: the glutamine synthetase (GS) activity feedback control module with bifunctional enzyme cascades for catalyzing reversible reactions, and the GS synthesis feedback control module with positive and negative feedback loops. We develop a full-scale dynamic model that unifies the two modules, and we analyze its robustness and fine tuning with respect to internal and external perturbations. The GS activity control is added to the GS synthesis module to improve its transient response to ammonia depletion, compensating the tradeoffs of each module, but its robustness to internal perturbations is lost. These findings suggest some design principles necessary for the synthesis of life.

1 Introduction

To create or synthesize natural or artificial life, it is critically important to identify molecular interaction networks and to understand design principles for how biochemical networks generate particular cellular functions, such as replication, adaptation, metabolism, and stress response, and how they evolve complex systems. Such design principles underlying molecular processes should be explored in comparison with engineering control architectures [12, 24], which leads to an understanding of life as it could be. Elementary gene regulatory circuits, called network motifs, have been presented as building blocks of life [41], and combinations of them are suggested to lead to the emergence of new and complex functions. The biological design principles are keys to the creation or synthesis of life.

Cells are always exposed to various types of perturbations: environmental stresses, genetic changes, parameter uncertainty, and stochastic fluctuations. In particular, cells evolve various types of gene regulatory networks to survive environmental stresses. Robustness is the quality of being able to bear such perturbations [9, 43]. A system is said to be robust if it is capable of coping with internal and external perturbations with minimal damage. Different regulation strategies of feedforward and feedback loops and pathway redundancy can be employed to generate robustness.

To analyze the mechanisms by which engineering circuits provide particular functions, different methods such as block diagrams, transfer functions, and state equations have been established in control engineering. On the other hand, molecular interaction networks, which involve the transfer of signals, mass, and energy, are apparently similar to engineering control block diagrams or electrical circuit diagrams [29], but the biochemical network maps are not exactly duplicated by such engineering diagrams, due to their complexity and to the nonlinearity of the molecular interactions [12, 24]. Thus, we need new ways to intuitively analyze the architecture of a biochemical model and to identify the mechanisms by which network architecture is linked to specific functions.

Module decomposition is among the best ways for the analysis of a biochemical network. Cellular functions and biochemical events are coordinately carried out by modules, that is, groups of molecules interacting for a particular function. Modules can be classified in terms of various aspects, such as topology, elementary pathway, and control engineering architecture [10, 13, 39, 40]. From a control or regulation standpoint, module-based analysis has been presented that decomposes a biochemical network map into hierarchical modules: molecular, functional, and flux modules [12, 24]. If an engineering purpose is provided for a biochemical network, functional modules are readily identified and an engineering function can be assigned to each module. This provides an intuitive understanding of how the biochemical network of interest is made of hierarchical modules, facilitating the rational design of genetic manipulation for achieving the desired outcome [14, 38, 42].

Bacterial cells are always exposed to nutrient stresses such as sugars, phosphates, and ammonia; thus they evolve complex nutrient uptake modules. E. coli absolutely needs ammonia for synthesizing glutamine and glutamate, which are the sources of most nitrogen-containing compounds [37]. Therefore, E. coli evolved the ammonia assimilation system that takes up ammonia from the environment. To maintain the balance between carboneous and nitrogenous sources within cells against a change in an extracellular ammonia concentration, the ammonia assimilation system implements complex but highly structured modules: the glutamine synthetase (GS) activity feedback control module with bifunctional enzyme cascades for catalyzing reversible reactions, and the GS synthesis feedback control module with positive and negative feedback loops. Each module has been separately investigated to understand how it yields robustness to genetic and environmental perturbations. Dynamic models of the GS activity feedback module have been analyzed theoretically and computationally to reveal how the bifunctional enzyme cascades provide robustness with respect to internal perturbations [21, 34]. Mathematical models for the GS synthesis module have been analyzed, which suggest that positive feedback loops convert a gradual change in ammonia concentration into switchlike responses [7, 23, 28, 35]. However, few studies have investigated the molecular architecture for how these modules act in concert for enhanced robustness with respect to different types of perturbations or have identified the tradeoffs generated by such modular architectures.

The objective of this study is to develop a full-scale mathematical model that assembles those control modules, and to analyze its robust or fine-tuned properties to the internal and external perturbations. The module-based analysis and the subsequent sensitivity analysis elucidate the biological design principles of how these modules are assembled into a complex system for enhanced robustness.

2 Methods

2.1 Ammonia Assimilation System

A schematic diagram of the E. coli nitrogen assimilation system is shown in Figure 1. E. coli absolutely needs ammonia for synthesizing glutamine and glutamate, which are the primary products of ammonia assimilation, from which almost all nitrogen-containing compounds, including amino acids and nucleotides, are synthesized [36, 37]. Glutamine and glutamate are synthesized through glutamine synthetase (GS), glutamate synthase (GOGAT), and glutamate dehydrogenase (GDH) by adding ammonia to 2-ketoglutarate (2-KG), which is an intermediate of the TCA cycle. Among these enzymes, GS plays a critical role in connecting nitrogen metabolism to carbon metabolism [36]. In order to maintain the balanced metabolism between nitrogen and carbon, multiple negative and positive feedback loops control the activity or synthesis of glnA (GS), glnG (NRI), and glnL (NRII), and the transcription of other nitrogen-regulated (Ntr) genes (glnK, nac, and other genes responsible for transporting and degrading nitrogenous compounds).

Figure 1. 

A schematic diagram of the E. coli ammonia assimilation system: (A) Full model, (B) GS activity control flux module, (C) GS synthesis control flux module. The arrows indicate the reactions: binding reaction with a complex (), conversion reaction (), catalyzing the reaction (), inhibiting the reaction (), activating the reaction (), translation reaction (). UTUR indicates UTase/UR. The bold rectangles show the functional modules. Six functional modules are identified: plant, GS activity actuator, GS synthesis actuator, sensor, computer, and regulator. The higher-level flux modules—the GS activity flux module (blue thick arrow) and the GS synthesis flux module (red thick arrow)—are marked in (B) and (C), respectively.

Figure 1. 

A schematic diagram of the E. coli ammonia assimilation system: (A) Full model, (B) GS activity control flux module, (C) GS synthesis control flux module. The arrows indicate the reactions: binding reaction with a complex (), conversion reaction (), catalyzing the reaction (), inhibiting the reaction (), activating the reaction (), translation reaction (). UTUR indicates UTase/UR. The bold rectangles show the functional modules. Six functional modules are identified: plant, GS activity actuator, GS synthesis actuator, sensor, computer, and regulator. The higher-level flux modules—the GS activity flux module (blue thick arrow) and the GS synthesis flux module (red thick arrow)—are marked in (B) and (C), respectively.

The ammonia assimilation system is decomposed into three monocyclic systems: (i) PII-UTase/UR monocycle, (ii) GS-PI (ATase) monocycle, (iii) NRI-NRII monocycle (forming a positive feedback loop). Note that NRI and NRII form a two-component phosphorelay system. The PII-UTase/UR and GS-PI bicyclic system composed of (i) and (ii) regulates the activity of GS [37]. The PII-UTase/UR–NRI-NRII bicyclic system composed of (i) and (iii) is responsible for the synthesis of GS, NRI, NRII, and other Ntr proteins. These two bicyclic systems constitute major feedback loops for controlling the activity and synthesis of GS [17].

The adenylylation of GS is controlled in both the bicyclic and monocyclic systems. In the bicyclic loop, PI not only catalyzes the ATP-dependent addition of AMP to GS, with the release of inorganic pyrophosphate (PPi), but also catalyzes the phosphate-dependent removal of AMP from GS-AMP [37]. These two inverse reactions depend on whether PII is uridylylated. Uridylylation of PII is controlled by the bifunctional protein of UTase/UR, which monitors glutamine and 2-KG to catalyze the uridylylation or deuridylylation of PII. When 2-KG binds to UTase/UR, UTase catalyzes the UTP-dependent addition of UMP to PII. PII-UMP stimulates the deadenylylating activity of PI, resulting in deadenlylating GS [16]. When glutamine binds to UTase/UR, UR catalyzes the deuridylylation of PII-UMP, and unmodified PII makes PI adenylylate GS. GS activity is controlled by the adenylylation and deadenylylation reactions, called the bifunctional enzyme cascade for catalyzing reversible reactions [34]. In the monocyclic manner, 2-KG controls PII directly. PII with 2-KG does not activate the adenylylation activity of PI, showing that 2-KG controls the activity of PII without UTase/UR [19].

In the branch point of the dual bicyclic loops, PII, which shows activity for binding unphosphorylated NRII, transduces the information regarding glutamine and 2-KG to GS synthesis [36]. GlnK is a PII homolog in function, but the transcription regulation is governed by NRI-P, whereas PII is synthesized constitutively [3, 6]. NRII shows the reciprocal regulation of the kinase and phosphatase activities. Without PII or GlnK, autophosphorylated NRII can transfer its phosphate to NRI [18]. NRI is phosphorylated not only by NRII-P, but also directly by acetyl phosphate [30]. Dephosphorylated NRII controls the dephosphorylation of NRI-P. NRI is a transcriptional suppressor that functions in a σ70-dependent promoter. NRI-P forms multimers to strongly activate the expression of the glnALG operon encoding glnA (GS), glnL (NRII), glnG (NRI), and other Ntr genes in a σ54-dependent manner, which results in a higher level of GS, NRII, and NRI [11]. These form not only global negative feedback loops (UTase/UR, PII, NRII, NRI, GS), but also local positive feedback loops (NRII, NRI) responsible for enhanced activity of NRI-P. An excess amount of NRI-P binds to the governor to suppress Ntr gene expression [4].

2.2 Mathematical Modeling

In general, since the exact values of kinetic parameters are not measured in vivo and details of reaction networks are hard to fully identify, most of the kinetic parameters are provided nominal values so that the model reproduces most of the experimental observations and qualitative features. Thus, the model would be constructed at the coarse-grained level rather than at the level of the exact and full kinetics. Ideally, we would like to gain access to the activities of all-important molecular species. There is a strong need for methods that can handle molecular systems at an intermediate level without going all the way down to exact biochemical reactions. In this study, we apply forward and reverse engineering to building a dynamic model of the E. coli ammonia assimilation system, using CADLIVE [22, 23, 25] (see Tables 2 and 3 in  Appendix 1; see also  Appendix 2). Forward engineering constructs mathematical models from biochemical network maps; reverse engineering explores the values of kinetic parameters needed to fit experimental data. Note that this mathematical model is a great extension of and improvement on our previous model [23].

The ammonia assimilation system is the typical case where there are few kinetic data in vivo, while some kinetic parameters of the metabolic reactions regarding GS, GDH, and GOGAT are measured in vitro [1, 31–33]. Thus, the unknown values of kinetic parameters are estimated by genetic algorithms so that the model reproduces the key features of experimental data. The resultant mathematical model consists of 53 algebraic equations and 19 differential equations with 145 parameters. The two-phase partition (TPP) method is employed to divide the kinetics of molecular interactions into the molecular binding phase and the reaction phase, assuming that association and dissociation rates between proteins to be high compared with the rates of synthesis and degradation of mRNAs or proteins [8, 20]. This is a commonly used assumption or rule in biological systems.

2.3 Hierarchical Module Decomposition in Analogy to Control Engineering

A module has been characterized as a subsystem that possesses a function that is separable from that of other modules, in the sense that it is capable of maintaining most of its identity [12, 24, 38]. To provide a useful characterization of modularity, a multi-resolution scheme can then be used to assess different aspects of the modular decomposition, zooming out from the molecular description (molecular modules) to a block-diagram-like picture (functional modules). At a lower level of resolution, the components of a system can be divided into functional groups, which we refer to as functional modules. We sought analogies with the modules that are traditionally identified in control engineering schemes. The process to be controlled is identified, and the rest of the network is classified in terms of the function that it accomplishes to facilitate this regulation. A block represents each of these modules, and the interconnection of such blocks is frequently referred to as a block diagram. We define a flux module as a pathway that traces the mechanisms of interaction of a group of molecules involved together in the performance of a certain function. A flux module ideally connects functional modules to form an entity that possesses its own functionality in terms of control engineering. Each flux module cannot always be independent; the modules interact in a complex manner. A combination of flux modules may generate an emergent property such as oscillation, hysteresis, and irreversibility.

2.4 System Analysis

The main objective of the ammonia assimilation system is assumed to be to maintain the balance between nitrogen and carbon sources against a change in ammonia concentration. The ratio of glutamine concentration to 2-KG concentration (N/C ratio) is defined as the balance between nitrogenous and carboneous sources. This ratio is the output of this system.

Robustness in engineering design and analysis is measured by the system's ability to resume successful operation in the presence of signal and system uncertainties. Sensitivity analysis is the analysis of how model output varies with changes in model parameters. A model is said to be robust to a specific parameter if a change in the parameter hardly alters the model output. We quantify the system's robustness by calculating the steady-state logarithmic gain of the N/C ratio with respect to internal or external perturbation. The external perturbation is a change in ammonia concentration, while the internal one is a change in the synthesis rate constant of proteins.

External sensitivity is defined as the change in the N/C ratio with respect to a change in ammonia concentration:
formula
Internal sensitivity is defined as the change in the N/C ratio with respect to a change in the synthesis rate constant of a protein (k):
formula
In addition to those numerical sensitivities, the relative change in the N/C ratio (relative change) is defined with respect to an external perturbation, namely, a large change in ammonia concentration:
formula
This measure presents the relative change in the N/C ratio when the ammonia concentration is changed by −99 to 100%.

3 Results

3.1 Hierarchical Modular Structure

If an engineering purpose is provided for a biochemical network, functional modules are readily determined and an engineering function can be assigned to each module (see Section 2). To assign a particular function to proteins in terms of enhanced robustness of the N/C ratio, the biochemical network map was decomposed into functional and flux modules analogous to an engineering control system's block diagram [12, 24] (Figure 1A).

The GS, GOGAT, and GDH proteins that directly involve converting 2-KG to glutamine are taken as the plant to be controlled. In this model, we focus on GS, because it plays a major role in controlling the N/C ratio. UTase/UR is the sensor module to monitor the ratio of glutamine concentration to 2-KG concentration. The signal from the sensor is sent to the computer module, which consists of PII and GlnK. The computer is connected to two actuator modules: the GS activity actuator module and the regulator module connecting to the GS synthesis actuator module. In the GS activity actuator, the PI protein receives the signals from PII and GlnK to catalyze the adenylylation or deadenylylation of GS. The regulator module consists of the phosphorylation cascade of NRII and NRI. In the GS synthesis actuator module, phosphorylated NRI (NRI-P) leads to synthesis of the proteins: GS, NRI, NRII, and GlnK. The regulatory sites—the promoters of glnAp1, glnLp, glnAp2, and governor—are responsible for their gene expressions.

By connecting these functional modules, we identify many flux modules forming closed feedback loops, as shown in Table 4 and Figure 10, both in  Appendix 1. Although our description of these fluxes is qualitative, their functions can be easily identified in terms of control. Many feedback control flux modules are expected to act in concert to provide the robustness of the N/C ratio with respect to internal or external perturbation. In the hierarchical modular structures, functional modules are integrated into flux modules, which can be further assembled into a higher-level flux module. Finally, the two highest-level (most highly structured) flux modules—the GS activity control flux module and the GS synthesis control flux module—are identified (Figure 1B, C). In the former module, the N/C ratio sensed by UTase/UR is transmitted to the computer, which controls the actuator to alter the GS activity. This flux module does not require any gene expression, but functions at the protein interaction level. The GS activity control flux module consists of bifunctional enzyme cascades for catalyzing reversible reactions, where UTase/UR controls the urydylylation or deurydylylation of PII and GlnK, and then PII and GlnK control the adenylylation or deadenylylation of GS through PI. In the latter flux module, PII and GlnK in the computer module calculate the signal deriving from the sensor, and control the GS synthesis actuator through the amplifier regulator module (NRI and NRII). This module has typical positive and negative feedback regulations [44].

Note that PII has an additional or optional function to sense 2-KG to control the GS activity. This local feedback controls the GS activity in a different manner from the bifunctional enzyme cascade, which is included in the GS activity control flux module.

3.2 Dynamic Simulation of the Ammonia Assimilation System

We built the dynamic model of the ammonia assimilation system based on forward and reverse engineering (see Section 2 and Tables 2 and 3 in  Appendix 1), and then simulated the time course of the N/C ratio and GS concentration as shown in Figure 2A, where the ammonia concentration decreased to one-tenth over 1000 min. This simulation confirms a basic property of the ammonia assimilation system: A decrease in the ammonia concentration causes an abrupt decrease in the N/C ratio, but, as expected, the feedback system enhances GS synthesis, restoring the N/C ratio.

Figure 2. 

Dynamics for wild type and a knockout mutant. (A) The time course of the N/C ratio of glutamine to 2-ketoglutarate (solid line) and the total GS concentration (dotted line) in wild type are simulated. The ammonia concentration decreases to one-tenth over 1000 min. (B) Hysteresis of wild type. (C) Irreversible curves of a glnK knockout mutant. The steady-state level of the N/C ratio (solid lines) and the total GS concentration (dotted lines) are simulated with respect to the ammonia concentration in wild type (B) and a glnK knockout mutant (C). They are simulated while decreasing the ammonia concentration very gradually and then increasing it. For the wild type, the GS shows an abrupt increase at the low threshold concentration when the ammonia concentration decreases. In contrast, with the increase in ammonia concentration, the GS decreases rapidly at the high threshold concentration.

Figure 2. 

Dynamics for wild type and a knockout mutant. (A) The time course of the N/C ratio of glutamine to 2-ketoglutarate (solid line) and the total GS concentration (dotted line) in wild type are simulated. The ammonia concentration decreases to one-tenth over 1000 min. (B) Hysteresis of wild type. (C) Irreversible curves of a glnK knockout mutant. The steady-state level of the N/C ratio (solid lines) and the total GS concentration (dotted lines) are simulated with respect to the ammonia concentration in wild type (B) and a glnK knockout mutant (C). They are simulated while decreasing the ammonia concentration very gradually and then increasing it. For the wild type, the GS shows an abrupt increase at the low threshold concentration when the ammonia concentration decreases. In contrast, with the increase in ammonia concentration, the GS decreases rapidly at the high threshold concentration.

3.3 Validation of Knockout Mutants

In order to demonstrate the validity of the dynamic model, we compared the simulated results on knockout mutants with experimental data [2, 37], as shown in Table 1, where the GS concentrations were simulated in ammonia-rich and ammonia-limiting media. Note that the important thing is to compare the qualitative features of the simulated and experimental behaviors. These simulations qualitatively agreed with the experimental behaviors of the knockout mutants, ΔglnG (NRI), ΔglnL (NRII), ΔglnD (UTase/UR), ΔglnB (PII), and ΔglnK (GlnK). In the ΔglnG (NRI) mutant, the GS concentration was decreased from that of the wild type because GS synthesis was not enhanced. In both the ΔglnG (NRI) and ΔglnL (NRII) mutants, the signal of the N/C ratio was not transmitted to the glnA gene, so that the GS concentration was not changed at all, despite ammonia depletion. In the ΔglnD (UTase/UR) mutant, most of the signal of the N/C ratio was lost; only the signal of 2-KG was provided to PII, through 2-KG binding PII. Therefore, the GS concentration was still regulated to a small extent. In the ΔglnB (PII) mutant, the GS concentration increased in an ammonia-rich medium, because the positive feedback loop consisting of NRI and NRII was triggered. In the ΔglnK (GlnK) mutant, the GS concentration increased in an ammonia-rich medium, because GlnK has a similar function to PII, while the effect of GlnK is smaller than that of PII. These simulated results are rather consistent with experimental data, demonstrating the validity of the dynamic model.

Table 1. 

Model validation by experimental knockout mutants.

Mutant
NH3-rich
NH3-limiting
Experiment
Simulation [nM]
Experiment
Simulation [nM]
Wild type 150 197.01 1000 828.20 
ΔglnG (NRI) 70 142.86 50 142.86 
ΔglnL (NRII) 500 854.25 700 854.25 
ΔglnD (UTase/UR) 80 148.54 400 223.91 
ΔglnB (PII) 900 868.25 1000 866.59 
ΔglnK (GlnK) 200 233.20 1000 868.09 
Mutant
NH3-rich
NH3-limiting
Experiment
Simulation [nM]
Experiment
Simulation [nM]
Wild type 150 197.01 1000 828.20 
ΔglnG (NRI) 70 142.86 50 142.86 
ΔglnL (NRII) 500 854.25 700 854.25 
ΔglnD (UTase/UR) 80 148.54 400 223.91 
ΔglnB (PII) 900 868.25 1000 866.59 
ΔglnK (GlnK) 200 233.20 1000 868.09 

∗GS activity (GS transferase activity) in permeabilized cells. The results are the averages of numerous experiments [2, 37].

Simulated concentration of GS in a cell.

3.4 Validation of Hysteresis

The previous experiment showed that cells lacking GlnK show significant signs of the debilitating memory of starvation even after 10 h of starvation (runaway expression) [6]. Using the dynamic model, we analyzed the dynamics of the mutant lacking GlnK to understand the mechanism for runaway expression of the GS concentration. Figure 2B shows the steady-state concentration of GS and the N/C ratio as a function of ammonia concentration. They are simulated while the ammonia concentration is decreased very gradually and then increased. With the decrease in ammonia concentration, GS production showed an abrupt increase at the low threshold concentration. In contrast, with the increase in ammonia concentration the total GS concentration decreased rapidly at the high threshold concentration. The minimum concentration of ammonia necessary for enhancing GS synthesis is distinctly lower than the maximum concentration necessary to hold GS at a high level. The different thresholds indicate hysteresis and the existence of bistability or state transition, which can explain the runaway expression of the Ntr genes. In the mutant lacking GlnK, the simulation shows an almost irreversible state transition (Figure 2C) when the ammonia concentration increases.

The hysteresis curve was observed in the wild type, while the irreversible transition was seen in a GlnK knockout mutant model. A meaningful difference between PII and GlnK, a homolog of PII, is transcription regulation. The transcription of GlnK is controlled by NRI-P, whereas the transcription of PII occurs constitutively [36]. The increased synthesis of GlnK by NRI-P suppresses the phosphorylation of NRII by GlnK binding to NRII, thereby weakening the effect of the positive feedback loop on NRI-P-activated transcription of the Ntr genes. With the decrease in the GlnK level, the hysteresis comes close to an irreversible reaction. GlnK is responsible for regulating the irreversible response to a change in ammonia concentration. Thus, the experimental observation of runaway expression can be regarded as hysteresis or irreversibility. Generally hysteresis is generated by positive feedback regulation [44], whose mechanism in the GS synthesis control flux module is demonstrated by our theoretical model [28].

3.5 Transient Response Analysis

To identify the function of the two highest-level flux modules in terms of control engineering, we simulated the dynamic behavior of each module separately. The GS activity control flux module (UTase/UR, PI, PII, GlnK, and GS) is implemented through the protein interactions without any protein synthesis (Figure 1B). In contrast, the GS synthesis control flux module (UTase/UR, PII, GlnK, NRI, NRII, GS) requires gene expression for GS, NRI, NRII, and GlnK (Figure 1C). As shown in Figure 3, we simulated the dynamics of the N/C ratio with respect to changes in ammonia concentration in these regulator modules.

Figure 3. 

Transient response of the N/C ratio to a change in ammonia concentration for three modules. The N/C ratio is simulated. The ammonia concentration is decreased to one-fifth at 1000 min. (A) The GS activity control flux module that is made by setting the association constants between PII or GlnK and NRII, Kb[9,10,15,16], and the translation rate constant of GS, kp[1], to zero in the full model. (B) The GS synthesis control flux module that is made by setting the association constants between PII or GlnK and PI, Kb[5, 6], and the reaction rate constant of GS, kx[5,6,7,8,9], to zero in the full model. (C) The full model, which has the GS activity control module and the GS synthesis control module.

Figure 3. 

Transient response of the N/C ratio to a change in ammonia concentration for three modules. The N/C ratio is simulated. The ammonia concentration is decreased to one-fifth at 1000 min. (A) The GS activity control flux module that is made by setting the association constants between PII or GlnK and NRII, Kb[9,10,15,16], and the translation rate constant of GS, kp[1], to zero in the full model. (B) The GS synthesis control flux module that is made by setting the association constants between PII or GlnK and PI, Kb[5, 6], and the reaction rate constant of GS, kx[5,6,7,8,9], to zero in the full model. (C) The full model, which has the GS activity control module and the GS synthesis control module.

Figure 3A shows the time course of the N/C ratio in the GS activity control flux module, where it decreases with the decrease in ammonia concentration, and then recovers a little, achieving a steady state. The steady state level of the N/C ratio at the low ammonia concentration was much higher than that of the system with no feedback regulation (data not shown). In the synthesis control flux module alone (Figure 3B), the N/C ratio recovered significantly after ammonia depletion, and an overshoot of the N/C ratio was observed. Generally, since an overshoot wastes the system's resource due to excess protein synthesis, it should be suppressed. In the full system with both control modules (Figure 1A), the N/C ratio was substantially restored in response to ammonia depletion, while the overshoot of the N/C ratio was suppressed (Figure 3C). The addition of the GS activity flux module contributes to saving the cost of protein synthesis.

Generally, protein signal transduction is a fast process, while gene regulation is a slow one because transcription and translation are very slow compared with protein interaction. Thus, the GS activity control flux module is the fast system that regulates the GS activity without any gene expression. On the other hand, since the total amount of GS is limited, the N/C ratio cannot be restored so much against a large decrease in ammonia concentration. The limited control range (bandwidth) of ammonia concentration is a tradeoff for a fast reaction. In contrast, the GS synthesis flux module, which requires gene expression, shows a slow response, but substantially restores the N/C ratio against ammonia depletion. A large overshoot appears because the negative feedback loop in the GS synthesis module has a time delay due to its slow gene expression. Such an overshoot can be a tradeoff for the broad control range.

3.6 Robustness with Respect to External Changes

To explore the mechanism by which the modular architecture provides robustness to external perturbation, we simulated the relative change of the N/C ratio with respect to a broad change in ammonia concentration (Equation 3). The standard ammonia concentration is set to 100 μM, and the ammonia concentration is changed from −99% to +100%. As shown in Figure 4, the relative changes for three models—the full model, the GS activity control flux module, and the GS synthesis control flux module—are plotted with respect to a change in ammonia concentration. The activity control flux module showed the highest value of the relative changes. Namely, a large decrease in ammonia concentration caused the relative change to become very large. This is due to the limited total concentration of GS. On the other hand, in the GS synthesis control module and the full model, the relative changes were smaller (in absolute value). For a large decrease in ammonia concentration the GS synthesis control module increases the total amount of GS to maintain the N/C ratio. The GS synthesis control module is able to cope with the broad range of ammonia concentration. The full model shows the smallest relative changes (in absolute value) of all the three models, indicating the combination of the GS activity and the GS synthesis control flux module can act in concert to enhance the robustness against external change.

Figure 4. 

Relative change of the N/C ratio with respect to an external perturbation in three modules. This relative change of the N/C ratio is simulated at the steady state by varying the ammonia concentration for three modules: full model (solid line); GS activity control flux module (dotted line); GS synthesis control flux module (dot-dashed line). These modules are made in the same manner as in Figure 3.

Figure 4. 

Relative change of the N/C ratio with respect to an external perturbation in three modules. This relative change of the N/C ratio is simulated at the steady state by varying the ammonia concentration for three modules: full model (solid line); GS activity control flux module (dotted line); GS synthesis control flux module (dot-dashed line). These modules are made in the same manner as in Figure 3.

The dimples around −80% change are due to the high expression of GS caused by the positive feedback loop, reflecting the state transition of GS. In contrast, the GS activity control flux module does not show such a dimple, because it hardly involves the state transition.

3.7 Analysis of Sensitivity to Internal Changes

To explore the mechanism of how the modular architecture provides robustness to internal perturbation, we simulated the internal sensitivity of the N/C ratio to a change in the protein expression level (Equation 2), as shown in Figure 5. Figure 5A shows the internal sensitivity profile in the GS activity control flux module. The GS activity control flux module shows the robustness to internal perturbation, which is due to its bifunctional enzyme cascades for catalyzing reversible reactions. An increase in the total UTase/UR concentration enhances both the uridylylation and deuridylylation rates for PII and GlnK simultaneously, and an increase in the total concentration of these two proteins increases both the adenylation and deadenylation rates for GS at the same time. Consequently, a change in the total UTase/UR, PII, or GlnK level hardly alters the steady-state GS activity, that is, the ratio of GS to GS-AMP. Details of this mechanism will be described elsewhere.

Figure 5. 

Sensitivity of the N/C ratio to an internal perturbation in three modules. The internal sensitivity of the N/C ratio for each protein is simulated at the steady state with respect to a 1.05-fold change in the protein synthesis rate constant for three modules. (A) The GS activity control flux module, which consists of UTase/UR, PII, PI, and GlnK. Since it has neither NRI nor NRII, the sensitivity is zero. (B) The GS synthesis control flux module, which consists of UTase/UR, PII, GlnK, NRI, and NRII. Since the concentration of GlnK is set to be low compared with that of its homolog PII, the internal sensitivity of GlnK is small. Since this module contains no PI, its sensitivity is zero. (C) The full model that consists of the GS activity control flux module and the GS synthesis control flux module. These modules are made in the same manner as in Figure 3.

Figure 5. 

Sensitivity of the N/C ratio to an internal perturbation in three modules. The internal sensitivity of the N/C ratio for each protein is simulated at the steady state with respect to a 1.05-fold change in the protein synthesis rate constant for three modules. (A) The GS activity control flux module, which consists of UTase/UR, PII, PI, and GlnK. Since it has neither NRI nor NRII, the sensitivity is zero. (B) The GS synthesis control flux module, which consists of UTase/UR, PII, GlnK, NRI, and NRII. Since the concentration of GlnK is set to be low compared with that of its homolog PII, the internal sensitivity of GlnK is small. Since this module contains no PI, its sensitivity is zero. (C) The full model that consists of the GS activity control flux module and the GS synthesis control flux module. These modules are made in the same manner as in Figure 3.

Figure 5B shows the internal sensitivity of the GS synthesis control flux module. The internal sensitivity is much higher than that of the GS activity control module. This is caused by the positive feedback loops, where a perturbation to PII, NRI, and NRII can be amplified. Actually, the NRI and NRII proteins responsible for the positive feedback should not be controlled at the constant level against perturbation to their levels. If these protein levels were fixed constant against perturbation, the GS synthesis control flux module could not enhance GS synthesis. The high internal sensitivity is an inherent property of a positive feedback loop. On the other hand, the high internal sensitivity for PII is due to the fact that the PII level is a critical factor to turn on the positive feedback regulators (NRI and NRII).

Figure 5C shows the internal sensitivity in the full model with both the GS activity and GS synthesis control flux modules. The sensitivity profiles of PII, GlnK, GS, NRI, and NRII are similar to that of the GS synthesis control module. The sensitivity profiles of UTase/UR and PI correspond to that of the GS activity control flux module. Integration of the two modules hardly affects the sensitivity for UTase/UR and PI, whereas the sensitivity profile in the GS synthesis control flux module is inherited into the full model. Consequently, the robustness to changes in the PII level, achieved by the bifunctional enzyme cascade, is lost by the module integration.

3.7 Robustness Tradeoff in the GS Synthesis Module

The GS synthesis module enhances the robustness of the N/C ratio with respect to ammonia depletion, while it reduces the robustness with respect to the internal perturbation. This suggests a robustness tradeoff between the external and internal sensitivities. We investigated this possibility. While changing the value of the kinetic parameter that directly involved the NRII-NRI-glnAp2 positive feedback regulation (Table 4 in  Appendix 1), the external and internal sensitivities were simulated, as shown in Figure 6. With an increase in the strength of the positive feedback, the internal sensitivity increases whereas the external one decreases. This confirms a typical tradeoff between the two sensitivities.

Figure 6. 

A tradeoff between the external and internal sensitivities in the GS synthesis module. The internal sensitivity of the N/C ratio to NRI (dashed line) and NRII (dotted line) and the external sensitivity to ammonia depletion (solid line) are simulated, while the value of the association constant between NRI-P and the glnAp2 enhancer (Kb[17]), which directly involves the NRII-NRI-glnAp2 positive feedback regulation, is changed (Table 4). An increase in that value indicates enhancing the strength of the positive feedback.

Figure 6. 

A tradeoff between the external and internal sensitivities in the GS synthesis module. The internal sensitivity of the N/C ratio to NRI (dashed line) and NRII (dotted line) and the external sensitivity to ammonia depletion (solid line) are simulated, while the value of the association constant between NRI-P and the glnAp2 enhancer (Kb[17]), which directly involves the NRII-NRI-glnAp2 positive feedback regulation, is changed (Table 4). An increase in that value indicates enhancing the strength of the positive feedback.

3.9 Effect of Protein Levels on Robustness to a Change in Ammonia Concentration

To further explore the proteins critically responsible for robustness to external perturbation, we simulated the effect of protein levels on the external sensitivity (Equation 1), as shown in Figure 7. The UTase/UR level hardly influences the external sensitivity (Figure 7A), because the external sensitivity and GS concentration are almost constant. This is due to the fact that a change in the total UTase/UR level does not change the ratio of PII to PII-UMP, an inherent property of the bifunctional enzyme cascades for catalyzing reversible reactions [21, 34]. The external sensitivity gradually decreases with increase in the PI concentration (Figure 7B), because PI is indirectly involved in the GS synthesis control flux module. PI binds to PII, which sequesters PII away from binding to NRII, enhancing the NRII-NRI phosphorylation cascade, leading to enhanced GS synthesis. The resultant increase in GS enhances the robustness to external perturbation. An increase in the GlnK concentration gradually increases the external sensitivity (Figure 7C). This is because an increase in the GlnK concentration suppresses the NRII-NRI phosphorylation cascade, leading to decreased GS synthesis. As a feature common to PII, NRI, and NRII, their sensitivities show a deep dent, corresponding to the occurrence of the state transition of the GS level (Figure 7D, E, F). These protein levels are found to be critically responsible for the state transition.

Figure 7. 

Effect of protein levels on robustness to an external perturbation. The external sensitivity (solid lines) and GS concentration (dotted lines) are simulated at the steady state at different protein levels. For UTase/UR (UTUR) (A), PI (B), and PII (D), the protein levels are changed. For GlnK (C), NRI (E), and NRII (F), the protein synthesis rates are changed.

Figure 7. 

Effect of protein levels on robustness to an external perturbation. The external sensitivity (solid lines) and GS concentration (dotted lines) are simulated at the steady state at different protein levels. For UTase/UR (UTUR) (A), PI (B), and PII (D), the protein levels are changed. For GlnK (C), NRI (E), and NRII (F), the protein synthesis rates are changed.

PII, an integrator for two major flux modules, is a critical factor to trigger the state transition of GS (Figure 1). However, the PII level is not reported to be regulated by any feedback loop (e.g., autogenous feedback) [15]. If PII were robustly maintained at a constant level, the internal sensitivity for PII would decrease. This raises the question of why the PII protein is not tightly controlled at a constant level despite its importance for the state transition.

To further explore how the PII level affects the system, we simulated the GS concentration with respect to the two-dimensional space of the ammonia concentration and the PII level, as shown in Figure 8. The GS state transition occurs within a relatively broad range of the PII level, while the threshold value for the state transition is finely tuned but readily varied by a change in the PII concentration (Figure 8D). This suggests that the important thing for E. coli is not to determine the exact ammonia concentration for the state transition by fixing the PII level, but to robustly produce the state transition for ammonia depletion. In other words, the ammonia assimilation system has little influence in determining the precise threshold value for GS state transition.

Figure 8. 

Transition of the GS level in the two-dimensional space of PII and ammonia concentration. (A, B, C) The steady-state level of GS is simulated, while the total PII and ammonia concentrations are varied systematically. (D) The threshold value of the ammonia concentration for the GS state transition is plotted with respect to the relative concentration of PII.

Figure 8. 

Transition of the GS level in the two-dimensional space of PII and ammonia concentration. (A, B, C) The steady-state level of GS is simulated, while the total PII and ammonia concentrations are varied systematically. (D) The threshold value of the ammonia concentration for the GS state transition is plotted with respect to the relative concentration of PII.

4 Discussion

4.1 Robustness Tradeoff in Modular Architecture

We have presented a full dynamic model for the E. coli ammonia assimilation system. The module-based analysis was applied to this model to understand the mechanism by which the complex biochemical network achieves robustness to internal and external perturbations. By analogy to engineering control, the model is decomposed into hierarchical modular architectures. Out of many possible flux modules, the two highest-level flux modules are identified as key control modules: the GS activity control flux module and the GS synthesis control flux module.

The GS activity control flux module is a protein interaction network; thus it shows a fast response, but it is not suitable for coping with a large change in ammonia concentration, due to its limited GS level. The GS activity control flux module provides high robustness to internal perturbation or protein expression levels, probably due to its intrinsic structure of bifunctional enzyme cascades catalyzing reversible reactions [34]. In contrast, the GS synthesis control flux module involves a slow process that requires gene expression, where time delay in the negative feedback controls produces an overshoot in transient response. The GS synthesis control flux module provides robustness of the N/C ratio with respect to a change in ammonia concentration. For a large decrease in it, the GS synthesis module enhances the synthesis of NRI and NRII through their positive feedback regulation to produce a large amount of GS, resulting in maintaining the N/C ratio. This architecture provides switchlike behavior that shows in the state transition of GS with respect to a change in ammonia concentration, called hysteresis or bistability [44]. The high internal sensitivity to NRI and NRII is a tradeoff for the enhanced robustness to ammonia depletion.

The tradeoff between internal and external sensitivity is identified for typical feedback loops, for example, the positive feedback loop presented in this study, and the negative feedback loop in the heat shock response [24]. Such a tradeoff seems reasonable, because a low internal sensitivity confines allowable parameter spaces narrowly, resulting in decreased robustness to external perturbations. Does this tradeoff extend to all biological feedback loops? One example to the contrary can be noted. If some particular components, such as reversible reactions catalyzed by bifunctional enzymes (as in the GS activity control module), are employed to form the feedback loops, they can provide robustness to change in the total concentration of enzymes (internal perturbation), while preserving the robustness to external changes. Most of the mechanisms by which different types of tradeoffs are generated by biological architectures remain to be revealed, and an understanding of them is important.

The GS activity control flux module and GS synthesis control flux module are combined by an integrator protein, PII, to take in the advantages of both the flux modules—nonovershoot (fast adjustment) and a broad control range—thereby compensating their tradeoffs. The activity flux module adjusts the GS activity at a fast rate, suppressing the overshoot. The GS synthesis module enables the response to a large change in ammonia concentration. On the other hand, module integration deteriorates the robustness of the N/C ratio to changes in the PII level, achieved by the bifunctional enzyme cascade. This suggests that the robustness to external perturbation would be required more than robustness to internal changes.

4.2 A Critical Integrator in Two Major Feedback Modules

The PII protein is a critical factor to switch the state transition of GS, but the PII level is not regulated for constancy by any feedback loops (e.g., autogenous negative feedback). Note that if the PII level were tightly fixed, the robustness to internal perturbation would be enhanced. Actually, the N/C ratio and GS concentration are rather sensitive to perturbation of the PII level (Figure 8), that is, the threshold value of ammonia concentration for the state transition is finely tuned. This suggests that the exact threshold value is not critical in a real environment. Since not only ammonia but also many other factors (substrates or stresses) affect the assimilation system, it may be pointless to precisely determine the GS concentration just for each ammonia concentration. The important thing is to robustly produce the state transition for coping with ammonia depletion.

In many cases, specific exact values are not required in biological systems. For bacterial chemotaxis, the occurrence of exact adaptation is robust to various perturbations, but the response time and the steady-state level are finely tuned [5]. For the bacteriolysis reaction of the lambda phage, the selection of bacteriolysis or lysogenization must be robust to cope with environmental changes, while the threshold values are finely tuned for the occurrence of such selection [27].

4.3 Biological Design Features

We confirmed a universal principle that complexity enhances robustness at the systems level, as observed in other biological systems [24]. Use of complex regulation strategies is likely to be a specifically designed solution to different aspects and requirements of robustness rather than the result of evolutionary accidents that gave birth to complicated regulatory loops. The regulatory structure has crucial elements, which are orchestrated to address the numerous and sometimes conflicting design requirements [12, 43].

To make clear some differences between the ammonia assimilation model (i.e., biological design features) and engineering systems, we superimpose its functional modules on a classical control block diagram, as shown in Figure 9. This superposition is helpful in qualitatively analyzing the dynamic architecture of the biochemical network in comparison with engineering control architectures. First, we find that the functional modules are mutually interacting (crosstalked). For example, the NRI and NRII levels in the computer module alter the GS synthesis in the actuator, while the synthesis of these proteins is directly amplified by the actuator. In a general engineering control block diagram, a signal passes in only one direction. Second, the output value (N/C ratio) of the biological system is definitely sensitive to internal perturbation to the computer (NRI and NRII), while an engineering digital computer is extremely robust with respect to perturbations or noise. The high internal sensitivity for those components responsible for positive feedback is a tradeoff with their robustness to external perturbation (Figure 6), which results from the fact that biological computation is constrained by the implementation of molecular interactions. Third, a high internal sensitivity for a critical integrator (PII) is tolerated, suggesting that the threshold value of ammonia concentration for state transition is finely tuned, not a serious problem. Finally, the input-output characteristic of the E. coli system is far different from that of established engineering controllers. The ammonia assimilation system has no explicit input value (target value of the N/C ratio) and performs no constant-variable control (Figure 2B), while engineering systems, such as a proportional-integral-derivative (PID) controller, adjust the output value to the target (input) one against external perturbation.

Figure 9. 

Superposition of functional modules on a typical control block diagram. Red indicates the biological design features that are different from those in a typical engineering system.

Figure 9. 

Superposition of functional modules on a typical control block diagram. Red indicates the biological design features that are different from those in a typical engineering system.

4.4 Relationship Between Biological Design and Artificial Life

The above design features underlying molecular processes are critically important in the rational design of a biological system. In general, for achieving de novo synthesis of a biological system, it is necessary to make a list of elementary network components, including network motifs [41], and to understand how they are assembled to generate particular cellular functions, by systematically analyzing lots of model networks [26]. Actually, the ammonia assimilation system can be decomposed into typical reactions and components: reversible reactions catalyzed by a bifunctional enzyme (UTase/UR, PII, GlnK), a two-component phosphorelay system composed of NRI and NRII, cooperative binding of transcription factors, negative feedback loops, and positive feedback loops. These components could be assembled or evolved to enhance robustness under a given environment rather than by accidents. This suggests that biological networks can be regarded as an optimal combination of elementary components. If that is true, we can synthesize a cellular system in the bottom-up manner that is characteristic of engineering and artificial life. In biology, it is critically important to list all the elementary components, to understand the features generated by the combination of them, and to present the design principles for their combination to survive restrictive but varying environments. An understanding of these principles will be helpful for research on artificial life that aims at finding simple rules for creating life.

Acknowledgments

This work was supported by a Grant-in-Aid for Scientific Research on Priority Areas (“Systems Genomics”) and partially by a Grant-in-Aid for Scientific Research (B) (22300101, 2010) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. Koichi Masaki and Kazuhiro Maeda were supported by Research Fellowships from the Japan Society for the Promotion of Science for Young Scientists.

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Appendix 1: Supplementary Tables and Figure

Table 2. 

Full mathematical equations of the E. coli ammonia assimilation system. (Subscript o refers to the total.)

Binding phase 
 
[UTUR]o = [UTUR] + [UTUR:glutamine] + [UTUR:aKG] + [P2:UMP:UTUR:aKG] + [UTUR:glutamine:P2-UMP] 
 
   + [GlnK:UMP:aKG:UTUR] + [GlnK-UMP:glutamine:UTUR] + [aKG:P2-UMP:glutamine:UTUR] 
 
   + [aKG:P2:UMP:aKG:UTUR] 
 
[P2]o = [P2] + [P2:UMP:UTUR:aKG] + [P1:P2] + [GS:AMP:P1:P2] + [NR2:P2] + [NR2-P:P2:ADP] + [NR1-P:NR2:P2] 
 
  + [aKG:P2] + [aKG:P2:UMP:aKG:UTUR] 
 
[P2-UMP]o = [P2-UMP] + [UTUR:glutamine:P2-UMP] + [P1:P2-UMP] + [GS-AMP:P1:P2-UMP] + [aKG:P2-UMP:P1] 
 
    + [aKG:P2-UMP:glutamine:UTUR] + [GS-AMP:aKG:P2-UMP:P1] + [aKG:P2-UMP] 
 
[P1]o = [P1] + [P1:P2] + [P1:P2-UMP] + [GS:AMP:P1:P2] + [GS-AMP:P1:P2-UMP] + [GS-AMP:P1:GlnK-UMP] 
 
  + [GS:AMP:P1:GlnK] + [GlnK-UMP:P1] + [GlnK:P1] + [aKG:P2-UMP:P1] + [GS-AMP:aKG:P2-UMP:P1] 
 
[GS]o = [GS] + [GS:AMP:P1:P2] + [GS:AMP:P1:GlnK] 
 
[GS-AMP]o = [GS-AMP] + [GS-AMP:P1:P2-UMP] + [GS-AMP:P1:GlnK-UMP] + [GS-AMP:aKG:P2-UMP:P1] 
 
[NR2]o = [NR2] + [NR2:P2] + [NR2:ATP] + [NR1-P:NR2:P2] + [NR2:GlnK] + [NR1-P:NR2:GlnK] 
 
[NR2-P]o = [NR2-P] + [NR2-P:P2:ADP] + [NR1:NR2-P] + [NR2-P:ADP:GlnK] 
 
[NR1]o = [NR1] + [NR1:NR2-P] + [NR1:acetyl] + [NR1:Enhancer(glnAp1)] + [NR1:Enhancer(glnLp1)] 
 
[NR1-P]o = [NR1-P] + [NR1-P:NR2:P2] + 4[NR1-P(4):Enhancer(glnAp2)] + [NR1-P:NR2:GlnK] + [NR1-P:Enhancer(GlnK)] 
   + [NR1-P:governor] + [NR1-P:Enhancer(Nac)] 
 
[Enhancer(glnAp1)]o = [Enhancer(glnAp1)] + [NR1:Enhancer(glnAp1)] 
 
[Enhancer(glnLp1)]o = [Enhancer(glnLp1)] + [NR1:Enhancer(glnLp1)] 
 
[Enhancer(glnAp2)]o = [Enhancer(glnAp2)] + [NR1-P(4):Enhancer(glnAp2)] 
 
[GlnK]o = [GlnK] + [GlnK:P1] + [NR2:GlnK] + [GS:AMP:P1:GlnK] + [GlnK:UMP:aKG:UTUR] + [NR2-P:ADP:GlnK] 
   + [NR1-P:NR2:GlnK] 
 
[GlnK-UMP]o = [GlnK-UMP] + [GlnK-UMP:P1] + [GS-AMP:P1:GlnK-UMP] + GlnK-UMP:glutamine:UTUR] 
 
[governor]o = [governor] + [NR1-P:governor
 
[Enhancer(GlnK)]o = [Enhancer(GlnK)] + [NR1-P:Enhancer(GlnK)] 
 
[Enhancer(Nac)]o = [Enhancer(Nac)] + [NR1-P:Enhancer(Nac)] 
 
[UTUR:glutamine] = Kb[1][UTUR][glutamine] 
 
[UTUR:aKG] = Kb[2][UTUR][aKG] 
 
[P2:UMP:UTUR:aKG] = Kb[3][UTUR:aKG][P2][UMP] 
 
[UTUR:glutamine:P2-UMP] = Kb[4][UTUR:glutamine][P2-UMP] 
 
[P1:P2] = Kb[5][P1][P2] 
 
[P1:P2-UMP] = Kb[6][P1][P2-UMP] 
 
[GS:AMP:P1:P2] = Kb[7][P1:P2][GS][AMP] 
 
[GS-AMP:P1:P2-UMP] = Kb[8][P1:P2-UMP][GS-AMP] 
 
[NR2:P2] = Kb[9][NR2][P2] 
 
[NR2:ATP] = Kb[10][NR2][ATP] 
 
[NR2-P:P2:ADP] = Kb[11][P2][NR2-P][ADP] 
 
[NR1:NR2-P] = Kb[12][NR1][NR2-P] 
 
[NR1:acetyl] = Kb[13][NR1][acetyl] 
 
[NR1-P:NR2:P2] = Kb[14][NR2:P2][NR1-P] 
 
[NR1:Enhancer(glnAp1)] = Kb[15][NR1][Enhancer(glnAp1)] 
 
[NR1:Enhancer(glnLp1)] = Kb[16][NR1][Enhancer(glnLp1)] 
 
[NR1-P(4):Enhancer(glnAp2)] = Kb[17]4[NR1-P]4[Enhancer(glnAp2)] 
 
[GS-AMP:P1:GlnK-UMP] = Kb[18][GlnK-UMP:P1][GS-AMP] 
 
[GS:AMP:P1:GlnK] = Kb[19][GS][GlnK:P1][AMP] 
 
[NR2-P:ADP:GlnK] = Kb[20][NR2-P][GlnK][ADP] 
 
[NR1-P:NR2:GlnK] = Kb[21][NR1-P][NR2:GlnK] 
 
[GlnK:UMP:aKG:UTUR] = Kb[22][GlnK][UTUR:aKG][UMP] 
 
[GlnK-UMP:glutamine:UTUR] = Kb[23][GlnK-UMP][UTUR:glutamine] 
 
[GlnK-UMP:P1] = Kb[24][GlnK-UMP][P1] 
 
[GlnK:P1] = Kb[25][GlnK][P1] 
 
[NR1-P:Enhancer(GlnK)] = Kb[26][NR1-P][Enhancer(GlnK)] 
 
[NR1-P:governor] = Kb[27][NR1-P][governor
 
[NR2:GlnK] = Kb[28][GlnK][NR2] 
 
[NR1-P:Enhancer(Nac)] = Kb[29][Enhancer(Nac)][NR1-P] 
 
[aKG:P2] = Kb[30]3[aKG]3[P2] 
 
[aKG:P2-UMP:P1] = Kb[31][aKG:P2-UMP][P1] 
 
[GS-AMP:aKG:P2-UMP:P1] = Kb[32][aKG:P2-UMP:P1][GS-AMP] 
 
[aKG:P2-UMP:glutamine:UTUR] = Kb[33][aKG:P2-UMP][UTUR:glutamine] 
 
[aKG:P2:UMP:aKG:UTUR] = Kb[34][aKG:P2][UTUR:aKG][UMP] 
 
[aKG:P2-UMP] = Kb[35]3[aKG]3[P2-UMP] 
 
 
Reaction phase 
 
= + km[1]([NR-P(4) : Enhancer(glnAp2)]/[Enhancer(glnAp2)]o) · [DNA(GS)] 
   · (1.3 − [NR-P : governor]/[governor]o) + km[2](1.0 − ([NR1 : Enhancer(glnAp1)]/[Enhancer(glnAp1)]o)) 
   · [DNA(GS)] − kmd[2][mRNA(GS)] 
 
= + km[3](1.0 − ([NR1:Enhancer(glnLp1)]/[Enhancer(glnLp1)]o)) · [DNA(NR1)] 
   + km[4]([NR1-P(4) : Enhancer(glnAp2)]/[Enhancer(glnAp2)]o) 
   · [DNA(NR1)] · (1.3 − [NR-P : governor]/[governor]o) − kmd[2][mRNA(NR1)] 
 
= + km[5](1:0 − ([NR1 : Enhancer(glnLp1)]/[Enhancer(glnLp1)]o)) · [DNA(NR2)] 
   + km[6]([NR1-P(4) : Enhancer(glnAp2)]/[Enhancer(glnAp2)]o) · [DNA(NR2)] 
   · (1.3 − [NR-P : governor]/[governor]o) − kmd[3][mRNA(NR2)] 
 
= + km[7]([NR1-P : Enhancer(GlnK)]/[Enhancer(GlnK)]o)[DNA(GlnK)] − kmd[4][mRNA(GlnK)] 
 
= + km[8]([NR1-P : Enhancer(Nac)]/[Enhancer(Nac)]o)[DNA(nac)] − kmd[5][mRNA(nac)] 
 
= − kx[1][P2 : UMP : UTUR : aKG] + kx[2][UTUR : glutamine : P2-UMP] 
   − kx[3][aKG : P2 : UMP : aKG : UTUR] + kx[4][aKG : P2-UMP : glutamine : UTUR] 
 
= ;− kx[2][UTUR : glutamine : P2-UMP] + kx[1][P2 : UMP : UTUR : aKG] 
   + kx[3][aKG : P2 : UMP : aKG : UTUR] ;− kx[4][aKG : P2-UMP : glutamine : UTUR] 
 
= −kx[5][GS : AMP : P1 : P2] − kpd[1][GS : AMP : P1 : P2] + kx[6][GS-AMP : P1 : P2-UMP] 
   + kp[1][mRNA(GS)] − kpd[2][GS] + kx[7][GS-AMP : P1 : GlnK-UMP] − kx[8][GS : AMP : P1 : GlnK] 
   − kpd[3][GS : AMP : P1 : GlnK] + kx[9][GS-AMP : aKG : P2-UMP : P1] 
 
= −kx[6][GS-AMP : P1 : P2-UMP] − kpd[4][GS-AMP : P1 : P2-UMP] + kx[5][GS : AMP: P1 : P2] 
   + kx[8][GS : AMP : P1 : GlnK] − kx[7][GS-AMP : P1 : GlnK-UMP] − kpd[5][GS-AMP] 
   − kpd[6][GS-AMP : P1 : GlnK-UMP] − kpd[7][GS-AMP : aKG : P2-UMP : P1] 
   − kx[9][GS-AMP : aKG : P2-UMP : P1] 
 
= − kpd[8][NR2 : P2] − kx[10][NR2 : ATP] − kpd[9][NR2 : ATP] + kx[11][NR2-P : P2 : ADP] 
   + kx[12][NR1 : NR2-P] + kp[2][mRNA(NR2)] − kpd[10][NR2] − kpd[11][NR1-P : NR2 : P2] 
   + kx[13][NR2-P : ADP : GlnK] − kpd[12][NR2 : GlnK] + kpd[13][NR1-P : NR2 : GlnK] 
 
= − kx[11][NR2-P : P2 : ADP] − kpd[14][NR2-P : P2 : ADP] − kx[12][NR1 : NR2-P] 
   − kpd[15][NR1 : NR2-P] + kx[10][NR2 : ATP] − kpd[16][NR2-P] − kx[13][NR2-P : ADP : GlnK] 
   − kpd[17][NR2-P : ADP : GlnK] 
 
= − kx[14][NR1 : NR2-P] − kpd[18][NR1 : NR2-P] − kx[15][NR1 : acetyl] − kpd[19][NR1 : acetyl] 
   − kpd[20][NR1 : Enhancer(glnAp1)] − kpd[21][NR1 : Enhancer(glnLp1)] + kx[16][NR1-P : NR2 : P2] 
   + kp[3][mRNA(NR1)] − kpd[22][NR1] + kx[17][NR1-P : NR2 : GlnK] 
 
= − kx[16][NR1-P : NR2 : P2] − kpd[23][NR1-P : NR2 : P2] + kx[14][NR1 : NR2-P] + kx[15][NR1 : acetyl] 
   − kdp[24] · 4[NR1-P(4) : Enhancer(glnAp2)] − kpd[25][NR1-P] − kpd[26][NR1-P : governor
   − kpd[27][NR1-P : Enhancer(GlnK)] − kx[17][NR1-P : NR2 : GlnK] − kpd[28][NR1-P : NR2 : GlnK] 
   − kpd[29][NR1-P : Enhancer(Nac)] 
 
= − kp[4][mRNA(GlnK)] + kx[18][GlnK-UMP : glutamine : UTUR] − kpd[30][GlnK] − kpd[31][GlnK : P1] 
   − kpd[32][NR2 : GlnK] − kx[19][GlnK : UMP : aKG : UTUR] − kpd[33][GS : AMP : GlnK : P1] 
   − kpd[34][NR1-P : NR2 : GlnK] − kpd[35][GlnK : UMP : aKG : UTUR] − kpd[36][NR2-P : ADP : GlnK] 
 
= + kx[19][GlnK : UMP : aKG : UTUR] − kpd[37][GS-AMP : GlnK-UMP : P1] 
   − kpd[38][GlnK-UMP : P1] − kx[18][GlnK-UMP : glutamine : UTUR] − kpd[39][GlnK-UMP] 
   − kpd[40][GlnK-UMP : glutamine : UTUR] 
 
= + kp[5][mRNA(nac)] − kpd[41][nac] 
 
Metabolic reactions 
 
 
 
 
 
 
Binding phase 
 
[UTUR]o = [UTUR] + [UTUR:glutamine] + [UTUR:aKG] + [P2:UMP:UTUR:aKG] + [UTUR:glutamine:P2-UMP] 
 
   + [GlnK:UMP:aKG:UTUR] + [GlnK-UMP:glutamine:UTUR] + [aKG:P2-UMP:glutamine:UTUR] 
 
   + [aKG:P2:UMP:aKG:UTUR] 
 
[P2]o = [P2] + [P2:UMP:UTUR:aKG] + [P1:P2] + [GS:AMP:P1:P2] + [NR2:P2] + [NR2-P:P2:ADP] + [NR1-P:NR2:P2] 
 
  + [aKG:P2] + [aKG:P2:UMP:aKG:UTUR] 
 
[P2-UMP]o = [P2-UMP] + [UTUR:glutamine:P2-UMP] + [P1:P2-UMP] + [GS-AMP:P1:P2-UMP] + [aKG:P2-UMP:P1] 
 
    + [aKG:P2-UMP:glutamine:UTUR] + [GS-AMP:aKG:P2-UMP:P1] + [aKG:P2-UMP] 
 
[P1]o = [P1] + [P1:P2] + [P1:P2-UMP] + [GS:AMP:P1:P2] + [GS-AMP:P1:P2-UMP] + [GS-AMP:P1:GlnK-UMP] 
 
  + [GS:AMP:P1:GlnK] + [GlnK-UMP:P1] + [GlnK:P1] + [aKG:P2-UMP:P1] + [GS-AMP:aKG:P2-UMP:P1] 
 
[GS]o = [GS] + [GS:AMP:P1:P2] + [GS:AMP:P1:GlnK] 
 
[GS-AMP]o = [GS-AMP] + [GS-AMP:P1:P2-UMP] + [GS-AMP:P1:GlnK-UMP] + [GS-AMP:aKG:P2-UMP:P1] 
 
[NR2]o = [NR2] + [NR2:P2] + [NR2:ATP] + [NR1-P:NR2:P2] + [NR2:GlnK] + [NR1-P:NR2:GlnK] 
 
[NR2-P]o = [NR2-P] + [NR2-P:P2:ADP] + [NR1:NR2-P] + [NR2-P:ADP:GlnK] 
 
[NR1]o = [NR1] + [NR1:NR2-P] + [NR1:acetyl] + [NR1:Enhancer(glnAp1)] + [NR1:Enhancer(glnLp1)] 
 
[NR1-P]o = [NR1-P] + [NR1-P:NR2:P2] + 4[NR1-P(4):Enhancer(glnAp2)] + [NR1-P:NR2:GlnK] + [NR1-P:Enhancer(GlnK)] 
   + [NR1-P:governor] + [NR1-P:Enhancer(Nac)] 
 
[Enhancer(glnAp1)]o = [Enhancer(glnAp1)] + [NR1:Enhancer(glnAp1)] 
 
[Enhancer(glnLp1)]o = [Enhancer(glnLp1)] + [NR1:Enhancer(glnLp1)] 
 
[Enhancer(glnAp2)]o = [Enhancer(glnAp2)] + [NR1-P(4):Enhancer(glnAp2)] 
 
[GlnK]o = [GlnK] + [GlnK:P1] + [NR2:GlnK] + [GS:AMP:P1:GlnK] + [GlnK:UMP:aKG:UTUR] + [NR2-P:ADP:GlnK] 
   + [NR1-P:NR2:GlnK] 
 
[GlnK-UMP]o = [GlnK-UMP] + [GlnK-UMP:P1] + [GS-AMP:P1:GlnK-UMP] + GlnK-UMP:glutamine:UTUR] 
 
[governor]o = [governor] + [NR1-P:governor
 
[Enhancer(GlnK)]o = [Enhancer(GlnK)] + [NR1-P:Enhancer(GlnK)] 
 
[Enhancer(Nac)]o = [Enhancer(Nac)] + [NR1-P:Enhancer(Nac)] 
 
[UTUR:glutamine] = Kb[1][UTUR][glutamine] 
 
[UTUR:aKG] = Kb[2][UTUR][aKG] 
 
[P2:UMP:UTUR:aKG] = Kb[3][UTUR:aKG][P2][UMP] 
 
[UTUR:glutamine:P2-UMP] = Kb[4][UTUR:glutamine][P2-UMP] 
 
[P1:P2] = Kb[5][P1][P2] 
 
[P1:P2-UMP] = Kb[6][P1][P2-UMP] 
 
[GS:AMP:P1:P2] = Kb[7][P1:P2][GS][AMP] 
 
[GS-AMP:P1:P2-UMP] = Kb[8][P1:P2-UMP][GS-AMP] 
 
[NR2:P2] = Kb[9][NR2][P2] 
 
[NR2:ATP] = Kb[10][NR2][ATP] 
 
[NR2-P:P2:ADP] = Kb[11][P2][NR2-P][ADP] 
 
[NR1:NR2-P] = Kb[12][NR1][NR2-P] 
 
[NR1:acetyl] = Kb[13][NR1][acetyl] 
 
[NR1-P:NR2:P2] = Kb[14][NR2:P2][NR1-P] 
 
[NR1:Enhancer(glnAp1)] = Kb[15][NR1][Enhancer(glnAp1)] 
 
[NR1:Enhancer(glnLp1)] = Kb[16][NR1][Enhancer(glnLp1)] 
 
[NR1-P(4):Enhancer(glnAp2)] = Kb[17]4[NR1-P]4[Enhancer(glnAp2)] 
 
[GS-AMP:P1:GlnK-UMP] = Kb[18][GlnK-UMP:P1][GS-AMP] 
 
[GS:AMP:P1:GlnK] = Kb[19][GS][GlnK:P1][AMP] 
 
[NR2-P:ADP:GlnK] = Kb[20][NR2-P][GlnK][ADP] 
 
[NR1-P:NR2:GlnK] = Kb[21][NR1-P][NR2:GlnK] 
 
[GlnK:UMP:aKG:UTUR] = Kb[22][GlnK][UTUR:aKG][UMP] 
 
[GlnK-UMP:glutamine:UTUR] = Kb[23][GlnK-UMP][UTUR:glutamine] 
 
[GlnK-UMP:P1] = Kb[24][GlnK-UMP][P1] 
 
[GlnK:P1] = Kb[25][GlnK][P1] 
 
[NR1-P:Enhancer(GlnK)] = Kb[26][NR1-P][Enhancer(GlnK)] 
 
[NR1-P:governor] = Kb[27][NR1-P][governor
 
[NR2:GlnK] = Kb[28][GlnK][NR2] 
 
[NR1-P:Enhancer(Nac)] = Kb[29][Enhancer(Nac)][NR1-P] 
 
[aKG:P2] = Kb[30]3[aKG]3[P2] 
 
[aKG:P2-UMP:P1] = Kb[31][aKG:P2-UMP][P1] 
 
[GS-AMP:aKG:P2-UMP:P1] = Kb[32][aKG:P2-UMP:P1][GS-AMP] 
 
[aKG:P2-UMP:glutamine:UTUR] = Kb[33][aKG:P2-UMP][UTUR:glutamine] 
 
[aKG:P2:UMP:aKG:UTUR] = Kb[34][aKG:P2][UTUR:aKG][UMP] 
 
[aKG:P2-UMP] = Kb[35]3[aKG]3[P2-UMP] 
 
 
Reaction phase 
 
= + km[1]([NR-P(4) : Enhancer(glnAp2)]/[Enhancer(glnAp2)]o) · [DNA(GS)] 
   · (1.3 − [NR-P : governor]/[governor]o) + km[2](1.0 − ([NR1 : Enhancer(glnAp1)]/[Enhancer(glnAp1)]o)) 
   · [DNA(GS)] − kmd[2][mRNA(GS)] 
 
= + km[3](1.0 − ([NR1:Enhancer(glnLp1)]/[Enhancer(glnLp1)]o)) · [DNA(NR1)] 
   + km[4]([NR1-P(4) : Enhancer(glnAp2)]/[Enhancer(glnAp2)]o) 
   · [DNA(NR1)] · (1.3 − [NR-P : governor]/[governor]o) − kmd[2][mRNA(NR1)] 
 
= + km[5](1:0 − ([NR1 : Enhancer(glnLp1)]/[Enhancer(glnLp1)]o)) · [DNA(NR2)] 
   + km[6]([NR1-P(4) : Enhancer(glnAp2)]/[Enhancer(glnAp2)]o) · [DNA(NR2)] 
   · (1.3 − [NR-P : governor]/[governor]o) − kmd[3][mRNA(NR2)] 
 
= + km[7]([NR1-P : Enhancer(GlnK)]/[Enhancer(GlnK)]o)[DNA(GlnK)] − kmd[4][mRNA(GlnK)] 
 
= + km[8]([NR1-P : Enhancer(Nac)]/[Enhancer(Nac)]o)[DNA(nac)] − kmd[5][mRNA(nac)] 
 
= − kx[1][P2 : UMP : UTUR : aKG] + kx[2][UTUR : glutamine : P2-UMP] 
   − kx[3][aKG : P2 : UMP : aKG : UTUR] + kx[4][aKG : P2-UMP : glutamine : UTUR] 
 
= ;− kx[2][UTUR : glutamine : P2-UMP] + kx[1][P2 : UMP : UTUR : aKG] 
   + kx[3][aKG : P2 : UMP : aKG : UTUR] ;− kx[4][aKG : P2-UMP : glutamine : UTUR] 
 
= −kx[5][GS : AMP : P1 : P2] − kpd[1][GS : AMP : P1 : P2] + kx[6][GS-AMP : P1 : P2-UMP] 
   + kp[1][mRNA(GS)] − kpd[2][GS] + kx[7][GS-AMP : P1 : GlnK-UMP] − kx[8][GS : AMP : P1 : GlnK] 
   − kpd[3][GS : AMP : P1 : GlnK] + kx[9][GS-AMP : aKG : P2-UMP : P1] 
 
= −kx[6][GS-AMP : P1 : P2-UMP] − kpd[4][GS-AMP : P1 : P2-UMP] + kx[5][GS : AMP: P1 : P2] 
   + kx[8][GS : AMP : P1 : GlnK] − kx[7][GS-AMP : P1 : GlnK-UMP] − kpd[5][GS-AMP] 
   − kpd[6][GS-AMP : P1 : GlnK-UMP] − kpd[7][GS-AMP : aKG : P2-UMP : P1] 
   − kx[9][GS-AMP : aKG : P2-UMP : P1] 
 
= − kpd[8][NR2 : P2] − kx[10][NR2 : ATP] − kpd[9][NR2 : ATP] + kx[11][NR2-P : P2 : ADP] 
   + kx[12][NR1 : NR2-P] + kp[2][mRNA(NR2)] − kpd[10][NR2] − kpd[11][NR1-P : NR2 : P2] 
   + kx[13][NR2-P : ADP : GlnK] − kpd[12][NR2 : GlnK] + kpd[13][NR1-P : NR2 : GlnK] 
 
= − kx[11][NR2-P : P2 : ADP] − kpd[14][NR2-P : P2 : ADP] − kx[12][NR1 : NR2-P] 
   − kpd[15][NR1 : NR2-P] + kx[10][NR2 : ATP] − kpd[16][NR2-P] − kx[13][NR2-P : ADP : GlnK] 
   − kpd[17][NR2-P : ADP : GlnK] 
 
= − kx[14][NR1 : NR2-P] − kpd[18][NR1 : NR2-P] − kx[15][NR1 : acetyl] − kpd[19][NR1 : acetyl] 
   − kpd[20][NR1 : Enhancer(glnAp1)] − kpd[21][NR1 : Enhancer(glnLp1)] + kx[16][NR1-P : NR2 : P2] 
   + kp[3][mRNA(NR1)] − kpd[22][NR1] + kx[17][NR1-P : NR2 : GlnK] 
 
= − kx[16][NR1-P : NR2 : P2] − kpd[23][NR1-P : NR2 : P2] + kx[14][NR1 : NR2-P] + kx[15][NR1 : acetyl] 
   − kdp[24] · 4[NR1-P(4) : Enhancer(glnAp2)] − kpd[25][NR1-P] − kpd[26][NR1-P : governor
   − kpd[27][NR1-P : Enhancer(GlnK)] − kx[17][NR1-P : NR2 : GlnK] − kpd[28][NR1-P : NR2 : GlnK] 
   − kpd[29][NR1-P : Enhancer(Nac)] 
 
= − kp[4][mRNA(GlnK)] + kx[18][GlnK-UMP : glutamine : UTUR] − kpd[30][GlnK] − kpd[31][GlnK : P1] 
   − kpd[32][NR2 : GlnK] − kx[19][GlnK : UMP : aKG : UTUR] − kpd[33][GS : AMP : GlnK : P1] 
   − kpd[34][NR1-P : NR2 : GlnK] − kpd[35][GlnK : UMP : aKG : UTUR] − kpd[36][NR2-P : ADP : GlnK] 
 
= + kx[19][GlnK : UMP : aKG : UTUR] − kpd[37][GS-AMP : GlnK-UMP : P1] 
   − kpd[38][GlnK-UMP : P1] − kx[18][GlnK-UMP : glutamine : UTUR] − kpd[39][GlnK-UMP] 
   − kpd[40][GlnK-UMP : glutamine : UTUR] 
 
= + kp[5][mRNA(nac)] − kpd[41][nac] 
 
Metabolic reactions 
 
 
 
 
 
 
Table 3

Parameters for a full mathematical model of the E. coli nitrogen assimilation system. The symbols of “:” and “-” indicate binding complexes and modification, respectively. For example, the molecules of PI:PII, NRI-P:Enhancer, and NRI-P:NRII are binding complexes, whereas NRI-P is phosphorylated NRI.

(a) Components used in the model
Component
Definition
Concentration (nM)
UTUR UTase/UR 2.763105 × 10−6 
P2 PII 5.418544 × 10−1 
P2-UMP Uridylated PII 6.748318 × 10−1 
GlnK GlnK 7.964620 × 10 
GlnK-UMP Uridylated GlnK 4.724517 × 10 
GS Glutamine synthetase 1.535636 × 10−2 
GS-AMP Adenylylated GS 4.309176 × 10 
GOGAT Glutamate synthase 6.000000 × 10−2 
GDH Glutamine dehydrogenase 1.000000 × 10−2 
P1 PI, adenylyltransferase 3.427711 
NR1 NRI 4.687643 × 10−1 
NR1-P Phosphorylated NRI 6.554415 × 10 
NR2 NRII 1.214602 × 10−2 
NR2-P phosphorylated NRII 1.075513 × 10−2 
nac  3.447690 × 10−1 
Enhancer (glnAp1Enhancer 9.552226 × 10−1 
Enhancer (glnLp1Enhancer 9.552226 × 10−1 
Enhancer (glnAp2Enhancer 9.315546 × 10−1 
Enhancer (GlnK) Enhancer 1.502762 × 10−2 
Enhancer (Nac) Enhancer 9.552226 × 10−1 
AMP  1.00 × 10−6 
ADP  1.00 × 10−6 
ATP  1.00 × 10−6 
UMP  1.00 × 10−6 
mRNA(GS) mRNA of GS 3.447689 × 10−1 
mRNA(NR1) mRNA of NRI 2.811910 × 10−1 
mRNA(NR2) mRNA of NRII 2.811910 × 10−1 
mRNA(GlnK) mRNA of GlnK 2.462431 × 10−1 
mRNA(nac) mRNA of nac 1.537818 × 10−2 
DNA1(GS) Gene encoding GS 1.00 
DNA1(NR1) Gene encoding NRI 1.00 
DNA1(NR2) Gene encoding NRII 1.00 
DNA(GlnK) Gene encoding GlnK 1.00 
DNA(nac) Gene encoding of nac 1.00 
Ammonia  1.000000 × 10−5 
aKG 2-Ketoglutarate (2-KG) 3.486786 × 10−6 
Glutamine  3.983876 × 10−5 
Glutamate  3.065781 × 10−7 
UTUR:glutamine Glutamine-bound UTase/UR 1.100787 
UTUR:aKG 2-Ketoglutarate-bound UTase/UR 9.634354 × 10 
P2:UMP:UTUR:aKG PII:UMP:UTase/UR: 2-ketoglutarate 5.220417 × 10−1 
UTUR:glutamine:P2-UMP UTase/UR:glutamine:PII-UMP 7.428459 × 10−1 
P1:P2 PI:PII 1.857320 × 10 
P1:P2-UMP PI:PII-UMP 2.313128 × 10−4 
NR2:P2 NRII:PII 6.581371 × 10−2 
NR2:ATP NRII:ATP 1.214602 
NR2-P:P2:ADP NRII-P:PII:ADP 5.827713 × 10−1 
NR1:NR2-P NRI:NRII-P 5.041620 × 10−3 
NR1:acetyl NRI:acetylphosphate 4.687643 × 101 
NR1-P:NR2:P2 NRI-P:NRII:PII 4.313704 × 10 
NR1:Enhancer(glnAp1NRI:Enhancer(glnAp1) 4.477742 × 10−2 
NR1:Enhancer(glnLp1NRI:Enhancer(glnLp1) 4.477742 × 10−2 
NR1-P(4):Enhancer(glnAp2NRI-P(4):Enhancer(glnAp2) 6.844537 × 10−2 
GS:AMP:P1:P2 GS:AMP:PI:PII 2.852168 × 10−1 
GS-AMP:P1:P2-UMP GS-AMP:PI:PII-UMP 9.967678 × 10−7 
NR2:GlnK NRII:GlnK 4.848405 
GS-AMP:P1:GlnK-UMP GS-AMP:PI:GlnK-UMP 6.978399 × 10−2 
GS:AMP:P1:GlnK GS:AMP:PI:GlnK 4.192351 × 10−4 
NR2-P:ADP:GlnK NRII-P:ADP:GlnK 8.566050 × 10−2 
NR1-P:NR2:GlnK NRI-P:NRII:GlnK 3.177846 
GlnK:UMP:aKG:UTUR GlnK:UMP: 2-ketoglutarate:UTase/UR 7.673397 × 10−1 
GlnK-UMP:glutamine:UTUR GlnK-UMP:glutamine:UTase/UR 5.200685 × 10−1 
GlnK-UMP:P1 GlnK-UMP:PI 1.619428 
GlnK:P1 GlnK:PI 2.730041 
NR1-P:Enhancer(GlnK) NRI-P:Enhancer(GlnK) 9.849724 × 10−1 
NR1-P:Enhancer(Nac) NRI-P:Enhancer(Nac) 6.151237 × 10−2 
governor Governor of glnAp2 9.384876 × 10−1 
NR1-P:governor NRI-P:governor 6.151237 × 10−2 
aKG:P2 aKG:PII 2.296986 × 10 
aKG:P2-UMP:P1 2-Ketoglutarate:PII-UMP:PI 9.805630 × 10−3 
GS-AMP:aKG:P2-UMP:P1 GS-AMP: 2-Ketoglutarate:PII-UMP:PI 4.225419 × 10−5 
aKG:P2-UMP:glutamine:UTUR 2-Ketoglutarate:PII-UMP:glutamine: UTase/UR 3.149013 × 10−3 
aKG:P2:UMP:aKG:UTUR 2-Ketoglutarate:PII:UMP:aKG: UTase/UR 2.212998 × 10−4 
aKG:P2-UMP 2-Ketoglutarate:PII-UMP 2.860693 × 10 
(b) Biochemical parameters used in the model 
Parameter Definition Value Reference 
Kb[25] Association constant between GlnK and P1 107 M−1 Optimized 
Kb[26] Association constant between NR1-P and Enhancer(GlnK) 109 M−1 Optimized 
Kb[27] Association constant between NR1-P and governor 106 M−1 Optimized 
Kb[28] Association constant between NR2 and GlnK 109.7 M−1 Optimized 
Kb[29] Association constant between NR1-P and Enhancer(Nac) 106 M−1 Optimized 
Kb[30] Association constant between aKG and P2 103 M−3 Optimized 
Kb[31] Association constant between aKG:P2-UMP and P1 105 M−1 Optimized 
Kb[32] Association constant between GS-AMP and aKG:P2-UMP:P1 105 M−1 Optimized 
Kb[33] Association constant between aKG:P2-UMP and glutamine:UTUR 105 M−1 Optimized 
Kb[34] Association constant between aKG:P2, aKG:UTUR, and UMP 105 M−2 Optimized 
Kb[35] Association constant between aKG and P2-UMP 103 M−3 Optimized 
km[1] Transcription rate constant of mRNA(GS) enhanced by glnAp2 0.15 min−1 Assumed 
km[2–3] Transcription rate constant 0.03 min−1 Assumed 
km[4] Transcription rate constant of mRNA(NR1) enhanced by glnAp2 0.06 min−1 Assumed 
km[5] Transcription rate constant 0.03 min−1 Assumed 
km[6] Transcription rate constant of mRNA(NR2) enhanced by glnAp2 0.06 min−1 Assumed 
km[7–11] Transcription rate constant 0.03 min−1 Assumed 
kmd[1–5] mRNA degradation rate constant 0.12 min−1 Assumed 
kp[1–5] Translation rate constant 20.0 min−1 Assumed 
kpd[1–41] Protein degradation rate constant 0.035 min−1 [1] 
kx[1] Reaction rate constant 10.0 min−1 Assumed 
kx[2–19] Reaction rate constant 7.0 min−1 Assumed 
Q[1] Synthesis rate of the glutamine(GS) 3 × 10−2min−1 Assumed 
Q[2] Synthesis rate of the glutamate(GOGAT) 9 × 10−3min−1 Assumed 
Q[3] Synthesis rate of the glutamate(GDH) 1.8 × 10−1 min−1 Assumed 
Q[4] Outflow and inflow rate of the aKG(TCA) 1.056 × 10−1 min−1 Assumed 
kf [1] Outflow rate of the glutamine 2.82 × 104 min−1 [2] 
kf [2] Outflow rate of the glutamate 1.8 × 106 min−1 Assumed 
kf [3] Inflow rate of the aKG 2.184 × 104 min−1 [4] 
kr[1]  2.88 × 103 min−1 [2] 
K[1]  715.2 [3] 
ki[1]  3.6 × 10−3 min−1 Assumed 
Km[1] Michaelis constant of aKG 6.0 × 10−3[2] 
Km[2] Michaelis constant of NADPH 1.3 × 10−5[2] 
Km[3] Michaelis constant of ammonia 3.3 3.3 × 10−3[2] 
Km[4] Michaelis constant of glutamate 3.8 × 10−3[2] 
Km[5] Michaelis constant of NADP+ 6.1 × 10−6[2] Km[ 
Km[6] Michaelis constant of glutamate 3.7 × 10−3[3] 
Km[7] Michaelis constant of glutamine 2.5 × 10−3[3] 
Km[8] Michaelis constant of ATP 5.0 × 10−4[3] 
Km[9] Michaelis constant of ADP 4.4 × 10−5[3] 
Km[10] Michaelis constant of ammonia 6.0 × 10−5[3] 
Km[11] Michaelis constant of Pi 3.0 × 10−3[3] 
Km[12] Michaelis constant of glutamine 2.5 × 10−4[4] 
Km[13] Michaelis constant of aKG 7.3 × 10−6[4] 
Km[14] Michaelis constant of NADPH 7.7 × 10−6[4] 
Km[15] Michaelis constant of ammonia 3.6 × 10−5Assumed 
Km[16] Michaelis constant of glutamine 1.0 × 10−2Assumed 
(a) Components used in the model
Component
Definition
Concentration (nM)
UTUR UTase/UR 2.763105 × 10−6 
P2 PII 5.418544 × 10−1 
P2-UMP Uridylated PII 6.748318 × 10−1 
GlnK GlnK 7.964620 × 10 
GlnK-UMP Uridylated GlnK 4.724517 × 10 
GS Glutamine synthetase 1.535636 × 10−2 
GS-AMP Adenylylated GS 4.309176 × 10 
GOGAT Glutamate synthase 6.000000 × 10−2 
GDH Glutamine dehydrogenase 1.000000 × 10−2 
P1 PI, adenylyltransferase 3.427711 
NR1 NRI 4.687643 × 10−1 
NR1-P Phosphorylated NRI 6.554415 × 10 
NR2 NRII 1.214602 × 10−2 
NR2-P phosphorylated NRII 1.075513 × 10−2 
nac  3.447690 × 10−1 
Enhancer (glnAp1Enhancer 9.552226 × 10−1 
Enhancer (glnLp1Enhancer 9.552226 × 10−1 
Enhancer (glnAp2Enhancer 9.315546 × 10−1 
Enhancer (GlnK) Enhancer 1.502762 × 10−2 
Enhancer (Nac) Enhancer 9.552226 × 10−1 
AMP  1.00 × 10−6 
ADP  1.00 × 10−6 
ATP  1.00 × 10−6 
UMP  1.00 × 10−6 
mRNA(GS) mRNA of GS 3.447689 × 10−1 
mRNA(NR1) mRNA of NRI 2.811910 × 10−1 
mRNA(NR2) mRNA of NRII 2.811910 × 10−1 
mRNA(GlnK) mRNA of GlnK 2.462431 × 10−1 
mRNA(nac) mRNA of nac 1.537818 × 10−2 
DNA1(GS) Gene encoding GS 1.00 
DNA1(NR1) Gene encoding NRI 1.00 
DNA1(NR2) Gene encoding NRII 1.00 
DNA(GlnK) Gene encoding GlnK 1.00 
DNA(nac) Gene encoding of nac 1.00 
Ammonia  1.000000 × 10−5 
aKG 2-Ketoglutarate (2-KG) 3.486786 × 10−6 
Glutamine  3.983876 × 10−5 
Glutamate  3.065781 × 10−7 
UTUR:glutamine Glutamine-bound UTase/UR 1.100787 
UTUR:aKG 2-Ketoglutarate-bound UTase/UR 9.634354 × 10 
P2:UMP:UTUR:aKG PII:UMP:UTase/UR: 2-ketoglutarate 5.220417 × 10−1 
UTUR:glutamine:P2-UMP UTase/UR:glutamine:PII-UMP 7.428459 × 10−1 
P1:P2 PI:PII 1.857320 × 10 
P1:P2-UMP PI:PII-UMP 2.313128 × 10−4 
NR2:P2 NRII:PII 6.581371 × 10−2 
NR2:ATP NRII:ATP 1.214602 
NR2-P:P2:ADP NRII-P:PII:ADP 5.827713 × 10−1 
NR1:NR2-P NRI:NRII-P 5.041620 × 10−3 
NR1:acetyl NRI:acetylphosphate 4.687643 × 101 
NR1-P:NR2:P2 NRI-P:NRII:PII 4.313704 × 10 
NR1:Enhancer(glnAp1NRI:Enhancer(glnAp1) 4.477742 × 10−2 
NR1:Enhancer(glnLp1NRI:Enhancer(glnLp1) 4.477742 × 10−2 
NR1-P(4):Enhancer(glnAp2NRI-P(4):Enhancer(glnAp2) 6.844537 × 10−2 
GS:AMP:P1:P2 GS:AMP:PI:PII 2.852168 × 10−1 
GS-AMP:P1:P2-UMP GS-AMP:PI:PII-UMP 9.967678 × 10−7 
NR2:GlnK NRII:GlnK 4.848405 
GS-AMP:P1:GlnK-UMP GS-AMP:PI:GlnK-UMP 6.978399 × 10−2 
GS:AMP:P1:GlnK GS:AMP:PI:GlnK 4.192351 × 10−4 
NR2-P:ADP:GlnK NRII-P:ADP:GlnK 8.566050 × 10−2 
NR1-P:NR2:GlnK NRI-P:NRII:GlnK 3.177846 
GlnK:UMP:aKG:UTUR GlnK:UMP: 2-ketoglutarate:UTase/UR 7.673397 × 10−1 
GlnK-UMP:glutamine:UTUR GlnK-UMP:glutamine:UTase/UR 5.200685 × 10−1 
GlnK-UMP:P1 GlnK-UMP:PI 1.619428 
GlnK:P1 GlnK:PI 2.730041 
NR1-P:Enhancer(GlnK) NRI-P:Enhancer(GlnK) 9.849724 × 10−1 
NR1-P:Enhancer(Nac) NRI-P:Enhancer(Nac) 6.151237 × 10−2 
governor Governor of glnAp2 9.384876 × 10−1 
NR1-P:governor NRI-P:governor 6.151237 × 10−2 
aKG:P2 aKG:PII 2.296986 × 10 
aKG:P2-UMP:P1 2-Ketoglutarate:PII-UMP:PI 9.805630 × 10−3 
GS-AMP:aKG:P2-UMP:P1 GS-AMP: 2-Ketoglutarate:PII-UMP:PI 4.225419 × 10−5 
aKG:P2-UMP:glutamine:UTUR 2-Ketoglutarate:PII-UMP:glutamine: UTase/UR 3.149013 × 10−3 
aKG:P2:UMP:aKG:UTUR 2-Ketoglutarate:PII:UMP:aKG: UTase/UR 2.212998 × 10−4 
aKG:P2-UMP 2-Ketoglutarate:PII-UMP 2.860693 × 10 
(b) Biochemical parameters used in the model 
Parameter Definition Value Reference 
Kb[25] Association constant between GlnK and P1 107 M−1 Optimized 
Kb[26] Association constant between NR1-P and Enhancer(GlnK) 109 M−1 Optimized 
Kb[27] Association constant between NR1-P and governor 106 M−1 Optimized 
Kb[28] Association constant between NR2 and GlnK 109.7 M−1 Optimized 
Kb[29] Association constant between NR1-P and Enhancer(Nac) 106 M−1 Optimized 
Kb[30] Association constant between aKG and P2 103 M−3 Optimized 
Kb[31] Association constant between aKG:P2-UMP and P1 105 M−1 Optimized 
Kb[32] Association constant between GS-AMP and aKG:P2-UMP:P1 105 M−1 Optimized 
Kb[33] Association constant between aKG:P2-UMP and glutamine:UTUR 105 M−1 Optimized 
Kb[34] Association constant between aKG:P2, aKG:UTUR, and UMP 105 M−2 Optimized 
Kb[35] Association constant between aKG and P2-UMP 103 M−3 Optimized 
km[1] Transcription rate constant of mRNA(GS) enhanced by glnAp2 0.15 min−1 Assumed 
km[2–3] Transcription rate constant 0.03 min−1 Assumed 
km[4] Transcription rate constant of mRNA(NR1) enhanced by glnAp2 0.06 min−1 Assumed 
km[5] Transcription rate constant 0.03 min−1 Assumed 
km[6] Transcription rate constant of mRNA(NR2) enhanced by glnAp2 0.06 min−1 Assumed 
km[7–11] Transcription rate constant 0.03 min−1 Assumed 
kmd[1–5] mRNA degradation rate constant 0.12 min−1 Assumed 
kp[1–5] Translation rate constant 20.0 min−1 Assumed 
kpd[1–41] Protein degradation rate constant 0.035 min−1 [1] 
kx[1] Reaction rate constant 10.0 min−1 Assumed 
kx[2–19] Reaction rate constant 7.0 min−1 Assumed 
Q[1] Synthesis rate of the glutamine(GS) 3 × 10−2min−1 Assumed 
Q[2] Synthesis rate of the glutamate(GOGAT) 9 × 10−3min−1 Assumed 
Q[3] Synthesis rate of the glutamate(GDH) 1.8 × 10−1 min−1 Assumed 
Q[4] Outflow and inflow rate of the aKG(TCA) 1.056 × 10−1 min−1 Assumed 
kf [1] Outflow rate of the glutamine 2.82 × 104 min−1 [2] 
kf [2] Outflow rate of the glutamate 1.8 × 106 min−1 Assumed 
kf [3] Inflow rate of the aKG 2.184 × 104 min−1 [4] 
kr[1]  2.88 × 103 min−1 [2] 
K[1]  715.2 [3] 
ki[1]  3.6 × 10−3 min−1 Assumed 
Km[1] Michaelis constant of aKG 6.0 × 10−3[2] 
Km[2] Michaelis constant of NADPH 1.3 × 10−5[2] 
Km[3] Michaelis constant of ammonia 3.3 3.3 × 10−3[2] 
Km[4] Michaelis constant of glutamate 3.8 × 10−3[2] 
Km[5] Michaelis constant of NADP+ 6.1 × 10−6[2] Km[ 
Km[6] Michaelis constant of glutamate 3.7 × 10−3[3] 
Km[7] Michaelis constant of glutamine 2.5 × 10−3[3] 
Km[8] Michaelis constant of ATP 5.0 × 10−4[3] 
Km[9] Michaelis constant of ADP 4.4 × 10−5[3] 
Km[10] Michaelis constant of ammonia 6.0 × 10−5[3] 
Km[11] Michaelis constant of Pi 3.0 × 10−3[3] 
Km[12] Michaelis constant of glutamine 2.5 × 10−4[4] 
Km[13] Michaelis constant of aKG 7.3 × 10−6[4] 
Km[14] Michaelis constant of NADPH 7.7 × 10−6[4] 
Km[15] Michaelis constant of ammonia 3.6 × 10−5Assumed 
Km[16] Michaelis constant of glutamine 1.0 × 10−2Assumed 

∗Kb[1–35] are the parameters optimized by genetic algorithms. The values of the other parameters are estimated based on the vast literature or on biological validity. See Section 2 in the text.

[1] Bremer, H., & Dennis, P. P. (1996). Modulation of chemical composition and other parameters of the cell by growth rate. In F. C. Neidhardt (Ed.), Escherichia coli and salmonella: Cellular and molecular biology (pp. 1553–1569). Washington: ACM Press. [2] McPherson, M. J., Baron, A. J., Jones, K. M., Price, G. J., & Wootton, J. C. (1988). Multiple interactions of lysine-128 of Escherichia coli glutamate dehydrogenase revealed by site-directed mutagenesis studies. Protein Engineering, 2(2), 147–152. [3] Meek, T. D., & Villafranca, J. J. (1980). Kinetic mechanism of Escherichia coli glutamine synthetase. Biochemistry, 19(24), 5513–5519. [4] Miller, R. E., & Stadtman, E. R. (1972). Glutamate synthase from Escherichia coli. An iron-sulfide flavoprotein. Journal of Biological Chemistry, 247(22), 7407–7419.

Table 4. 

Flux module identification in the E. coli ammonia assimilation system.

No.
Flux module
1GS activity control
1.1  PII symmetric bidirectional negative feedback 
1.2  GlnK symmetric bidirectional negative feedback 
1.3  One-directional 2-KG negative feedback 
GS synthesis control 
2.1  PII-(glnAp, glnLp) negative feedback 
2.2  GlnK-(glnAp, glnLp) negative feedback 
2.3  PII-glnAp2 negative feedback 
2.4  GlnK-glnAp2 negative feedback 
2.5  NRII-NRI-glnAp2 positive feedback 
2.6  PII-governor positive feedback 
2.7  GlnK-governor positive feedback 
GS synthesis feedforward 
No.
Flux module
1GS activity control
1.1  PII symmetric bidirectional negative feedback 
1.2  GlnK symmetric bidirectional negative feedback 
1.3  One-directional 2-KG negative feedback 
GS synthesis control 
2.1  PII-(glnAp, glnLp) negative feedback 
2.2  GlnK-(glnAp, glnLp) negative feedback 
2.3  PII-glnAp2 negative feedback 
2.4  GlnK-glnAp2 negative feedback 
2.5  NRII-NRI-glnAp2 positive feedback 
2.6  PII-governor positive feedback 
2.7  GlnK-governor positive feedback 
GS synthesis feedforward 

∗The numbers corresponds to the flux modules as shown in Figure 10.

Figure 10. 

(a) Flux modules in the E. coli ammonia assimilation system. The numbers correspond to the flux modules as shown in Table 4. 1.1: The PII symmetric bidirectional negative feedback module. The signal of the N/C ratio is transmitted to control the GS activity through UTase/UR (UTUR), PII, and PI. 1.2: The GlnK symmetric bidirectional negative feedback module. The signal of the N/C ratio is transmitted to control the GS activity through UTUR, GlnK, and PI. 1.3: The one-directional 2-KG negative feedback module. This controls the GS activity by binding of 2-KG to PII. (b) 2.1: The PII-(glnAp1, glnLp) negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by non-phosphorylated NRI through PII, NRII, and NRI. 2.2: The GlnK-(glnAp1, glnLp) negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by non-phosphorylated NRI through GlnK, NRII, and NRI. 2.3: The PII-glnAp2 negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by phosphorylated NRI (NRI-P) through PII, NRII, and NRI. (c) 2.4: The GlnK-glnAp2 negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by phosphorylated NRI (NRI-P) through GlnK, NRII, and NRI. 2.5: The NRII-NRI-glnAp2 positive feedback module. This positively increases the synthesis of NRII and NRI by phosphorylated NRI (NRI-P) binding to the glnAp2. 2.6: The PII-governor positive feedback module. This suppresses the synthesis of GS, NRI, and NRII by NRI-P binding to the governor of the glnAp2. (d) 2.7: The GlnK-governor positive feedback module. This suppresses the synthesis of GS, NRI, and NRII by NRI-P binding to the governor of the glnAp2. 3: The GS synthesis feedforward module. Acetylphosphate activates NRI to produce GS.

Figure 10. 

(a) Flux modules in the E. coli ammonia assimilation system. The numbers correspond to the flux modules as shown in Table 4. 1.1: The PII symmetric bidirectional negative feedback module. The signal of the N/C ratio is transmitted to control the GS activity through UTase/UR (UTUR), PII, and PI. 1.2: The GlnK symmetric bidirectional negative feedback module. The signal of the N/C ratio is transmitted to control the GS activity through UTUR, GlnK, and PI. 1.3: The one-directional 2-KG negative feedback module. This controls the GS activity by binding of 2-KG to PII. (b) 2.1: The PII-(glnAp1, glnLp) negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by non-phosphorylated NRI through PII, NRII, and NRI. 2.2: The GlnK-(glnAp1, glnLp) negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by non-phosphorylated NRI through GlnK, NRII, and NRI. 2.3: The PII-glnAp2 negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by phosphorylated NRI (NRI-P) through PII, NRII, and NRI. (c) 2.4: The GlnK-glnAp2 negative feedback module. The N/C ratio is sensed by UTase/UR to control the synthesis of GS by phosphorylated NRI (NRI-P) through GlnK, NRII, and NRI. 2.5: The NRII-NRI-glnAp2 positive feedback module. This positively increases the synthesis of NRII and NRI by phosphorylated NRI (NRI-P) binding to the glnAp2. 2.6: The PII-governor positive feedback module. This suppresses the synthesis of GS, NRI, and NRII by NRI-P binding to the governor of the glnAp2. (d) 2.7: The GlnK-governor positive feedback module. This suppresses the synthesis of GS, NRI, and NRII by NRI-P binding to the governor of the glnAp2. 3: The GS synthesis feedforward module. Acetylphosphate activates NRI to produce GS.

Appendix 2: Instructions for Use of the CADLIVE Dynamic Simulator

1. Log in to the CADLIVE dynamic simulator (http://kurata23.bio.kyutech.ac.jp/Life/index.html or http://www.cadlive.jp). LOGINID: cadlive; PASSWORD: simulator.

2. Click [Simulator].

3. Click [Math model data].

4. Select the title: DEMO (ammonia assimilation system in Escherichia coli).

5. Click [Go simulator].

6. Select [Go to [Select Analysis Type]]. Do not select [Go to [Edit Model Data]]. This requires input of all the parameter data.

7. Click [Submit].

8. Input parameters to be displayed and then click [Confirm].

9. Click [Submit] and then click [Confirm].

10. Click [Submit] for simulation.

11. Click [Graph] for displaying the simulated time courses.

12. Input the molecule index to be displayed and then click [Submit].

13. Click [Re-calc] and [Draw graph] after setting the parameters.

14. Click [Close] to finalize the display.

To finalize the simulator, click [Close] or [Quit].

Author notes

Corresponding author.

∗∗

Department of Bioscience and Bioinformatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan. E-mail: kurata@bio.kyutech.ac.jp (H.K.); k_masaki@lake.ocn.ne.jp (K.M.); h791008k@bio.kyutech.ac.jp (K.M.)