Quantitative Coarse-Grained Modeling of Microbial Metabolism: Uncovering Growth Laws, Environmental Responses, and Host-Circuit Couplings

Abstract

Microbial cell metabolism is highly complex, involving a vast number of molecular species and multi-level regulatory processes. To address this complexity and decode cellular behavior, coarse-grained modeling provides a powerful quantitative framework by grouping similar biological entities while preserving essential network properties. This approach offers insights into intracellular regulation and resource allocation, enables systematic analysis of natural biological systems, and generates predictions for synthetic engineering outcomes. In this chapter, we present a comprehensive overview of coarse-grained modeling, from model construction to applications, and demonstrate its utility in uncovering microbial metabolic behavior.

1. Introduction

Microorganisms are widely adopted as synthetic platforms for metabolic engineering and have been extensively studied for their capacity to produce valuable compounds. This synthetic capacity fundamentally relies on how cells allocate their limited resources, particularly proteins, which can account for more than half of the cellular dry biomass—up to ~55% in E. coli [1]. Proteins are the primary functional components that sustain cellular activities: enzymes drive the catabolism and anabolism of essential nutrients, while RNA polymerases and ribosomes enable the synthesis of these enzymes. Therefore, understanding how cells allocate protein resources across different functions is critical for explaining how they adapt and thrive under diverse growth conditions.

To understand how protein allocation is controlled, it is necessary to examine the underlying layers of cellular regulation. However, the cellular metabolism is inherently complex, involving multiple regulatory mechanisms such as allosteric control, signaling transduction, transcriptional regulation, and protein translation (Figure 1). These layers do not act independently; instead, they communicate and coordinate to provide the cell with optimal strategies in response to environmental cues. As a result, experimental approaches alone often struggle to disentangle these overlapping layers, making mathematical modeling an essential tool for generating quantitative insights into how regulation shapes cellular behavior.

Figure 1. System-level overview of multi-level regulation coordination.

Cellular metabolic responses are coordinated through three interconnected layers: the metabolic layer, the signaling regulatory layer, and the gene and protein expression layer. This coordination begins with the metabolic layer, which senses changes in fluxes. These changes trigger signaling pathways that regulate transcription factors and subsequent gene expression. The resulting gene and protein expression determines the cellular protein and enzyme distribution, which redirect the metabolic fluxes.

Here, we present a coarse-grained mathematical approach to capture the essence of different regulatory layers and their interconnections, which together shape intracellular dynamics. This method follows a top-down analysis, where essential behaviors of metabolic networks and regulatory systems are represented by selecting key metabolites, proteins, and regulatory factors. The term “coarse-grained” originates from grouping proteins into functional categories, providing an intuitive view of how cells allocate resources [2,3]. This principle can be extended beyond proteins to other cellular components, such as metabolites and regulatory factors [4]. By simplifying complex cellular networks, coarse-grained modeling enables systematic analysis of how different regulatory layers coordinate to drive cellular dynamics and adaptation.

In the following sections, we provide detailed instructions on the construction and application of coarse-grained models. Section 2 describes the step-by-step design of the modeling framework, including the selection of representative sectors and units, the organization of reactions, and the derivation of kinetic equations for the essential network. Section 3 illustrates how this framework can be applied to study cellular resource allocation under different contexts, such as changes in growth rate, nutrient composition, environmental conditions, and the expression of synthetic gene constructs. The presented modeling examples demonstrate how coarse-grained modeling serves as a useful platform for quantitatively understanding and systematically exploring cellular resource investment strategies across diverse conditions. By revealing how cells manage internal resources and adapt to environmental pressures, the coarse-grained modeling framework provides a systems-level perspective on resource allocation and metabolic flux distribution. Such insights are valuable for guiding strategies to control cellular metabolism, improve the robustness of engineered systems, and accelerate progress in metabolic engineering and synthetic biology.

2. Methods

In this section, we provide step-by-step instructions for constructing a coarse-grained kinetic modeling framework that captures the essential regulatory coordination within the cellular system (Figure 2). The framework begins with the description of single-cell behavior. We first demonstrate how the complex whole-cell regulatory network can be simplified into a core essential network. Next, we illustrate how different layers of regulation can be organized into distinct modules. Then, we present one module as an example to demonstrate how a series of network reactions can be translated into a set of ordinary differential equations that describe molecular dynamics over time. We subsequently explain how these modules can be integrated to represent the dynamic response of a single cell. Finally, we extend the framework from single-cell to homogeneous population dynamics, thereby enabling direct comparison with bulk experimental measurements.

Figure 2. Schematic diagram of coarse-grained model construction.

The framework is developed through five main steps: (i) selecting coarse-grained representative units for metabolites, proteins, and regulations to construct an essential network of the whole system; (ii) separating different layers of dynamics and regulation into distinct modules, where the output of one module serves as the input for another; (iii) listing all biochemical reactions within each module and compiling them into reaction tables; (iv) formulating rate equations to describe molecular changes, applying the protein conservation law, and deriving kinetic equations; and (v) combining all modules into integrated equation sets to simulate overall system behavior under homogeneous population dynamics. The resulting simulations can then be directly compared with bulk experimental measurements.

2.1. Step 1: Essential Network Construction

The main idea of coarse graining is to group components into broader units, which reduces modeling complexity while preserving the system behaviors of interest. Therefore, before simplification, we should always ask: What specific question are we addressing, and how detailed are the measurements available to us? Incorporating unmeasured metabolites or overly fine detail can overcomplicate the framework and hinder parameter estimation. Since this is a top-down approach, the first step is to decide the lowest level of detail we want the model to represent. The guiding principle is to keep the framework as simple as possible.

In the following, we illustrate how to construct a minimal essential network that captures our interest in understanding central carbon metabolism in baker’s yeast (S. cerevisiae). Despite its simplicity, the network represents the major fluxes from glucose uptake to biomass formation, bioproduct synthesis, and energy generation. To capture these fluxes, we first identify the key components of the network, with changes in their abundance enabling quantification of flux distributions. We then define relative proteome sectors that determine reaction activities. Finally, we incorporate regulators and signaling factors that govern flux sensing and the expression of enzymes and proteins. Together, this streamlined design enables the representation of core carbon flux–related reactions and their regulatory mechanisms.

2.1.1. Representative metabolites

In this step, we select representative units that most effectively capture the flux distribution of interest. Because the focus is on how carbon influx is partitioned into energy, biomass, and bioproducts, six core metabolites are identified. Glucose represents the primary carbon source entering the system. Precursors mark the key intermediate branching points where carbon is directed toward bioproduct, energy, or biomass formation (see Note 1). ATP functions as the universal energy currency, driving diverse reactions and supporting cellular growth. Amino acids serve as the building blocks required for biomass synthesis (see Note 2). Bioproducts, such as ethanol, represent carbon sinks that accumulate as metabolites not directly utilized by the cell. Finally, proteins reflect the accumulation of cellular biomass (see Note 3). To ensure that these representatives are accurately defined, it is useful to consult curated metabolic maps such as KEGG (https://www.genome.jp/kegg/pathway.html) to verify stoichiometry and identify essential pathways.

2.1.2. Representative proteome sectors

Once the main components of the network are defined, the next step is to assign representative reactions that govern the conversion between them. Note that the rate of each reaction depends not only on substrate and product concentrations but also on enzyme availability. For this reason, it is important to quantify both metabolite levels and enzyme concentrations. To account for how enzyme’s availability changes the rate of representative reactions, proteins are organized into representative sectors that regulate these processes.

It is reported that proteins belonging to the same functional pathways often display similar dynamic patterns [5]. Because yeast typically expresses around 6,000 proteins, grouping them into functional sectors is critical for reducing complexity. Gene Ontology (GO) annotations provide a systematic way to classify proteins according to their functions. The Gene Ontology database (https://current.geneontology.org/products/pages/downloads.html) offers annotation files (.gaf) for commonly studied organisms. By combining these annotations with GO Slim term mappings (see Note 4), proteins can be grouped into representative sectors that capture their functional roles. Visualization tools such as Proteomaps (https://www.proteomaps.net/) further help illustrate this grouping process [2].

In the illustrated example, six representative reactions are considered: glucose to precursor, glucose to ATP, precursor to product, product to precursor, precursor to amino acids, and amino acids to proteins. Accordingly, proteins are categorized into six functional sectors, each governing one of these reactions, plus an additional sector that encompasses proteins responsible for other essential cellular functions (Z sector in the illustration). Using GO terms, protein functions associated with the representative reactions can be grouped into these sectors. Within the six functional sectors, we further distinguish between enzyme-based proteins (E sector) and ribosomal-based proteins (R sector). The E sector supports carbon metabolism and the synthesis of essential building blocks, while the R sector governs transcriptional and translational processes, enabling protein production and contributing to cellular growth.

To summarize, this step shows that proteins can be grouped into representative sectors that govern specific reactions in the essential network, while further classification into R, E, and Z sectors provides a higher-level understanding of their functional roles.

2.1.3. Representative regulators

Moving forward, this step is to understand how metabolite signals influence the regulation of gene expression. To achieve this, we examine the major regulatory pathways and signaling networks. The BioCyc database (https://biocyc.org/) provides curated regulatory networks for many common organisms, including transcription factors and their regulons. For instance, the E. coli regulatory overview maps are available at https://ecocyc.org/overviewsWeb/regOv.shtml?orgid=ECOLI. Nevertheless, literature searches are often necessary to capture the full spectrum of metabolic signals and the pathways they regulate.

After constructing a comprehensive map of the regulatory and signaling network, we identify the key regulators that respond to the representative metabolites. Using the baker’s yeast example, cAMP–PKA and SNF1 are the primary regulators of glucose availability [6], while amino acid concentrations influence TORC1 signaling [7,8], which plays a central role in biomass formation. We then map how these signaling factors exert control, either through direct regulation of enzyme activity [9] or through transcriptional [10] and translational control [10–12] of specific functional proteome sectors. Furthermore, cAMP–PKA and SNF1 can be grouped together, as they often produce opposing regulatory effects on the same representative pathways [13].

Altogether, this mapping provides a framework for understanding how representative metabolites trigger regulatory signals, which in turn modulate protein activity and abundance. At this stage, we have constructed a coarse-grained essential network that captures the topology of the original complex whole-cell system.

2.2. Step 2: Module Design and Integration

From the construction of the essential network, it becomes clear that cellular responses can be viewed across three layers: changes in metabolic flux, the buildup of signaling regulators, and the resulting alterations in protein abundance and activity. These layers can then be separated into individual modules for a clearer understanding of their interactions.

Each module represents a certain level of regulation and dynamics. The metabolism module captures the biochemical reactions and how changes in reaction rates alter the concentrations of representative metabolites. The signaling regulatory module responds to metabolite signals, where the active fraction of signaling molecules builds up [13,14]. The protein expression module reflects the downstream outcomes of signaling, altering gene expression and thereby reallocating proteome sectors. In addition, signaling molecules can directly modify protein activity through post-translational regulation and, in turn, affects metabolic reaction rates.

This modular design provides a systematic view of the cellular interconnections. Each module represents a specific level of regulation, with the output of one module serving as the input to the next. Taken together, the modules capture how the entire system responds to environmental and nutritional signals and adjusts cellular fluxes accordingly.

2.3. Step 3: Biochemical Network Reactions

Once the system has been divided into modules, the next step is to specify the reactions within each module. These reactions describe how molecular concentrations change over time and offer the fundamentals for establishing the corresponding rate equations. In practice, this certain step consists of two main tasks: identifying the reactions and organizing them into a reaction table. Because reaction specification and rate equation formulation are closely connected, they are always performed together to construct a set of ODE-based equations for each module. Repeating these two-steps process across all modules yields a complete set of ODEs for the entire cellular system.

Here, we provide an example of the protein expression module to illustrate how a set of mechanistic reactions (Step 3) can be transformed into ODEs (Step 4) to describe molecular dynamics within the module. Specifically, we demonstrate how the tRNA charging process and the activity of signaling molecules influence ribosomal protein partitioning and, in turn, shape the translational outcomes of different proteome sectors.

2.3.1. List of reactions

The process of protein translation is driven by a few key reactions. The first is tRNA charging, which determines the supply of charged tRNAs available for translation. The second concerns the role of charged and uncharged tRNAs during elongation: charged tRNAs support peptide chain elongation, whereas uncharged tRNAs can bind to ribosomes and cause translation stalling. These reactions are closely coupled to ribosome state transitions and directly influence the translational output of mRNAs. In the following, we outline the reactions underlying tRNA charging and ribosome partitioning in translation.

The tRNA aminoacylation process requires amino acids, ATP, and uncharged tRNA. Catalyzed by aminoacyl-tRNA synthetases, the reaction proceeds in two steps: first, the amino acid is activated by ATP to form an aminoacyl-adenylate (AA-AMP) intermediate; second, the activated amino acid is transferred to the corresponding tRNA, producing charged tRNA (aminoacyl-tRNA) and releasing AMP. This process can be described as a sequential series of four reactions.

• 1
  Amino acid and ATP binding/unbinding to the synthetase

  $$E+Aa+ATP\underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons }}E\cdot Aa\cdot ATP$$
• 2
  Formation of aminoacyl adenylate (AA–AMP) and release of pyrophosphate

  $$E\cdot Aa\cdot ATP\stackrel{k_2}{\to }E\cdot Aa\cdot AMP+PP$$
• 3
  tRNA binding/unbinding to the synthetase

  $$E\cdot Aa\cdot AMP+t_u\underset{k_{-3}}{\stackrel{k_3}{\rightleftharpoons }}E\cdot Aa\cdot AMP\cdot t_u$$
• 4
  Transfer of the amino acid to the tRNA molecule, releasing charged tRNA and AMP

  $$E\cdot Aa\cdot AMP\cdot t_u\stackrel{k_4}{\to }E+t_c+AMP$$

Here, $E$ denotes the aminoacyl-tRNA synthetase, $Aa$ the amino acid, and $t_u$ and $t_c$ the uncharged and charged tRNAs, respectively.

Next, we consider ribosome state transitions, which encompass translation initiation, elongation, and stalling. Translation initiation within a given proteome sector depends on both the concentration of the corresponding mRNA and the translational activity of ribosomes, both of which are regulated by TORC1 and Snf1. Ribosome states are further influenced by the binding of charged or uncharged tRNAs. Binding of a charged tRNA allows the ribosome to elongate the peptide chain; once elongation is completed, the ribosome is ready to bind another tRNA molecule. In contrast, binding of an uncharged tRNA can lead to ribosome stalling, thereby interrupting translation. Further details are listed in the following reactions.

• 5
  Ribosome binding/unbinding to the mRNA, preparing the ribosome for tRNA recruitment

  $$R_f+M_j\underset{k_{-5}\left(s^{*}\right)}{\stackrel{k_5\left({\tau }^{*}\right)}{\rightleftharpoons }}{R}_{o,j}$$
• 6
  Charged tRNA binding/unbinding to the ribosome to enable peptide chain elongation

  $$R_{o,j}+t_c\underset{k_{-6}}{\stackrel{k_6}{\rightleftharpoons }}{R}_{e,j}$$
• 7
  Peptide elongation followed by the release of uncharged tRNA

  $$R_{e,j}+(\mathit{\text{peptide}})\stackrel{k_7(\mathit{\text{ATP}})}{\to }{R}_{o,j}+t_u+\left({\mathit{\text{peptide}}}^{+1}\right)$$
• 8
  Uncharged tRNA binding/unbinding to the ribosome, which leads to the ribosome stalling

  $$R_{o,j}+t_u\stackrel{k_8}{\underset{k_{-8}}{\rightleftharpoons }}{R}_{s,j}$$

Where $R_f$ stands for the free ribosome, while $R_{o,j},R_{e,j},R_{s,j}$ represent the empty, elongating, and stalled ribosomes, respectively, associated with mRNA of protein sector type $j.M_j$ refers to the mRNA corresponding to protein sector type $j$. The symbol ($·$) indicates functional dependence; ${\tau }^{*}$ and $s^{*}$ stands for the active states of the signaling transduction molecule TORC1 and Snf1, respectively.

2.3.2. Reactions table

Based on the reactions outlined above, we can now organize them into reaction tables to systematically track the consumption and formation of each molecule. This representation will facilitate the subsequent analysis of molecular changes. In the tables that follow, each reaction is listed alongside its corresponding molecular changes and reaction rate. For simplicity, we assume that each molecule contributes in a first-order relationship to the reaction rate. The tRNA charging reactions (1–4) are summarized in Table 1, while the ribosome partitioning reactions (5–8) are presented in Table 2.

Table 1.

tRNA charging reactions and rates

Reaction	Molecule										Rate
	$E$	$Aa$	$ATP$	$E·Aa·ATP$	$E·Aa·AMP$	$PP$	$E·Aa·AMP·t_u$	$t_u$	$t_c$	$AMP$
$E+Aa+ATP\stackrel{k_1}{\to }E·Aa·ATP$	−1	−1	−1	1							$k_1\left[E\right]\left[Aa\right]\left[ATP\right]$
$E·Aa·ATP\stackrel{k_{-1}}{\to }E+Aa+ATP$	1	1	1	−1							$k_{-1}[E·Aa·ATP]$
$E·Aa·ATP\stackrel{k_2}{\to }E·Aa·AMP+PP$				−1	1	1					$k_2[E·Aa·ATP]$
$E·Aa·AMP+t_u\stackrel{k_3}{\to }E·Aa·AMP·t_u$					−1		1	−1			$k_3\left[E·Aa·AMP\right]\left[t_u\right]$
$E·Aa·AMP·t_u\stackrel{k_{-3}}{\to }E·Aa·AMP+t_u$					1		−1	1			$k_{-3}\left[E·Aa·AMP\cdot t_u\right]$
$E·Aa·AMP·t_u\stackrel{k_4}{\to }E+t_c+AMP$	1						−1		1	1	$k_4\left[E·Aa·AMP\cdot t_u\right]$

Table 2.

Ribosome partitioning reactions and rates

Reaction	Molecule							Rate
	$R_f$	$M_j$	$R_{o,j}$	$R_{e,j}$	$t_c$	$t_u$	$R_{s,j}$
$R_f+M_j\stackrel{k_5\left({\tau }^{*}\right)}{\to }{R}_{o,j}$	−1	−1	1					$k_5\left({\tau }^{*}\right)\left[R_f\right]\left[M_j\right]$
$R_{o,j}\stackrel{k_{-5}\left(s^{*}\right)}{\to }{R}_f+M_j$	1	1	−1					$k_{-5}\left(s^{*}\right)\left[R_{o,j}\right]$
$R_{o,j}+t_c\stackrel{k_6}{\to }{R}_{e,j}$			−1	1	−1			$k_6\left[R_{o,j}\right]\left[t_c\right]$
$R_{e,j}\stackrel{k_{-6}}{\to }{R}_{o,j}+t_c$			1	−1	1			$k_{-6}\left[R_{e,j}\right]$
$R_{e,j}+\left(\mathit{\text{peptide}}\right)\stackrel{k_7\left(ATP\right)}{\to }{R}_{o,j}+t_u+\left(\mathit{\text{peptid}}{e}^{+1}\right)$			1	−1		1		$k_7\left(ATP\right)\left[R_{e,j}\right]$
$R_{o,j}+t_u\stackrel{k_8}{\to }{R}_{s,j}$			−1			−1	1	$k_8\left[R_{o,j}\right]\left[t_u\right]$
$R_{s,j}\stackrel{k_{-8}}{\to }{R}_{o,j}+t_u$			1			1	−1	$k_{-8}\left[R_{s,j}\right]$

2.4. Step 4: Equations for Biochemical Reactions

With the reaction tables in place, we can now track the changes for each molecule. However, a model that includes every reaction rate and molecule becomes computationally intensive and requires numerous parameters. Therefore, we further simplify the model by applying time-scale domain knowledge, using the rapid equilibrium assumption or the quasi-steady-state assumption. This approach allows us to derive Michaelis-Menten type rate laws from the detailed reaction networks. We will demonstrate this simplification process by applying it to both the tRNA charging and ribosome partitioning processes.

2.4.1. Assumptions and derivations

For the tRNA charging process, the following assumptions can be made:

1. The binding and unbinding of molecules occur much faster than the other reactions.

2. The molecules dynamics are in equilibrium, meaning that the rate of aminoacyl adenylate formation are equal to the rate of charged tRNA synthesis.

From assumption (1), we apply the rapid equilibrium approximation to reactions (1) and (3), leading to the association and dissociation reaction relationships as:

$$k_1\left[E\right]\left[Aa\right]\left[ATP\right]=k_{-1}[E·Aa·ATP]$$ (1)

$$k_3\left[E·Aa·AMP\right]\left[t_u\right]=k_{-3}\left[E·Aa·AMP·t_u\right]$$ (2)

These can be rearranged as:

$$\left[E·Aa·ATP\right]=K_1\left[E\right]\left[Aa\right]\left[ATP\right],\text{where}{K}_1=\frac{k_1}{k_{-1}}$$ (3)

$$\left[E·Aa·AMP\cdot t_u\right]=K_3\left[E·Aa·AMP\right][t_u],\text{where}{K}_3=\frac{k_3}{k_{-3}}$$ (4)

From assumption (2), we relate reactions (2) and (4) as:

$$k_2[E·Aa·ATP]=k_4\left[E·Aa·AMP\cdot t_u\right]$$ (5)

which simplifies to:

$$\left[E·Aa·AMP\cdot t_u\right]=K_2\left[E·Aa·ATP\right],\text{where}{K}_2=\frac{k_2}{k_4}$$ (6)

Applying the conservation law for the total aminoacyl-tRNA synthetase:

$$\left[E_{tot}\right]=\left[E\right]+\left[E·Aa·ATP\right]+\left[E·Aa·AMP\right]+\left[E·Aa·AMP\cdot t_u\right]$$ (7)

Substituting equations (3), (4), and (6) into (7), we derive the pseudo-steady-state concentrations of each complex:

$$\left[E\right]=\frac{1}{1+K_1\left[Aa\right]\left[ATP\right]+\frac{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}{K_3\left[t_u\right]}+{K_{2}K}_1\left[Aa\right]\left[ATP\right]}[E_{tot}]$$ (8)

$$\left[E·Aa·ATP\right]=\frac{K_1\left[Aa\right]\left[ATP\right]}{1+K_1\left[Aa\right]\left[ATP\right]+\frac{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}{K_3\left[t_u\right]}+{K_{2}K}_1\left[Aa\right]\left[ATP\right]}[E_{tot}]$$ (9)

$$\left[E·Aa·AMP\right]=\frac{\frac{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}{K_3\left[t_u\right]}}{1+K_1\left[Aa\right]\left[ATP\right]+\frac{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}{K_3\left[t_u\right]}+{K_{2}K}_1\left[Aa\right]\left[ATP\right]}[E_{tot}]$$ (10)

$$\left[E·Aa·AMP\cdot t_u\right]=\frac{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}{1+K_1\left[Aa\right]\left[ATP\right]+\frac{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}{K_3\left[t_u\right]}+{K_{2}K}_1\left[Aa\right]\left[ATP\right]}[E_{tot}]$$ (11)

Finally, since our primary interest is the formation of charged tRNA, which directly impacts ribosome partitioning and peptide elongation, we obtain the synthesis rate as a Michaelis–Menten–type expression derived from reaction (4) and equation (11):

$$J_{t_c,syn}=k_4\left[E·Aa·AMP\cdot t_u\right]=\frac{k_4\left[E_{tot}\right]}{1+\frac{1}{K_2}+\frac{1}{{K_{2}K}_1\left[Aa\right]\left[ATP\right]}+\frac{1}{K_3[t_u]}}$$ (12)

This rate law explicitly demonstrates the dependence of charged tRNA synthesis on the availability of three substrates, amino acids, ATP, and uncharged tRNA, as well as on enzyme concentration (also see Note 5).

For the ribosome partitioning process, we assume that ribosome dynamics are balanced across all states, meaning each ribosome type can be treated under a quasi–steady-state approximation. Accordingly, the dynamics of ribosomes can be written as:

$$\frac{d\left[R_f\right]}{dt}={-k}_5\left({\tau }^{*}\right)\left[R_f\right]\left[M_j\right]+k_{-5}\left(s^{*}\right)\left[R_{o,j}\right]=0$$ (13)

$$\frac{d\left[R_{o,j}\right]}{dt}=k_5\left({\tau }^{*}\right)\left[R_f\right]\left[M_j\right]-k_{-5}\left(s^{*}\right)\left[R_{o,j}\right]-k_6\left[R_{o,j}\right]\left[t_c\right]+k_{-6}\left[R_{e,j}\right]+k_7\left(ATP\right)\left[R_{e,j}\right]-k_8\left[R_{o,j}\right]\left[t_u\right]+k_{-8}\left[R_{s,j}\right]=0$$ (14)

$$\frac{d\left[R_{e,j}\right]}{dt}=k_6\left[R_{o,j}\right]\left[t_c\right]-k_{-6}\left[R_{e,j}\right]-k_7\left(ATP\right)\left[R_{e,j}\right]=0$$ (15)

$$\frac{d\left[R_{s,j}\right]}{dt}=k_8\left[R_{o,j}\right]\left[t_u\right]-k_{-8}\left[R_{s,j}\right]=0$$ (16)

Rearranging equations (13–16), we obtain:

$$\left[R_{o,j}\right]=K_5({\tau }^{*},s^{*})\left[R_f\right]\left[M_j\right],\text{where}{K}_5\left({\tau }^{*},s^{*}\right)=\frac{k_5\left({\tau }^{*}\right)}{k_{-5}\left(s^{*}\right)}$$ (17)

$$\left[R_{e,j}\right]=K_5\left({\tau }^{*},s^{*}\right)K_6\left(ATP\right)\left[R_f\right]\left[M_j\right]\left[t_c\right],\text{where}{K}_6(ATP)=\frac{k_6}{k_{-6}+k_7(ATP)}$$ (18)

$$[R_{s,j}]=K_5\left({\tau }^{*},s^{*}\right)K_8\left[R_f\right]\left[M_j\right][t_u],\text{where}{K}_8=\frac{k_8}{k_{-8}}$$ (19)

From the ribosome conservation law:

$$\left[R_f\right]=\left[R_{tot}\right]-{\sum }_j\left[R_{o,j}\right]-{\sum }_j\left[R_{e,j}\right]-{\sum }_j\left[R_{s,j}\right]$$ (20)

The protein synthesis rate of a type-$j$ protein can then be defined as the peptide elongation rate divided by the amino acid length $l_j$ of the protein:

$$J_{P_j,syn}=\frac{k_7\left(ATP\right)\left[R_{e,j}\right]}{l_j}$$ (21)

The utilization rate of charged tRNA during peptide elongation is:

$$J_{t_c,deg}=-{\sum }_j{k}_7\left(ATP\right)\left[R_{e,j}\right]$$ (22)

From (equations 18), (21), and (22), the results indicate that protein translation rates are regulated by ${\tau }^{*},s^{*}$, and ATP. These factors influence the equilibrium constants ($K_5,K_6$) as well as the rate constant ($k_7$). In addition, the concentration of $\left[M_j\right]$ for each protein type $j$ reflects transcriptional output, which is also regulated by ${\tau }^{*},s^{*}$ (see Note 6).

2.4.2. Differential equations for module dynamics

We successfully derived the rates for two key processes: tRNA charging and its utilization during peptide elongation, and the overall protein synthesis rate. By focusing on the formation of the final products (charged tRNAs and proteins), we implicitly integrated all other reactions into the rate functions of these core processes. Based on these derived rates and considering growth-induced dilution ($\lambda$), we can now formulate the following differential equations to describe the dynamics of the final products within the protein expression module, where the $\lambda$ stands for the growth rate:

$$\frac{d\left[t_c\right]}{dt}=J_{t_c,syn}-J_{t_c,deg}-\lambda \left[t_c\right]$$ (23)

$$\frac{d\left[P_j\right]}{dt}=J_{P_j,syn}-\lambda \left[P_j\right]$$ (24)

2.5. Step 5: From Single Cell to Population Growth

By following the procedures outlined in Steps 3 and 4, each module can be expressed as a set of differential equations. To obtain a system-level view, these module equations are then integrated such that the output of one module serves as the input to another. For instance, in the protein expression module, the dynamics of different protein sectors are described by equation (24). The resulting protein sector concentrations can then feed into the metabolism module, where they regulate the corresponding reaction rates.

Up to this point, our discussion has focused on the single-cell level and its intracellular biochemical reactions. From an experimental perspective, however, cellular growth is typically monitored at the population level, such as through optical density measurements. This raises a key question: how can protein synthesis rate (biomass formation at the single-cell level) be connected to the observed population growth rate? To enable such comparisons, the single-cell framework must be extended to describe population-level dynamics. In this step, we demonstrate how to derive the overall population growth rate from biomass synthesis rate at the single-cell level.

First, we define the cellular biomass as:

$$M_{\mathit{\text{cell}}}=\left(a\left[P_{tot}\right]\right)V_{\mathit{\text{cell}}}=\mathit{\text{cons}}t.$$ (25)

where biomass is assumed to be proportional to the total protein concentration $\left[P_{tot}\right]$ within the cell. In addition, the cellular biomass $M_{\mathit{\text{cell}}}$, the total protein concentration $\left[P_{tot}\right]$, and the cell volume $V_{\mathit{\text{cell}}}$ are set as constants.

From equation (24–25), the rate of change of cellular biomass can be written as:

$$\frac{d{M}_{\mathit{\text{cell}}}}{dt}=a{V}_{\mathit{\text{cell}}}\frac{d\left[P_{tot}\right]}{dt}=a{V}_{\mathit{\text{cell}}}({\sum }_j{J}_{P_j,syn}-\lambda {\sum }_j\left[P_j\right])=0$$ (26)

Rearranging equation (26), the cellular growth rate is derived as:

$$\lambda =\frac{\sum _j{J}_{P_j,syn}}{\sum _j\left[P_j\right]}=\frac{\sum _j{J}_{P_j,syn}}{\left[P_{tot}\right]}$$ (27)

Thus, cellular growth is directly proportional to the total protein synthesis rate, while the total protein concentration $\left[P_{tot}\right]$ remains constant.

At the system level, the change in total biomass $M_{all}$ can be written as:

$$\frac{d{M}_{all}}{dt}=\lambda {\cdot M}_{all}$$ (28)

where the system biomass is the product of cellular biomass and cell number $N$,

$$M_{all}=M_{\mathit{\text{cell}}}\cdot N$$ (29)

Taking the derivative on both side of equation (29) gives:

$$\frac{d{M}_{all}}{dt}=M_{\mathit{\text{cell}}}\cdot \frac{dN}{dt}$$ (30)

By comparing equations (28–29) with equation (30), we arrive at the population growth equation (see Note 7):

$$\frac{dN}{dt}=\lambda N$$ (31)

3. Models and Extensions

3.1. Growth Laws: Steady-state and Dynamic Controls

With the essential network constructed, the coarse-grained model helps elucidate how different levels of regulation coordinate under varying conditions. By grouping proteins according to their functions, the model reveals how cells allocate resources to adapt to environmental cues. A central focus is the relationship between the abundance of ribosomal and non-ribosomal proteins and the cellular growth rate. Ribosomal protein concentration directly determines the overall protein synthesis rate, which in turn is proportional to the cellular growth rate. As described in Section 2.1.2, proteins can be grouped based on functional annotations, and in Section 2.4.1, we showed how ribosome abundance and ribosome states determine protein synthesis. Building on this, Section 2.5 demonstrated that protein synthesis is explicitly proportional to growth rate. Together, the coarse-grained model provides a framework to investigate trade-offs between ribosomal and enzymatic proteins (Figure 3, left panel).

Figure 3. Model applications and extensions.

(i) The constructed coarse-grained modeling framework can be applied to investigate cellular trade-offs, such as proteome allocation across different growth rates and environmental conditions. In addition, the framework can be extended at selected breakpoints to provide deeper insights into specific pathways. For example, (ii) integrating essential nutrients other than glucose, such as ammonium and amino acids, allows the model to capture how medium composition influences cellular growth. (iii) Zooming into glycolysis enables the framework to distinguish metabolic breakpoints between the upper and lower glycolytic profiles and to identify key rate-limiting steps. (iv) Incorporating external gene expression helps quantify the metabolic burden imposed by heterologous circuits, improve understanding of host–circuit interactions, and guide strategies for mitigating these coupled effects.

It is widely studied that cellular proteome allocation shifts in response to changes in growth rate, nutritional conditions, and stress. Under steady-state conditions, where a cell grows at a constant exponential rate, its proteome composition is carefully adapted to reach maximal growth. In E. coli, two empirical growth laws have been discovered, showing a linear relationship between the growth rate and ribosomal abundance under both varying nutritional conditions and antibiotic exposure [3]. Modeling analyses have further shown that this relationship emerges from a robust regulatory strategy that tunes fluxes to achieve optimal growth [15,16]. Similarly, a linear growth law relating growth to ribosomal proteins has been discovered for S. cerevisiae under carbon-limited nutritional conditions [17]. Modeling results for S. cerevisiae have also confirmed that the system achieves maximal growth allocation, as perturbations to regulatory factors led to lower growth rate [18].

Moving beyond steady-state conditions, recent studies have begun to explore how microorganisms navigate fluctuating environments. Nutritional upshifts and downshifts are used to identify the dynamic allocation strategies of the proteome, which reveal how cells partition their resources in non-constant conditions [19–21]. Key phenomena during these transitions include the hierarchical utilization of carbon sources [22,23] and the diauxic shifting events [24]. A common metric for evaluating a cell’s adaptability is the duration of the lag phase during these transitions [24,25]. A longer transition time indicates a lower adaptation ability, and researchers have identified that the allocation of protein resources for ribosomal synthesis or for stress responses becomes a central trade-off [26]. This is an area of ongoing research, and modeling continues to be an invaluable tool for understanding the underlying feedback control mechanisms of dynamic proteome allocation strategies under varying conditions.

3.2. System input Extensions: Medium Nutritional Composition

In the following sections, we present several examples of extending the coarse-grained model to incorporate additional pathways into the system. While earlier discussions have focused primarily on carbon source transitions, recent studies have begun to explore other nutritional requirements that shape cellular macromolecular composition. For example, under phosphorus-limited conditions, the synthesis of phosphorus-rich compounds is restricted, resulting in reduced nucleotide pools and slower growth [27]. Likewise, nitrogen limitation constrains the synthesis of nitrogen-rich compounds, which can decrease overall protein synthesis rates or alter protein composition [28]. To fully capture the trade-offs cell face under these conditions, it is essential to integrate multiple nutrient inputs into the modeling framework.

Here, we provide an example of incorporating nitrogen as another potential limiting resource (Figure 3, top right panel). This extension enables the model to account for how nitrogen availability shapes cellular behavior by treating nitrogen as an additional competing or non-competing substrate in the reactions. For example, under nutrient-rich conditions, the flux from precursors to amino acids is often modeled without considering ammonium concentration. However, when ammonium becomes limiting, its concentration directly constrains amino acid synthesis rates [27]. To capture this effect, two modifications are required: introducing a differential equation to track ammonium concentration over time, and adding an ammonium-dependent term to the relevant Michaelis–Menten rate equations. By incorporating nitrogen-related entries in this way, the model can systematically explore the cellular behavioral landscape across varying carbon and nitrogen compositions in the culture medium [29].

In addition, to compare rich and minimal media, the model can be expanded to include amino acid uptake pathways from the culture, allowing cells to acquire amino acids directly from the environment rather than relying solely on de novo synthesis. As shown in Figure 3 (top right panel), modeling results demonstrate that rich media lead to faster glucose uptake and greater biomass accumulation compared to minimal media. This modeling framework not only enables the dissection of proteome allocation but also allows for tracking changes in cellular composition and growth rates under different nutritional conditions [17,24,30]. Such insights provide valuable guidance for medium optimization and dynamic control in bioreactors [31].

3.3. Pathway Extensions: One Precursor to Four Precursors

Another area of study has focused on understanding how carbon flux is partitioned between assimilation for biomass formation and catabolism for energy production during environmental fluctuations. To better quantify this distribution, the coarse-grained model can be “zoomed in” to expand the one-step reaction from glucose to precursor into a more detailed, four-step pathway (Figure 3, middle right panel). In this refinement, representative upper-glycolysis precursors (fructose-6-phosphate and fructose-1,6-bisphosphate) and lower-glycolysis precursors (phosphoenolpyruvate and pyruvate) are selected, and the proteome sectors in the original one-step model are subdivided to govern distinct sets of reactions [24]. In addition, regulatory effects on protein activities and the signaling roles of these precursor metabolites are re-analyzed. The newly constructed essential network can then be described by a corresponding set of equations, following the same steps outlined in Section 2.

This approach quantitatively identifies rate-limiting steps and clarifies how key carbon-based precursors are distributed toward amino acid, ATP, and other product synthesis [24]. As shown in Figure 3, the significant concentration shifts of precursors 2 and 4 across different growth rates indicate their sensitivity for growth-related control. Quantifying these precursors pinpoints the most limited precursor pools and rate-limiting proteome sectors, which enhance the understanding of cellular resource allocation strategies [25]. Furthermore, flux distributions can be validated by externally supplementing precursor inputs, either through isotope tracing [30] or intermediate supplementation [32]. These analyses can ultimately guide enzyme overexpression strategies to overcome bottlenecks and improve bioproduction [33].

3.4. Module Extensions: Synthetic Circuits Incorporation

Although simulations can help us understand and optimize native cellular investment strategies, introducing external gene constructs often disrupts normal fluxes and produces unexpected results [34,35]. One major challenge is the host–circuit coupling effect, where the expression of integrated genes is constrained by the limited capacity of the host’s transcriptional and translational machinery [36,37].

Coarse-grained modeling helps analyze how heterologous gene expression affects cells. In this framework, translation of external genes can be represented as the formation of a new protein sector, which diverts amino acid resources away from native protein synthesis (Figure 3, lower right panel) [38,39]. As the rate of additional protein synthesis increases, the fractions of ribosomal and enzymatic proteins decline, leading to reduced glycolysis and slower overall growth. This phenomenon, known as metabolic burden [40], also impacts synthetic constructs, which result in unexpected behaviors such as bistability or oscillations [38,41]. Current research efforts focus on developing context-aware feedback controllers to mitigate these undesirable outcomes and improve the robustness of synthetic designs [42,43].

4. Notes

1. Representative precursors are often selected as pyruvate or acetyl-CoA, since they function as key metabolic intermediates and central branching points in cellular metabolism. As the end product of glycolysis, pyruvate serves as a hub that connects multiple pathways. In the mitochondria, pyruvate can be converted to acetyl-CoA to fuel the TCA cycle, generating ATP and providing precursors such as oxaloacetate and α-ketoglutarate for amino acid biosynthesis. In the cytosol, pyruvate can also be directed toward fermentation, producing compounds such as ethanol and acetate. Acetyl-CoA itself represents another crucial branching point. In mitochondria, it supports the TCA cycle and amino acid synthesis, while in the cytosol it provides the building blocks for lipid and acetate production.

2. Glutamine is selected as a representative amino acid to reflect growth-limiting conditions. Glutamine concentration has been shown to closely correlate with cellular growth rate under nitrogen limitation in both E. coli and S. cerevisiae, where its synthesis via glutamine synthetase is tightly regulated by external ammonia availability [8,44]. Moreover, under nitrogen limitation, translation elongation is slowed at glutamine codons [45].

3. The overall protein composition of a cell is typically assumed to be constant; therefore, a higher protein accumulation rate reflects a faster cellular growth rate (see Section 2.5 for detailed derivations).

4. GO Slim is a curated subset of the Gene Ontology (GO) that provides a simplified view by grouping detailed terms into broader, high-level categories. These terms are drawn from the three main GO domains: Biological Process, Molecular Function, and Cellular Component. GO Slim terms are selected to represent major branches of the ontology, allowing researchers to summarize large gene annotation sets. For example, the go_slim_mapping.tab file from the Saccharomyces Genome Database provides a widely used mapping for yeasts (http://sgd-archive.yeastgenome.org/curation/literature/) [46].

5. Based on the conservation of tRNA, the relationship between uncharged tRNA $\left(t_u\right)$ and charged tRNA ${(t}_c)$ can be written as:

   $$\left[t_u\right]=\left[t_{tot}\right]-\left[t_c\right]$$ (32)

   where $\left[t_{tot}\right]$ stands for the total concentration of tRNA and is assumed to be proportional to the concentration of ribosomal components.

6. In equations (17, 18, 19, 21, and 22), protein translation rates are expressed as functions of ${\tau }^{*},s^{*}$, and ATP. The equilibrium constants ($K_5,K_6$) and the rate constant ($k_7$) are regulated parameters, denoted as $(\cdot )$, and can be represented explicitly with Hill functions. Similarly, $\left[M_j\right]$, the mRNA concentration for protein sector $j$, reflects transcriptional expression and is also regulated by ${\tau }^{*}$ and $s^{*}$. Thus, $\left[M_j\right]$ should be seen as $\left[M_j({\tau }^{*},s^{*})\right]$, with its transcriptional regulation also described by Hill functions.

7. Note that $\lambda$ is not a constant; it is a time-dependent variable that changes with the protein translation rate. This reflects the fact that cellular growth is dynamically coupled to intracellular processes. When $\lambda$ is assumed constant, cellular population growth can be approximated by the classical exponential formml:

   $$N\left(t\right)=N_0{e}^{\lambda t}$$ (33)

   This formulation has been used in microbial physiology to describe exponential growth under balanced conditions (e.g., in batch culture during the exponential phase). In our framework, however, $\lambda$ varies over time, as it is determined by protein synthesis dynamics.