1. Introduction

2. Main

3. Conclusions

4. Appendix

4.2. Appendix B: Systems Biology Approaches to Understand Resource Allocation

The coupling of activity between different intracellular molecular processes, constrained by the cellular resource budget, dictates phenotype. Many systems biology approaches combine high-throughput measurements with computational algorithms to model the interplay between intracellular processes and understand phenotype.

Phenomenological models, for example, are coarse-grained representations that identify quantitative relationships between resource constraints, cellular activity, and phenotype. Such models have described relationships between gene expression and growth(Scott et al., 2010), proteome allocation under resource limitation(Hui et al., 2015), and macromolecular concentration homeostasis(Lin and Amir, 2018). Describing such relationships in a few meaningful parameters that are robust across contexts, these models shed light on principles of resource allocation. However, they lack molecular details.

Thus, models that integrate resource availability with mechanisms are necessary. Kinetic ordinary differential equations (ODEs) are commonly used to model gene expression (Appendix B Figure a). Such approaches have been extended to account for gene expression noise via a “two-state model” that can account for random, non-constitutive transcription(Munsky et al., 2012). However, these ODE models require extensive measurements of kinetic parameters and cannot account for resource availability. In contrast, genome-scale metabolic models (M-Models) map out all metabolic pathways of the cell, represent them mathematically using their stoichiometric coefficients(Lewis et al., 2012), and use flux balance analysis (FBA) to simulate the reaction fluxes(Lewis et al., 2012; Orth et al., 2010). These fluxes optimize a cellular objective that represents a phenotype of interest (often growth rate(Feist and Palsson, 2010)) while directly accounting for nutrient and bioenergy resource constraints. In mammalian cells, FBA has explored numerous questions, including polyamine metabolism in T-helper 17 cell pathogenicity(Wagner et al., 2021) and enzymes that affect migration, but not proliferation, of cancer cells(Yizhak et al., 2014).

M-Models do not comprehensively account for machinery resources. They can partially do so using methods to overlay gene expression measurements that prune the metabolic network(Opdam et al., 2017), but this only incorporates machinery in a binary manner. Other methods have constrained M-Models by more explicitly minimizing machinery costs. Parsimonious FBA, for example, minimizes the sum of fluxes throughout the network as a secondary objective. Here, the total flux is a proxy for the global “effort” or resource cost of biological activity(Holzhütter, 2004; Lewis et al., 2010a). Another method estimates the extent of total reaction flux that is backwards as a proxy for machinery costs, with higher backwards flux requiring a larger enzyme budget to drive the reaction forward(Noor et al., 2014). Finally, a number of strategies constrain reaction fluxes by machinery abundance at varying resolution(Noor et al., 2016),(Mori et al., 2016),(Domenzain et al., 2022; Sánchez et al., 2017). Machinery coupling is not limited to canonical catalytic enzymes, but can be extended to transport(Sahoo et al., 2014), gene expression(Schwanhäusser et al., 2011), and protein secretion(Gutierrez et al., 2020).

Such methods that incorporate individual enzymes into genome-scale models differentiate between nutrient- and machinery-limiting conditions. However, to explicitly account for the entire machinery budget and costs, genome-scale models of metabolism and expression (ME-Models) add gene synthesis reactions for each enzyme(Thiele et al., 2009) and couple them to metabolism (Appendix B Figure b)(Lerman et al., 2012; O’Brien et al., 2013). In prokaryotes, ME-Models(Dahal et al., 2021) have shed light on proteome allocation strategies under oxidative(Yang et al., 2019), acidic(Du et al., 2019), and thermal(Chen et al., 2017) stress. In eukaryotes, the ME-Modeling framework accounts for compartment-specific proteome constraints, enabling analysis of metabolic shifts, machinery protein costs, and proteome reallocation(Elsemman et al., 2022). Resource Balance Analysis (RBA) models employ similar concepts to encode gene expression and metabolism while also accounting for spatial constraints(“Cell design in bacteria as a convex optimization problem,” 2011). While RBA is limited to prokaryotes, a theoretical roadmap for eukaryotes exists(Goelzer and Fromion, 2019). Finally, whole-cell models (WCMs) additionally characterize kinetics and encompass more cellular processes(Macklin et al., 2020). By modeling each intracellular system separately then connecting them via common inputs and outputs, WCMs can account for resources shared between multiple intracellular subsystems(Karr et al., 2012) and interconnectivity of molecular processes, e.g. the effect of glycolytic flux for producing dNTPs used in chromosomal replication(Thornburg et al., 2022).

Appendix B Figure: Modeling cellular activity from individual reactions to global networks. a | A pathway with a set of connected molecules can be decomposed into its individual reactions. These reactions can be used for high-resolution modeling of a particular process, incorporating detailed kinetic parameters. Globally, all the reaction pathways form a network defining some molecular process such as metabolism. b | Processes can be coupled (black arrows) using first-principles to create more comprehensive genome-scale models such as ME-models. Signaling and secretion incorporate extracellular interactions, providing a basis for modeling multicellular systems.

Appendix B Figure: Modeling cellular activity from individual reactions to global networks. a | A pathway with a set of connected molecules can be decomposed into its individual reactions. These reactions can be used for high-resolution modeling of a particular process, incorporating detailed kinetic parameters. Globally, all the reaction pathways form a network defining some molecular process such as metabolism. b | Processes can be coupled (black arrows) using first-principles to create more comprehensive genome-scale models such as ME-models. Signaling and secretion incorporate extracellular interactions, providing a basis for modeling multicellular systems.

For details on how these modeling approaches are implemented and practical use-cases, please see the following references: network dynamics(Alon, 2006), M-models(Palsson, 2015), and whole-cell models(Covert, 2017).

4.5. Appendix E: Modeling Machinery Activity and Abundance to Couple Protein with Metabolism

As an example of systems biology approaches, we model here enzyme-catalyzed reactions. The assumptions regarding their reaction kinetics are used to constrain models with machinery costs and couple gene expression to metabolism.

For conversion of substrate (S) to product (P) by a machinery enzyme (E), Michaelis-Menten kinetics is often assumed:

$$v=k_{cat}[E]\frac{[S]}{[S]+K_M}$$

(1)

Under in vitro, fully saturated conditions ([S] >> K_M), the maximal reaction rate is achieved:

$$v_{max}=k_{cat}[E]$$

(2)

In equation (2), reaction rates depend on machinery abundance and activity. However, in vivo observed reaction rates often deviate from this maximum value due to saturating, thermodynamic, and regulatory effects:

$$v_{obs}=k_{cat}[E]\mathsf{\eta }(\mathrm{C})=v_{max}$$

(3)

η(C) is a context-dependent scaling parameter that accounts for the discrepancy between v_max and v_obs due to in vivo variables such as metabolite concentration, pH, and molecular crowding. η(C) can be decomposed into saturating, thermodynamic, and regulatory components. Saturating and thermodynamic components can only decrease v_obs relative to v_max. However, regulatory effects such as allostery and post-translational modifications can cause v_obs > v_max.

4.5.1. Saturation

Focusing on saturating effects, if substrate concentration is no longer sufficiently high such that K_M is not negligible, then enzymes are not fully saturated. In this case, we can represent total enzyme concentration as:

[E] = [E]_used + [E]_free

(A1)

From Michaelis-Menten derivations:

$$[\mathrm{E}{]}_{\mathrm{used}}\equiv [\mathrm{ES}]=\frac{[E][S]}{[S]+K_M}$$

(A2)

We assume the scaling factor as proportional to the fraction of used enzyme :

$${\mathsf{\eta }}_{\mathrm{saturation}}(\mathrm{C})=\frac{{[E]}_{used}}{[\mathrm{E}]}=\frac{[S]}{[S]+K_M}$$

(A3)

Thus, $\frac{[S]}{[S]+K_M}$is the enzyme saturation scaling factor. From equations (3) and (A3) (and also shown in the Michaelis-Menten derivation) Equation (1) can also be written as:

$$v_{obs}=k_{cat}{[E]}_{\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{d}}$$

(A4)

Defining the capacity utilization of a reaction as the ratio between the observed and maximal reaction rate yields the following equality from equations (2) and (A4):

$$\frac{v_{obs}}{v_{max}}=\frac{{[E]}_{used}}{[\mathrm{E}]}$$

(A5)

Note that the capacity utilization and saturation scaling factor are equivalent and analogous concepts, and many of these equations are different conceptual representations of the same mathematical equivalencies. Broadly, equation (A5) indicates that with saturating effects, there is some fraction of unused enzyme that is decreasing the overall reaction rate.

4.5.2. Resource Loading

Due to resource loading, multiple substrates may be competing for a shared machinery resource. Resultant resource-limitation can be modeled in enzyme-catalyzed reactions by simply adding a resource term R that scales the reaction rate. Let’s take the subscript i to represent the substrate of interest. Thus, equation (1) can be represented as follows:

$$v_i=k_{cat,i}[E]\frac{[S_i]}{[S_i]+K_{M,i}}$$

(B1)

Separately, given an inhibitor represented by the subscript j, competitive inhibition in Michaelis-Menten is typically modeled as follows:

$$v_i=k_{cat,i}[E]\frac{[S_i]}{K_{M,i}(1+\frac{[S_j]}{K_{Mj}})+[S_i]}$$

(B2)

For formatting, let’s rewrite equation (B2) as follows:

$$v_i=k_{cat,i}[E]\frac{[S_i]}{K_{M,i}(1+\frac{[S_i]}{K_{M,i}}+\frac{[S_j]}{K_{Mj}})}$$

(B3)

Let’s take all possible substrates of the machinery E to be represented by the subscript j. Then, we can expand on this competitive inhibition model to derive a reaction flux term that accounts for resource loading. First, let’s define a term Y to represent all competing substrates:

$$Y={\sum }_{j\ne i}\frac{[S_j]}{K_{M,j}}$$

(B4)

Then, using this definition in (B4) to extend (B3) to account for all competing substrates j ≠ i, we can get:

$$v_i=k_{cat,i}[E]\frac{[S_i]}{K_{M,i}(1+\frac{[S_i]}{K_{M,i}}+Y)}$$

(B5)

Ultimately, substituting (B4) into (B5) gives us a reaction rate term that can account for resource competition:

$$v_i=k_{cat,i}[E]\frac{[S_i]}{K_{M,i}(1+{\sum }_j\frac{[S_j]}{K_{M,j}})}$$

(B6)

From equation (B6) and equation (3), we can define scaling due to resource competition as follows:

$${\mathsf{\eta }}_{\mathrm{competition}}(\mathrm{C})=\frac{[S_i]}{K_{M,i}(1+{\sum }_j\frac{[S_j]}{K_{M,j}})}$$

(B7)

Note that the saturation term (A3) can be related to resource competition effects (B7). We can derive a resource competition term R which relates η_competition(C) to η_saturation(C):

η_competition(C) = η_saturation(C)*R

(B8)

To derive a term for R, we can substitute (B4) back into (B7) as follows:

$${\mathsf{\eta }}_{\mathrm{competition}}(\mathrm{C})=\frac{[S_i]}{K_{M,i}+[S_i]+Y}$$

(B9)

This can be rearranged as follows:

$$\begin{gathered} {\mathsf{\eta }}_{\mathrm{competition}}(\mathrm{C})=\frac{[S_i]}{K_{M,i}+[S_i]}*\frac{1}{1+\frac{Y}{K_{M,i}+[S_i]}} \\ =\frac{[S_i]}{K_{M,i}+[S_i]}*\frac{1}{\frac{K_{M,i}+[S_i]+Y}{K_{M,i}+[S_i]}} \\ =\frac{[S_i]}{K_{M,i}+[S_i]}*\frac{K_{M,i}+[S_i]+Y}{K_{M,i}+[S_i]} \end{gathered}$$

(B10)

Substituting (B4) back into the derivation for (B10) gives:

$${\mathsf{\eta }}_{\mathrm{competition}}(\mathrm{C})=\frac{[S_i]}{K_{M,i}+[S_i]}*\frac{K_{M,i}(1+{\sum }_j\frac{[S_j]}{K_{M,j}})}{K_{M,i}+[S_i]}$$

(B11)

Finally, using equations (B8) and (B11) gives us:

$$R=\frac{K_{M,i}(1+{\sum }_j\frac{[S_j]}{K_{M,j}})}{K_{M,i}+[S_i]}$$

(B12)

Thus, R accounts for the effect of multiple substrates in the numerator normalized to the effects of a single substrate in the denominator. Together, equations (B8) and (B12) demonstrate that the reaction rate that accounts for resource competition in equation (B6) is simply a scaling of equation (1) in which saturation effects (derived in (A3)) include competitive inhibition from multiple substrates.

Furthermore, both (A3) and (B7) are a specific version of the context-dependent scaling parameter η(C) introduced in equation (3), and both depend on substrate concentration. Substrate concentration may be estimated by joint molecular abundance and cell volume(Bryan et al., 2014) measurements. The relative abundance of all competitive substrates, for example, may be estimated across multiple samples representing distinct contexts with omics data. Notably, subcellular compartmentalization alters reaction dynamics by influencing the availability of machinery and the abundance of machinery substrates. Furthermore, the spatial distribution of machinery across these compartments to maximize reaction flux has been demonstrated to be a relevant resource allocation principle(Giunta et al., 2022). Thus, the subcellular localization of machinery must be measured in the future(Mah et al., 2023). Overall, context-dependent modeling depends on the resolution at which technologies are capable of measuring the cells’ and machineries’ microenvironments. For example, when considering a cell’s communicatory partners, the local microenvironment of a receptor’s interacting ligand partners can be mapped at 1nm resolution(Geri et al., 2020). Thus, technological improvements that increase the throughput and resolution of context-dependent molecular measurements will improve model accuracy, and inversely, models will need to be expanded to better account for the additional information provided by such measurements.

4.5.3. Thermodynamics

Next, we will focus on thermodynamic effects. For simplicity, we will consider an isothermal, isobaric reaction of a substrate converted into product, disregarding Michaelis-Menten kinetics (conclusions are the same):

The reaction kinetics can be written as:

v_obs = v_f - v_r = k_f[S] - k_r[P]

(C1)

The reaction thermodynamics can be written as:

$$\mathsf{\Delta }\mathrm{G}=-\mathrm{RTln}\left(\frac{[S]K_{eq}}{[P]}\right)$$

(C2)

The kinetics and thermodynamics are related to each other via the equilibrium constant K_eq from the following:

$${\mathrm{K}}_{\mathrm{eq}}=\frac{k_f}{k_r}$$

(C3)

Thus, equations (C2) can be rewritten using equations (C1) and (C3) to link thermodynamics with kinetics as:

$$\mathsf{\Delta }\mathrm{G}=-\mathrm{RTln}\left(\frac{v_f}{v_r}\right)$$

(C4)

Equation (C4) holds for more complex reactions, including Michaelis-Menten kinetics and the Hill equation. From (C4), reactions far from equilibrium (ΔG << 0) tend not to have a high reverse flux. However, enzymes catalyzing reactions near equilibrium (ΔG ~ 0) face a higher reverse flux. In this sense, thermodynamic forces (e.g., intermediate metabolite concentrations) can affect machinery costs, with reactions that have a higher reverse flux requiring a larger enzyme investment per unit of forward flux.

Relevant Sources: (Rondelez, 2012); (Beard and Qian, 2007; Davidi et al., 2016; Dourado et al., 2021; Noor et al., 2016, 2014),(Chou and Talalay, 1977)

5. Glossary

6. Highlighted References

• PMID 22068332: This article demonstrates that proteome allocation strategies minimize machinery costs by high expression of proteins with conserved functions and lower expression of proteins involved in context-specific and multicellular processes such as cell signaling and communication.
• PMID 32859923: This article quantifies relative ATP levels in relation to genome-wide machinery abundance across three conditions, highlighting the metabolic flexibility of bioenergetic pathways, distinguishing between machinery- or substrate-limited reactions, and identifying relationships between bioenergetics and growth.
• PMID 25745177: This article uses a model of gene expression to show the variable contributions of macromolecular abundance, synthesis rates, and degradation rates on differential protein expression in non-steady-state conditions.
• PMID 33173034: This article uses a model of gene expression that accounts for resource loading to demonstrate that, in engineered mammalian circuits, transcriptional resources limit protein abundance.
• PMID 26575626: This article quantifies the energetic costs of various steps of gene expression, showing that protein translation consumes a considerable fraction of the energy consumed during gene expression, and it also argues that the bioenergy budget scales (across prokaryotes to eukaryotes) with cell size when accounting for the total lifetime of the cell.
• PMID 31519914: This article provides a unique example of mammalian cell-decision making that optimizes for non-growth phenotypes (in this case, motility) by choosing migratory paths that will be less energetically costly.
• PMID 27951527: This article introduces the concept of a gene-specific, tissue-independent PTR that indicates that coupling between transcript and protein processes is affected by gene-intrinsic features.
• PMID 31896772: This article creates a mammalian genome-scale model that couples metabolism with the secretory pathway, demonstrating that trade-offs occur between cell growth and protein secretion.
• PMID 30638811: This article identifies Pareto fronts from single-cell measurements in tissue, showing that cells have varying extents of specialization according to their spatial location, resulting in division of labor that optimizes tissue-level function.
• PMID 29398113: This article shows how cell communication via growth factor signaling works in combination with constraints to maintain tissue compositional homeostasis.
• PMID 35421362: This article demonstrates how cells employ combinatorial strategies to efficiently communicate by only expressing a small number of ligands.
• PMID 10878248: This article adapts supply-demand analysis from economics to metabolism, showing that reaction flux and metabolite concentration are independently controlled to maintain homeostatic concentrations of metabolites across contexts.
• PMID 22308336: This article presents a theory for how specialization is favored by evolution due to context-specificity and synergistic effects.
• PMID 14550310: This article shows that humans regulate diet to intake balanced macronutrient ratios and prioritize protein when a balanced diet is not possible.

7. Figure Captions

Abbreviations

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