A proteome optimal allocation model for elucidating effects of temperature on bacterial growth

Abstract

Although considerable research has been conducted on how bacteria respond to temperature, there are still few studies that use simple protein allocation models to characterize these responses. To bridge this gap, we utilize a protein allocation framework to delineate quantitative correlations among temperature, growth rate, and the fraction of functional protein sectors. By calibrating the model with experimental temperature-dependent growth curves of Escherichia coli, we predict how proteomic resources are reallocated across various temperatures. This approach enables us to elucidate, from a proteomic perspective, the temperature dependency of constitutive β-galactosidase expression and the variations in E. coli cell size. Our research deepens the understanding of bacterial physiological adaptation to temperature changes.

Introduction

Bacteria in natural environments encounter diverse stressors, among which temperature is a particularly critical factor. Even slight variations in temperature can significantly impact ecosystems¹. The effects of temperature on bacterial growth have been extensively studied for several decades^2–8. A consistent observation is that the temperature-dependent growth curves exhibit strikingly similar shapes² (Fig. 1) across nearly all bacterial species, irrespective of their optimal growth temperature. Within a moderate temperature range, commonly referred to as the normal range or Arrhenius region⁹, bacterial growth curves generally follow the Arrhenius equation:

$$\mathit{ln}\mu \propto -\frac{1}{T}.$$ 1

This relationship suggests that growth within this range can be described as governed by a single rate-limiting reaction, while intrinsic physiological properties remain relatively stable. However, when temperatures exceed this normal range—known as the heat region—or fall below it (the cold region), bacteria activate distinct stress response mechanisms^3–8, accompanied by a sharp decline in growth rate compared to the normal range (Fig. 1).

Fig. 1. Temperature dependence of E. coli growth rate.

The growth rate of E. coli in Neidhardt’s defined rich medium is plotted against the reciprocal of absolute temperature. Within the normal temperature range (21 °C − 37 °C), the logarithm of the growth rate (μ) exhibits a linear dependence on 1/T, consistent with Arrhenius-like behavior. Significant deviations from this linear trend occur in both the heat region and cold region. Vertical dashed lines mark the boundaries of the normal range at 21 °C and 37 °C. The inset shows the same data plotted as growth rate versus temperature directly. This pattern of temperature-dependent growth rate is consistent with observations in other bacterial species (data from refs. ^66,68).

After half a century of dedicated research, the heat shock and cold shock responses in E. coli have been extensively characterized^3–8,10. These findings provide a crucial foundation for temperature regulation in bioengineering^11,12 and food safety applications^13,14. Given the intrinsic complexity of bacterial temperature response mechanisms, it is both valuable and necessary to develop a simplified temperature-dependent model based on general principles, which will enhance our understanding of these processes¹⁰.

Previous models^15–20 developed to explain temperature-dependent growth curves have primarily focused on traits within the normal range, frequently assuming that a single rate-limiting enzymatic reaction governs bacterial growth. Within the heat region, high temperatures might destabilize enzymes, thereby inhibiting bacterial growth. Alternative approaches have been limited to fitting empirical formulas without providing deeper explanations^21–26. In this study, we apply a coarse-grained protein allocation model to quantitatively characterize the temperature-dependent growth curves of E. coli.

The process of cell growth can be viewed as a process of “self-replication,” and the cell is a “self-replicator.” A simple self-replicator is that ribosomes translate themselves as well as other proteins²⁷. In the model of protein allocation, proteins are categorized into classes based on their functions, and bacteria adjust the composition of functionally diverse protein sectors in response to environmental changes^28–35. Given the constraint of mass conservation, this model correlates the allocation ratios of different protein sectors with growth rates and environmental conditions, thereby offering a clear framework for delineating bacterial physiology. Such models have been further corroborated by quantitative proteomic data^36,37. Moreover, from the protein allocation perspective, researchers have proposed several quantitative growth laws that describe bacterial physiology under steady-state conditions^28–35,37.

It is reasonable to expect that in E. coli, the allocation of functionally distinct protein sectors depends on temperature^8,38. In our previous work²⁹, we developed a protein allocation model that explicitly incorporated a chaperone sector to account for protein allocation under nonlethal protein maturation stress. In the present study, we extend this framework to gain a deeper understanding of how temperature variations affect bacterial physiology. While there is conceptually analogous research³⁹, our model offers a distinct perspective and helps elucidate the temperature dependence of more physiological variables—such as β-galactosidase expression and cell size—directly from the principles of protein allocation.

We will first extend our previous model²⁹ by incorporating temperature dependence into its parameters, thereby quantitatively associating chaperone levels with cellular growth in E. coli. The extended model will then be calibrated against experimental data to account for variations in growth rate across a temperature range. Employing the temperature-dependent parameters obtained from this calibration, we will predict protein allocation across varying temperatures. Finally, by integrating documented correlations between protein allocation and specific physiological traits in E. coli from previous research^29,33,40, we will derive predictions for the variation of these physiological parameters with temperature. Our predictions can offer interpretative insights into the experimental observations, thereby advancing our understanding of how E. coli responds physiologically to changes in temperature.

Results

Flux and protein allocation in bacterial growth

We begin by describing the previous coarse-grained protein allocation model²⁹, which focuses on protein synthesis and quality control.

The first step in protein synthesis is the transformation of extracellular substrates (S) into intracellular amino acids (AA). This conversion is carried out by a set of proteins—comprising metabolic enzymes, amino acid synthesis enzymes, and transporters—collectively referred to as P₁-class proteins. The rate at which amino acids are produced by this class is denoted as v_S↦AA. Analogous notation will be used for corresponding parameters in the subsequent processes. It should be noted that even in bacteria cultured in nutrient-rich media, such as LB, amino acid synthesis and degradation continue to play a significant physiological role⁴¹. The reason is due to the preferential uptake of specific amino acids and the discrepancy between the amino acid composition of the medium and the physiological requirements of E. coli⁴¹.

Subsequently, ribosomes and associated auxiliary proteins, collectively referred to as R-class proteins, utilize genetic information to convert AAs into unfolded proteins (UP). This conversion occurs at a flux rate denoted as v_AA↦UP. To acquire biological function, these unfolded proteins must achieve the correct conformation. Molecular chaperones, categorized as P₂-class proteins, facilitate the folding process, transforming unfolded proteins into normal proteins (NP) at a flux rate v_UP↦NP. However, a fraction of proteins fails to achieve their native conformation, resulting in the production of aberrant proteins (AP) at a flux rate v_UP↦AP. These aberrant proteins can either be degraded into AAs by proteases, which are classified as P₃-class proteins with a flux rate v_AP↦AA, or they may form protein aggregates (PA) at a flux rate v_AP↦PA. Such aggregates can have cytotoxic effects on bacterial cells⁴².

In addition to the previously described functional protein classes, a distinct category known as Q-class proteins exists. The total cellular abundance of Q-class remains invariant across different growth conditions²⁸, as this class primarily comprises essential housekeeping proteins.

Within this framework, the proteome is divided into five functional categories: R, P₁, P₂, P₃, and Q (see Fig. 2a). The roles of these sectors in the protein synthesis and quality control network are summarized in the accompanying schematic diagram (see Fig. 2b).

Fig. 2. Schematic overview of the temperature-dependent protein allocation model.

a Protein sector classification and temperature-dependent allocation. The total proteome is partitioned into five functional classes (Q, R, P₁, P₂, and P₃) based on their roles in protein synthesis and quality control. Their mass fractions (ϕ_i) may shift with cultivation temperature. Under heat or cold stress, bacterial protein allocation undergoes significant alterations: the chaperone fraction (ϕ₂) increases, with a more pronounced rise under heat stress; the translation-related fraction (ϕ_r) decreases in both stress conditions; the amino acid supply fraction (ϕ₁) specifically increases in the cold; and the protease fraction (ϕ₃) remains relatively stable. The sector proportions presented are based on data from ref. ⁶⁴. b Coupling between metabolic fluxes and protein sectors. The schematic depicts the substrate-to-protein conversion process in E. coli, linking the flux rate per unit normal protein mass for each step to the mass fraction of its corresponding protein class. Color-coded arrows indicate the directional change of key parameters with a dual scheme: color for temperature change (red increasing, blue decreasing) and arrow direction for parameter change (upward increasing, downward decreasing). The abbreviations in the figure correspond to the following full terminology: substrate (S), amino acid (AA), unfolded protein (UP), normal protein (NP), aberrant protein (AP), and protein aggregate (PA).

Using the input-output relationships for each metabolite node, we derive the kinetic equations governing their dynamics:

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}{M}_{\mathrm{NP}}}{\mathrm{d}t}& =& v_{\mathrm{UP}\mapsto \mathrm{NP}},\hfill \\ \hfill \frac{\mathrm{d}{M}_{\mathrm{AA}}}{\mathrm{d}t}& =& v_{\mathrm{S}\mapsto \mathrm{AA}}+v_{\mathrm{AP}\mapsto \mathrm{AA}}-v_{\mathrm{AA}\mapsto \mathrm{UP}},\hfill \\ \hfill \frac{\mathrm{d}{M}_{\mathrm{UP}}}{\mathrm{d}t}& =& v_{\mathrm{AA}\mapsto \mathrm{UP}}-v_{\mathrm{UP}\mapsto \mathrm{NP}}-v_{\mathrm{UP}\mapsto \mathrm{AP}},\hfill \\ \hfill \frac{\mathrm{d}{M}_{\mathrm{AP}}}{\mathrm{d}t}& =& v_{\mathrm{UP}\mapsto \mathrm{AP}}-v_{\mathrm{AP}\mapsto \mathrm{AA}}-v_{\mathrm{AP}\mapsto \mathrm{PA}}.\hfill \end{array}$$ 2

Under steady-state growth conditions, all cellular components increase at the same rate, the growth rate μ, which gives:

$$\frac{\mathrm{d}{M}_i}{\mathrm{d}t}=\mu M_i,i\in \left\{\mathrm{NP},\mathrm{AA},\mathrm{UP},\mathrm{AP}\right\},$$ 3

and allows the growth rate to be expressed in terms of the fluxes,

$$\mu =\frac{1}{M_{\mathrm{NP}}}\frac{\mathrm{d}{M}_{\mathrm{NP}}}{\mathrm{d}t}=\frac{v_{\mathrm{UP}\mapsto \mathrm{NP}}}{M_{\mathrm{NP}}}.$$ 4

Here, we assume that protein aggregates do not inhibit growth beyond the wasteful consumption of cellular resources. Given that all component proportions remain invariant in steady-state, the kinetic equation for the ratio of intermediate metabolite mass to normal protein mass is given by

$$\begin{array}{ccc}\hfill 0& =& \frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{AA}}}{\mathrm{d}t}=J_{\mathrm{S}\mapsto \mathrm{AA}}+J_{\mathrm{AP}\mapsto \mathrm{AA}}-J_{\mathrm{AA}\mapsto \mathrm{UP}}-\mu {\mathrm{\Phi }}_{\mathrm{AA}},\hfill \\ \hfill 0& =& \frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{UP}}}{\mathrm{d}t}=J_{\mathrm{AA}\mapsto \mathrm{UP}}-J_{\mathrm{UP}\mapsto \mathrm{NP}}-J_{\mathrm{UP}\mapsto \mathrm{AP}}-\mu {\mathrm{\Phi }}_{\mathrm{UP}},\hfill \\ \hfill 0& =& \frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{AP}}}{\mathrm{d}t}=J_{\mathrm{UP}\mapsto \mathrm{AP}}-J_{\mathrm{AP}\mapsto \mathrm{AA}}-J_{\mathrm{AP}\mapsto \mathrm{PA}}-\mu {\mathrm{\Phi }}_{\mathrm{AP}}.\hfill \end{array}$$ 5

Here, ${\mathrm{\Phi }}_i=\frac{M_i}{M_{\mathrm{NP}}},i\in \left\{\mathrm{AA},\mathrm{UP},\mathrm{AP}\right\}$ and

$$J_j=\frac{v_j}{M_{\mathrm{NP}}},j\in \left\{\mathrm{S}\mapsto \mathrm{AA},\mathrm{AA}\mapsto \mathrm{UP},\mathrm{UP}\mapsto \mathrm{NP},\mathrm{AP}\mapsto \mathrm{AA},\mathrm{UP}\mapsto \mathrm{AP},\mathrm{AP}\mapsto \mathrm{PA}\right\},$$ 6

represents the flux rate per unit mass of normal proteins.

The protein synthesis and quality control network operates with high efficiency and economy⁴³. Therefore, in the steady-state exponential phase, intermediate metabolites are maintained at relatively low levels^44–46. The translation rate is primarily determined by the concentration of aminoacyl-tRNAs, rather than by free amino acids³⁴. Furthermore, excessive free amino acids could be toxic⁴⁷. As a result, the amino acid pool should be maintained at a level sufficient to meet aminoacylation requirements, constituting only about 1.5% of the dry cell weight^44,45,48. Similarly, unfolded or aberrant proteins, whose non-native conformations can interfere with cellular processes^49,50, should be regulated to remain a minimal fraction of the cell mass. All these indicate that the mass of intermediate metabolites is relatively small compared to that of normal proteins, and the dilution term can be neglected, enabling us to reformulate Eq. (5) as follows:

$$\begin{array}{c}{J}_{\mathrm{S}\mapsto \mathrm{AA}}-J_{\mathrm{AA}\mapsto \mathrm{UP}}+J_{\mathrm{AP}\mapsto \mathrm{AA}}\approx 0,\hfill \\ J_{\mathrm{AA}\mapsto \mathrm{UP}}-J_{\mathrm{UP}\mapsto \mathrm{NP}}-J_{\mathrm{UP}\mapsto \mathrm{AP}}\approx 0,\hfill \\ J_{\mathrm{UP}\mapsto \mathrm{AP}}-J_{\mathrm{AP}\mapsto \mathrm{AA}}-J_{\mathrm{AP}\mapsto \mathrm{PA}}\approx 0.\hfill \end{array}$$ 7

The influence of the dilution term on the model results is presented in Supplementary Note 6.

Newly synthesized normal proteins are allocated among the five functional categories based on cellular demands. The mass fraction of normal proteins allocated to class P_i is denoted by ϕ_i, i ∈ {r, 1, 2, 3, q}. Under steady-state conditions, ϕ_i equals the mass fraction of the respective protein category within the total normal protein, i.e., ${\varphi }_i=\frac{M_i}{M_{\mathrm{NP}}}$. In the following, we will express each reaction flux rate as a function of these allocation fractions ϕ_i.

During protein synthesis, E. coli precisely regulates the composition of R-class proteins, establishing a Michaelis-Menten relationship between the translation rate and the growth rate³⁴. When the growth rate is sufficiently high (μ > 0.5 h⁻¹), the relationship between the abundance of R-class proteins and the growth rate can be reliably modeled using a constant translation rate at a given temperature^28,34. This indicates that the synthesis flux rate of unfolded proteins is proportional to the mass fraction of active R-class proteins, expressed as:

$$v_{\mathrm{AA}\mapsto \mathrm{UP}}=k_r\left(M_r-M_0\right)\to J_{\mathrm{AA}\mapsto \mathrm{UP}}=k_r\left({\varphi }_r-{\varphi }_0\right),$$ 8

where k_r indicates the bacterial “translational capacity”²⁸, and M₀ is the mass of inactive R-class proteins, which do not engage in translation for various reasons³⁴, and ${\varphi }_0=\frac{M_0}{M_{\mathrm{NP}}}$ denotes the mass fraction of inactive R-class proteins. We assume that Eq. (8) is valid across the full range of growth rates considered.

A complex network of catabolic, anabolic, and transport reactions mediates the supply of amino acids. In batch culture, despite a gradual decline in substrate concentration due to consumption, E. coli maintains a relatively stable growth rate over an extended period⁵¹. This observation suggests that there exist enzymes that remain saturated, implying that the amino acid supply flux rate is proportional to the abundance of these rate-limiting enzymes. Thus, we treat these reactions as the bottleneck for growth²⁸. We further assume that the mass fraction of these rate-limiting enzymes is proportional to ϕ₁. This leads to the relationship:

$$v_{\mathrm{S}\mapsto \mathrm{AA}}=k_1{M}_1\to J_{\mathrm{S}\mapsto \mathrm{AA}}=k_1{\varphi }_1.$$ 9

Here, k₁ represents the bacterial “amino acid supply capacity,” which is contingent upon the nutritional quality of the medium. Specifically, a richer medium requires a smaller protein investment to maintain the same amino acid supply flux. The intercept term denotes the basal, growth-rate-independent expression level of enzymes in the amino acid supply process, which is considered part of the Q-sector⁵². This relationship has also been identified in other studies^28,37,53,54. The parameters k₁ and k_r are associated with the rates of their respective processes. Biochemical reaction rates are typically temperature-dependent. Consequently, both k₁ and k_r are expected to exhibit temperature dependence. As will be shown later, this dependence contributes to the variations in bacterial growth rate within the normal temperature range.

Consistent with the formulations established for other protein classes, we assume that the degradation flux rate of aberrant proteins is proportional to the abundance of P₃-class enzymes. This leads to the following expression:

$$v_{\mathrm{AP}\mapsto \mathrm{AA}}=k_3{M}_3\to J_{\mathrm{AP}\mapsto \mathrm{AA}}=k_3{\varphi }_3.$$ 10

Here, k₃ denotes the bacterial “aberrant protein degradation capacity,” and the intercept term is also incorporated into the Q-sector. As will be shown later, unless protein degradation is highly efficient, the impact of P₃-class protein is negligible within our modeling framework.

To facilitate proper protein folding, bacteria express molecular chaperones as auxiliary factors. Given the relatively low abundance of chaperones within E. coli (e.g., ϕ₂ ≈ 0.05 in glucose, ϕ₂ ≈ 0.08 in LB at 37 °C³⁶), this chaperone-assisted folding mechanism must be highly efficient, with the rate-limiting bottleneck residing in the quantity of unfolded proteins. We can assume that the flux rates for proper folding (v_UP↦NP) and misfolding (v_UP↦AP) are both proportional to the amount of unfolded proteins, i.e., v_UP↦NP ∝ M_UP, v_UP↦AP ∝ M_UP. The partitioning between these two competing pathways is determined by the availability of chaperones that facilitate proper folding.

Based on these characteristics described above, we employ a Michaelis-Menten-like formulation to model chaperone-assisted protein folding. The fluxes are given by:

$$\begin{array}{ccc}\hfill v_{\mathrm{UP}\mapsto \mathrm{NP}}& \hfill =\hfill & k_2{M}_{\mathrm{UP}}\frac{M_2}{M_2+{\mathrm{\Psi }}_f},\hfill \\ \hfill v_{\mathrm{UP}\mapsto \mathrm{AP}}& \hfill =\hfill & k_4{M}_{\mathrm{UP}}\frac{{\mathrm{\Psi }}_f}{M_2+{\mathrm{\Psi }}_f},\hfill \end{array}$$ 11

where k₂ and k₄ denote the rate constants for normal folding and aberrant folding, respectively, and Ψ_f represents the propensity for misfolding relative to normal folding. Within this formulation, an increase in chaperone abundance drives the factor $\frac{M_2}{M_2+{\mathrm{\Psi }}_f}$ closer to 1, thereby increasing the probability that unfolded proteins achieve their native conformation. We define:

$${\psi }_f=\frac{k_4{\mathrm{\Psi }}_f}{k_2{M}_{\mathrm{NP}}},$$ 12

referred to as “folding pressure,” reflecting the frustration level of proper protein folding. This quantity is influenced by the interplay between dynamics of folding and unfolding, as well as the kinetics of normal and aberrant folding processes. As will be demonstrated in the subsequent sections, variations in ψ_f are the primary factors driving significant physiological changes observed in bacteria under both heat and cold stress. Based on the relationship J_UP↦NP + J_UP↦AP = J_AA↦UP, we can express:

$$\begin{array}{ccc}\hfill J_{\mathrm{UP}\mapsto \mathrm{NP}}& \hfill =\hfill & J_{\mathrm{AA}\mapsto \mathrm{UP}}\frac{{\varphi }_2}{{\varphi }_2+{\psi }_f},\hfill \\ \hfill J_{\mathrm{UP}\mapsto \mathrm{AP}}& \hfill =\hfill & J_{\mathrm{AA}\mapsto \mathrm{UP}}\frac{{\psi }_f}{{\varphi }_2+{\psi }_f}.\hfill \end{array}$$ 13

Given that bacteria have evolved to maximize their growth rates during the exponential phase⁵⁵, it is assumed that their protein allocation is optimized for this purpose. Therefore, our model can be represented as the following constrained optimization problem:

$$\begin{array}{c}\mathrm{Maximize}:\mu \hfill \\ \mathrm{Subject}\mathrm{to}:\mu ={\displaystyle \frac{k_r({\varphi }_r-{\varphi }_0){\varphi }_2}{{\varphi }_2+{\psi }_f},}\hfill \\ {\varphi }_r+{\varphi }_1+{\varphi }_2+{\varphi }_3=1-{\varphi }_q,\hfill \\ k_1{\varphi }_1+k_3{\varphi }_3=k_r({\varphi }_r-{\varphi }_0),\hfill \\ {\displaystyle \frac{k_r({\varphi }_r-{\varphi }_0){\psi }_f}{{\varphi }_2+{\psi }_f}-k_3{\varphi }_3\ge 0,}\hfill \\ {\varphi }_r-{\varphi }_0,{\varphi }_1,{\varphi }_2,{\varphi }_3\ge 0.\hfill \end{array}$$ 14

The correlation between folding pressure and nutrient quality

The solution to this optimization problem (Eq. (14)) depends critically on the functional form of the folding pressure ψ_f.

Folding pressure is affected by various factors^29,56,57. In addition to temperature, it can be influenced by other environmental stresses, including osmotic pressure⁵⁶, oxidative stress²⁹, and pH^29,57. Notably, some proteomic datasets indicate that at the normal temperature, chaperone expression levels correlate with nutrient quality in the medium (Fig. 3a and refs. ^36,58,59). This observed correlation indicates a potential relationship between the amino acid supply capacity k₁ and the folding pressure ψ_f.

Fig. 3. Variation patterns of model parameters.

a Chaperone allocation scales with growth rate across nutrient conditions. Chaperone expression levels (ϕ₂) are plotted against growth rate (μ) from multiple proteomic datasets (see refs. ^36,58,59). The dark green and black dashed lines illustrate results derived from Eq. (16), using parameters chosen for strains BW25113 (ψ_f,1 = 0.0062h, k_f,0 = 0.0054) and NCM3722 (ψ_f,1 = 0.0024h, k_f,0 = 0.0053), respectively. b Temperature dependence of the translational capacity k_r. Open circles denote measured relative ER⁶⁶. A piecewise linear fit of the ER data (separately for T≥21 °C and T < 21 °C) is shown by the blue line. The green line represents the k_r value fitted from temperature-dependent growth data, and the red line is derived from the relative translation rate at 0 °C⁸³. Additional ER data from refs. ^70,94 are included for comparison. The abbreviation ER denotes the elongation rate of translation.

Furthermore, since unfolded protein accumulation leads to heat shock protein expression⁶⁰ and a reduction in translation rate can enhance chaperone activity⁶¹, we hypothesize that folding pressure ψ_f is associated with the concentration of unfolded proteins. Alternatively, extending this further, it could be associated with the translation flux rate, k_r(ϕ_r − ϕ₀). In light of the steady-state balance between amino acid production and consumption, k₁ϕ₁ + k₃ϕ₃ = k_r(ϕ_r − ϕ₀), this coupling could also explain the observed association between k₁ and ψ_f. Assuming an second-order protein misfolding process, we propose the following functional form for ψ_f (see Supplementary Note 3):

$${\psi }_f={\psi }_{f,0}+{\psi }_{f,1}{k}_r\left({\varphi }_r-{\varphi }_0\right).$$ 15

This indicates that an increase in cellular protein synthesis capacity requires a corresponding upregulation of chaperone demand to maintain proteostasis.

Under these constraints and with the assumption that the cost of acquiring amino acids from substrate is consistently lower than that of degrading aberrant proteins (k₃ < k₁), the optimization of protein allocation (Eq. (14)) yields the following results (see Method for details; for cases where k₃ > k₁, see Supplementary Note 4):

$$\begin{array}{ccc}{\mu }_{\mathrm{opt}}\hfill & =\hfill & \frac{k_1{k}_r{\varphi }^{\mathrm{**}}}{k_1+k_r}\frac{1}{{\left(\sqrt{\frac{k_1{k}_r}{k_1+k_r}{\psi }_{f,1}+k_{f,0}}+\sqrt{1+k_{f,0}}\right)}^2},\hfill \\ {\varphi }_{r,\mathrm{opt}}\hfill & =\hfill & \frac{k_1{\varphi }^{\mathrm{**}}}{k_1+k_r}\frac{\sqrt{1+k_{f,0}}}{\sqrt{\frac{k_1{k}_r}{k_1+k_r}{\psi }_{f,1}+k_{f,0}}+\sqrt{1+k_{f,0}}}+{\varphi }_0,\hfill \\ {\varphi }_{1,\mathrm{opt}}\hfill & =\hfill & \frac{k_r{\varphi }^{\mathrm{**}}}{k_1+k_r}\frac{\sqrt{1+k_{f,0}}}{\sqrt{\frac{k_1{k}_r}{k_1+k_r}{\psi }_{f,1}+k_{f,0}}+\sqrt{1+k_{f,0}}},\hfill \\ {\varphi }_{2,\mathrm{opt}}\hfill & =\hfill & {\varphi }^{\mathrm{**}}\frac{\sqrt{\frac{k_1{k}_r}{k_1+k_r}{\psi }_{f,1}+k_{f,0}}}{\sqrt{\frac{k_1{k}_r}{k_1+k_r}{\psi }_{f,1}+k_{f,0}}+\sqrt{1+k_{f,0}}},\hfill \\ {\varphi }_{3,\mathrm{opt}}\hfill & =\hfill & 0.\hfill \end{array}$$ 16

Here, $k_{f,0}=\frac{{\psi }_{f,0}}{{\varphi }^{\mathrm{**}}}$, ϕ^** = 1 − ϕ_q − ϕ₀ is the fraction of protein that can be allocated under different conditions.

From the perspective of Eq. (16), a reduction in translational capacity k_r—for instance, due to translation-inhibiting antibiotic treatment—should lead to a reduction in folding pressure ψ_f. This prediction appears consistent with the effects of chloramphenicol, which has been shown to inhibit the expression of heat shock proteins⁶² (primarily composed of chaperones). However, previous experimental studies⁶³ report an increase in the fraction of chaperone ψ₂ (Supplementary Fig. 1a) in response to the effects of kanamycin, another translational inhibitor used to validate protein allocation models²⁸. These observations indicate that the effect of antibiotics on translational capacity and folding pressure is likely compound-specific.

The strength of the relationship between folding pressure and nutrient quality varies across strains, and in some strains, this correlation is nearly negligible (Supplementary Fig. 1c and refs. ^64,65). Since confirming the presence and magnitude of such a relationship in the specific strains from which our growth rate data are derived is challenging, and given that nutrient-induced variations in chaperone fractions are relatively minor (approximately 0.04 in BW25113³⁶) compared to those of R-class proteins, we assume that ψ_f,1 = 0 in our subsequent analysis and abbreviate ψ_f,0 as ψ_f and k_f,0 as k_f. This is equivalent to treating folding pressure as constant at a given temperature. In fact, a non-zero ψ_f,1 is also capable of capturing the variations observed in growth rate across different temperatures (Supplementary Fig. 2a, Supplementary Table 1).

Temperature dependence of key parameters

To model temperature effects on E. coli physiology, we extend the aforementioned framework by incorporating the temperature dependence of critical kinetic parameters. The translational capacity k_r is directly associated with the peptide chain elongation rate (ER)²⁸. Assuming a stable translational system, where the rRNA fraction in total RNA, the mass ratio of R-class proteins to rRNA, and the composition of the R-class proteome remain constant, k_r is directly proportional to ER²⁸. Experimental data show that the ER of E. coli exhibits an Arrhenius-type temperature dependence in the normal and heat regions but declines more steeply in the cold region (Fig. 3b, and ref. ⁶⁶). For model simplicity, we assume that k_r follows the Arrhenius equation throughout the entire temperature range. Similarly, assuming that the amino acid supply is governed by a single rate-limiting enzymatic step, we propose that k₁ also exhibits Arrhenius-like temperature dependence. That is,

$$k_i=k_i\left(T_n\right)\exp \left[-\frac{E_i}{R}\left(\frac{1}{T}-\frac{1}{T_n}\right)\right],i\in \left\{1,r\right\},$$ 17

where E_i is the equivalent activation energy of k_i, R is the ideal gas constant, and T_n = 310.15K is the normal cultivation temperature for E. coli. We maintain the assumption that k₃ < k₁ holds at all temperatures. Consequently, the specific temperature dependence of k₃ does not influence the resulting optimal allocation (Eq. (16)).

Folding pressure increases with temperature due to the increased difficulty of achieving native protein conformations. Conversely, it also rises as temperature decreases because of slowed folding kinetics. Based on the definition of ψ_f (Eq. (12)), we assume that it takes the following functional form:

$$k_f=\frac{{\psi }_f}{{\varphi }^{\mathrm{**}}}\propto \frac{1}{k_2}\exp \left[-\frac{\mathrm{\Delta }G}{RT}\right],$$ 18

where k₂ is the folding rate, given by Eq. (17), and ΔG quantifies the thermodynamic stability of the folded state. This leads to the following expression⁴⁶:

$$k_f=k_f\left(T_n\right)\exp \left[-\frac{{E^{\prime }}_2\left(N\right)}{R}\left(\frac{1}{T}-\frac{1}{T_n}\right)+\frac{\mathrm{\Delta }{C}_p\left(N\right)}{R}\mathit{ln}\frac{T}{T_n}\right].$$ 19

Here, $E_2^{\mathrm{\prime }}=-E_2+\mathrm{\Delta }H(T_h,N)-\mathrm{\Delta }{C}_p\left(N\right)T_h$ and T_h = 373.5K is a reference temperature. In this expression, ΔH(T_h, N) and ΔC_p(N) are the enthalpy and heat capacity of unfolding for proteins of length N, respectively,

$$\begin{array}{ccc}\mathrm{\Delta }H(T_h,N)\hfill & =\hfill & (4.0N+143)\mathrm{kJ}/\mathrm{mol},\hfill \\ \mathrm{\Delta }{C}_p\left(N\right)\hfill & =\hfill & (0.048N+0.85)\mathrm{kJ}/(\mathrm{mol}\cdot \mathrm{K}).\hfill \end{array}$$ 20

This formula is derived from fitting in vitro data and does not account for the potential intracellular effect of protein stability⁶⁷. Therefore, the parameter N is not fixed to the mean polypeptide length derived from proteomic data but is treated as a free parameter.

In our model, the values of certain parameters at the normal temperature T_n have been determined in prior studies^28,29; these values are adopted directly from existing literature. We further assume that the parameters ϕ₀ and ϕ^** remain constant across the temperature range under investigation. The model is thus reduced to five adjustable parameters (k₁(T_n), E₁, E_r, E₂, N). For the specific culture medium, k₁(T_n) was also predetermined in prior studies²⁹.

During the fitting (see Method for details about the fitting procedure) of the temperature-dependent growth rate, the estimated values of E₁ and E_r are approximately equal throughout. Moreover, the covariance matrix obtained from the fitting indicated significant instability in these parameter estimates. This pattern suggests that a constraint of the form E₁ = E_r may be intrinsic to our model. The imposition of this constraint is empirically well justified, as it produces nearly identical fitting quality while substantially reducing parameter uncertainty (see Supplementary Note 5). Therefore, all fitting reported in the main text was performed under this constraint, reducing the model to four free parameters.

Variation in folding pressure causes the temperature-dependent growth curves to deviate from the Arrhenius law

Fitting our model to temperature-dependent growth data^66,68 successfully reproduces the experimental growth curves (Fig. 4a, parameters in Table 1). A significant finding is that within the normal range, where folding pressure remains relatively stable (Fig. 4b), the growth rate is primarily determined by the temperature dependence of amino acid supply capacity k₁ and translational capacity k_r,

$$\mu \propto {\varphi }^{**}\frac{k_1{k}_r}{k_1+k_r}.$$ 21

Within this range, the growth rate variation closely follows the Arrhenius law, as illustrated by the black dashed line in Fig. 4(a). Accordingly, the growth rate can be characterized by an effective activation energy,

$$E_{\mu }=\frac{k_r{E}_1+k_1{E}_r}{k_1+k_r}=E_r\approx 69.9\mathrm{kJ/mol}.$$ 22

Interestingly, this effective activation energy derived from our model aligns closely with the reported activation energy for ribosome-catalyzed peptide bond formation, which is 71 kJ/mol⁶⁹.

Fig. 4. Folding pressure significantly influences the variation in growth rate under heat and cold stress.

a Model fitting to temperature-dependent growth curves. Experimental growth data (points) are accurately described by the model (brown dashed line) using the fitted parameters (see Supplementary Table 1 for the coefficient of determination). The black dashed line shows the predicted growth rate if folding pressure were fixed at its T_n value (ψ_f = ψ_f(T_n)). All data correspond to the same culture medium, with k₁(T_n) = 20 h⁻¹²⁹ (data from refs. ^66,68 respectively). b Temperature dependence of folding pressure k_f = ψ_f/ϕ^**. Folding pressure remains relatively low within the normal temperature range but rises sharply outside this range. c Temperature-induced shift in flux toward aggregation. As folding pressure varies with temperature, the proportion of the net translation flux directed toward protein aggregates changes accordingly. At sufficiently high temperatures, the majority of the translation flux is diverted into aggregates.

Table 1.

The key parameters employed in this article

Parameter	Symbol	Value	Comment
normal culture temperature	Tn	310.15K(37 °C)
allocable protein mass fraction	ϕ**	0.48428	fixed
inactive R-class protein mass fraction	ϕ0	0.06628	fixed
mass ratio of R-class protein and RNA	ρ	0.7628	fixed
"AA supply capacity” under Tn for Neidhardt’s defined rich media + glucose	k1(Tn)	20 h−129
"translational capacity” under Tn without translation limitation	kr(Tn)	6 h−128
"AA supply capacity” under Tn for basal salts medium75 + succinate	k1(Tn)	1.40 h−1	calculated from the growth rate36, see Method for details
"folding pressure” at Tn	ψf(Tn)	0.006129	give kf(Tn) = 0.0126
activate energy of folding	E2	367 ± 53 kJ/mol	fit result
activate energy of “AA supply capacity” and “translational capacity”	E1 and Er	69.9 ± 3.0 kJ/mol	fit result
effective protein length in protein stability	N	558 ± 43	fit result

A comparison (Fig. 3b) shows that a subset of the experimental ER data⁷⁰ aligns well with our model-fitted predictions for k_r. This partial agreement also suggests that the temperature dependence of ER may be influenced by the specific experimental methodology employed.

The model fitting results indicate that, when E. coli is cultured in the heat region, the rapidly increasing folding pressure severely impairs the cell’s ability to achieve proper protein folding. As a result, a substantial fraction of newly translated proteins misfold and form aggregates (Fig. 4c). Consequently, although there is an enhanced capacity in other enzymatic processes, the overall growth rate declines sharply. Analogous behavior in the cold region: as temperature decreases, the concurrent increase in folding pressure and decrease in reaction capacities leads to a more pronounced decline in growth rate than that predicted by the Arrhenius relationship (Eq. (1), Fig. 4a, b).

The model demonstrates robust generalizability to other growth rate data. By refitting two parameters—the amino acid supply capacity k₁(T_n) and the activation energy E₁ = E_r—to independent experimental data^71,72, the model successfully reproduces temperature-dependent growth rate curves across various culture media (Fig. 5a).

Fig. 5. Model validation across growth conditions and proteomic data.

a Application to temperature-dependent growth curves in different media. The model was fitted to independent growth data (see refs. ^71,72), with E_r and k₁(T_n) treated as free parameters (for Trypticase-Soy, k₁(T_n) = 27.1 ± 4.6h⁻¹, E_r = 75.7 ± 4.4kJ/mol; for basic salts, k₁(T_n) = 9.55 ± 0.92 h⁻¹, E_r = 70.4 ± 5.3 kJ/mol). Other parameters were set to the values in Table 1. b Prediction of chaperone allocation ϕ₂. Model predictions from Eq. (16) accurately capture the trend of experimental chaperone mass fraction across temperatures (data from refs. ^36,56,64,68, respectively). c Prediction of other sector allocation. The model also predicts the temperature dependence of the translation-related (ϕ_r) and amino acid supply (ϕ₁) protein mass fractions. In comparison with the dataset⁶⁴, ϕ₀ = 0.17 was used, consistent with the reported ϕ₀ ≥ 0.1 in this dataset (data from refs. ^64,68, respectively).

The model captures effects of temperature on protein allocation

In our previous fitting of growth rates, we did not incorporate proteomic information. Based on the analytical solution derived from our model (Eq. (16)), we can utilize the parameters obtained from growth rate fitting to predict variations in protein allocation across different temperatures (Fig. 5b, c; see Method for classification about R and P₂).

As shown in Fig. 5b, c, the model predictions closely align with the observed variations⁶⁸ in both R-class and P₂-class proteins within the heat region. With increasing temperature, the folding pressure k_f increases sharply, prompting E. coli to upregulate chaperone expression to ensure proper protein folding and reduce metabolic waste. This adaptive response occurs at the expense of allocations to both amino acid supply and translational machinery. As a result, the chaperone fraction increases in the heat region, while the proportion of translation-associated proteins decreases. This trend is evident in the recent independent proteomic dataset⁶⁴, which shows substantial agreement with our model predictions within the normal and heat ranges, with only a nearly constant offset in ϕ_r. Such discrepancy potentially stems from variations in bacterial strains or differences in experimental conditions, as supported by the reported ϕ₀ > 0.1 in this studied proteomic data—approximately double the value of ϕ₀ = 0.066 adopted in our analysis.

Within the normal temperature range, folding pressure becomes negligible, resulting in a nearly constant chaperone allocation. Under the assumption that E₁ = E_r, the ratio $\frac{k_r}{k_1}$ is temperature-independent, resulting in approximately stable proportions of both R-class and P₁-class proteins. Consequently, the model predicts largely invariant protein allocation across this range, which is consistent with the widely reported experimental results⁶⁶.

However, within the cold region, the model predictions exhibit some discrepancies from the experimental trends. The observed discrepancy arises from our assumption that ψ_f increases as the folding rate decreases, resulting in predicted reductions in ϕ_r and increases in ϕ₂ (Eqs. ((16), (17), (18))). Although existing proteomic data^64,68 somewhat reflect this directional trend, the observed changes are less pronounced than the model predicts. The available dataset encompasses a limited temperature range in the cold region, which limits its ability to characterize proteomic shifts at lower temperatures comprehensively. Nonetheless, the discrepancies within the data range are relatively modest.

The model quantifies the temperature-dependent expression of β-galactosidase from a constitutive promoter

P₁-class proteins comprise constitutively expressed proteins²⁸ whose expression is regulated indirectly through competition for transcriptional and translational resources. A canonical example is the constitutively expressed β-galactosidase, which is widely employed as a reporter gene due to its easily measurable enzymatic activity. Prior research^28,53 indicates that under carbon or nitrogen limitation, or in the presence of translation inhibitors, the activity of constitutively expressed β-galactosidase closely aligns with the allocation trend of P₁-class proteins. In this study, we use the activity of constitutively expressed β-galactosidase (which is proportional to its amount per unit biomass) across different temperatures as a proxy to infer temperature-dependent changes in the relative abundance of P₁-class proteins.

Given that cellular mass is predominantly composed of proteins and RNA, and that RNA mass is proportional to M_R²⁸, we can establish a relationship between constitutively expressed β-galactosidase activity Z and the mass fractions of P₁-class proteins ϕ₁ and R-class proteins ϕ_r, represented as²⁹,

$$Z\propto \frac{{\varphi }_1}{\rho +{\varphi }_r},$$ 23

where ρ = 0.76 is the mass ratio of R-class protein to RNA.

We propose that the relationship established in Eq. (23) is valid across varying temperatures, with its proportionality coefficient remaining independent of temperature. The parameter ρ has been empirically⁷³ demonstrated to remain constant across a relatively wide temperature range. Since rRNA constitutes the vast majority of cellular RNA²⁸ and exhibits synchronized regulation with ribosomal proteins⁷⁴, it is reasonable to extrapolate that ρ stays constant throughout the experimental temperature spectrum, absent any evidence to the contrary. As shown in Fig. 6a, the model predictions generated from Eq. (23) using our previously determined temperature-dependent protein allocation closely align with the experimentally observed relative expression profile of the constitutively expressed β-galactosidase⁷⁵.

Fig. 6. Model predictions for temperature-dependent physiological traits.

a Temperature dependence of relative β-galactosidase activity. Experimental measurements⁷⁵ are compared with model predictions from Eq. (23), using k₁(T_n) = 1.40 h⁻¹. The predicted trend is robust to the value of k₁. b Cell volume variation with temperature in nutrient-rich medium. The model prediction based on Eq. (24) is plotted alongside experimental data⁷⁹. The model captures the overall temperature dependence of cell size.

In the cold region, both experimental data and model predictions show a decrease in the constitutive expression of β-galactosidase. This decline is mainly due to the reduced allocation of P₁-class proteins as the temperature decreases. Within the normal temperature range, the stability of both ϕ_r and ϕ₁ ensures a constant RNA mass fraction, leading to a relatively stable expression of β-galactosidase. In the heat region, our model predicts a decrease in constitutive β-galactosidase expression, driven by the reduction in ϕ₁. This prediction is awaiting future experimental validation.

The model captures effects of temperature on cell size

At a fixed temperature, bacterial cell size exhibits an exponential dependence on growth rate⁷⁶. Based on the threshold accumulation hypothesis for bacterial cell size control, protein allocation models have been developed to explain the relationship between cell size and nutrient quality^33,40 at a specific temperature. The analysis of correlations between cell size and the proportions of various protein sectors reveals that the division-related sector remains minimal across different nutrient conditions and is co-regulated with P₁-class proteins^33,77. This conclusion is further supported by dynamic single-cell experiments⁷⁸. Consequently, the relationship between E. coli cell size and the allocation ratio ϕ₁ can be mathematically represented by the equation:

$$V\propto \frac{1}{{\varphi }_1},$$ 24

which indicates an inverse proportionality between E. coli cell size and the proportion of P₁-class proteins.

Temperature affects bacterial cell size, causing morphological changes at both low and high temperatures^64,79,80. We hypothesize that the established relationship in Eq. (24) is valid across different temperatures and that the proportional coefficient is independent of temperature.

As shown in Fig. 6b, the model effectively captures the size characteristics of E. coli within a suitable temperature range⁷⁹. Cell volume increases when temperatures deviate from the normal range, and changes in folding pressure drive this response. At elevated temperatures (e.g., T ≥ 42 °C), our model predicts a rapid increase in bacterial cell volume, which may correspond to filamentation observed under heat stress⁸⁰. One possible explanation is that we consolidate all growth-rate-independent components of the P₁-class protein into the Q sector.

Recent studies⁸¹ indicate that high expression of the chaperone HtpG can interfere with the assembly of the cell division machinery and the formation of the FtsZ-ring, resulting in filamentation. Furthermore, the polymerization of FtsZ is additionally inhibited by chaperone ClpX⁸². These findings may provide a possible explanation for the validity of Eq. (24) across different temperatures: in our model with constant nutrient conditions, the reduction in ϕ₁ is predominantly driven by increased chaperone expression.

Discussion

By incorporating temperature dependence into the established protein allocation model and calibrating it with experimental growth data, we quantitatively characterize the relationship between growth rate of E. coli and temperature. This calibrated model enables us to predict temperature-dependent shifts in protein allocation. Furthermore, under the assumption that certain physiological parameters—such as ρ—are stable across temperature, we can explain the temperature dependence of cell volume and the constitutive expression of β-galactosidase.

Our model indicates that within the normal temperature range, folding pressure is low, implying a correspondingly low cellular demand for chaperones. The model fitting further supports the relationship E_r = E₁, suggesting that variations in growth rate within this regime are governed by the activation energies associated with protein translation and amino acid supply. Thus, the apparent “activation energy” for growth (E_μ) within the normal range is given by E_μ = E_r = E₁.

However, ER data (Fig. 3b) show a slight discrepancy between the activation energy of translation and the “activation energy” of the growth within the normal range⁶⁶. This discrepancy could stem from experimental methodology or may indicate a genuine difference between the activation energy for amino acid supply E₁ and that for translation E_r. If the latter holds, E_μ would represent a combined effect of E₁, E_r, and the nutrient quality of the culture medium, as formulated in Eq. (22). This interpretation is consistent with a recent study⁶⁴ reporting that E_μ varies with medium composition.

When temperature deviates from the normal range, our model suggests that increased folding pressure requires higher chaperone expression in E. coli. This reallocation of cellular resources reduces the proportion of R-class proteins, ultimately leading to a slower overall growth rate. While this mechanism provides a plausible explanation for growth limitation in the heat region, its relevance to the cold region remains less clear. At low temperatures, protein synthesis in E. coli is primarily constrained by the accumulation of inactive ribosomes⁶⁶ (Supplementary Note 1) and the stabilization of mRNA secondary structures⁶. Substantial evidence^66,83 further indicates an increased activation energy for protein synthesis at low temperature, which would lead to a steeper decline in translational capacity at lower temperatures. In contrast, although folding pressure could be a potential contributing factor⁸⁴, its role in the low-temperature physiology remains underexplored in the literature.

It is generally accepted that the physiological state of E. coli remains relatively stable across the normal temperature range, as demonstrated by several experimental studies^73,85. However, proteomic data^64,68 reveal subtle yet consistent variations in the abundance of translation-associated proteins even within this temperature range. This suggests that E. coli may continuously fine-tune its translational machinery in response to temperature changes, rather than maintaining a fixed state.

The role of chaperone proteins in the cold region remains poorly characterized. On one hand, many psychrophilic bacteria exhibit upregulation of chaperone genes⁸⁶, and the cold-induced inactivation of chaperones is known to impair E. coli growth at low temperatures^87,88. The heterologous expression of psychrophilic chaperones has even been shown to enhance E. coli growth at low temperatures⁸⁶. On the other hand, overexpression of heat shock proteins (primarily chaperones) in E. coli inhibits cellular growth⁸⁹. This apparent contradiction suggests that E. coli, as a mesophile, may lack efficient chaperone-mediated adaptation to cold stress. Potential missing adaptive mechanisms or other intrinsic bacterial limitations may result in the cells growing in a suboptimal state under certain environmental conditions⁵⁵. This may partially explain the discrepancies between our model predictions and experimental data.

The temperature dependence of E. coli cell volume remains incompletely characterized. Data from multiple studies indicate that the nutrient quality strongly influences volumetric changes in response to temperature elevation. When the cultivation temperature is increased from 37°C to 42°C, E. coli cell volume decreases in glucose-supplemented minimal medium⁹⁰, but increases in nutrient-rich medium⁷⁹ or in MR medium supplemented with glucose⁸⁰. Notably, the increase in cell volume observed in LB medium is primarily due to changes in cell width⁶⁴, whereas in MR medium, it is associated with an increase in cell length⁸⁰. Conversely, when the temperature decreases, cell volume increases in both minimal and complex media⁷⁹. These context-dependent results highlight the need for further experimental work to disentangle the interplay among bacterial cell size, temperature, and nutrient conditions.

A comprehensive protein allocation model should be capable of elucidating the physiological impacts of various orthogonal stresses on bacterial systems. As validated in our earlier work²⁹, the model demonstrates consistent applicability across various stress conditions. However, a more refined partitioning of the proteome may introduce correlations among parameters characterizing distinct functional sectors (Supplementary Fig. 1). For instance, while various translational inhibitors effectively validate a coarse-grained model consisting solely of the R, P₁, and Q sectors²⁸, they exert differential effects on chaperone expression levels⁵⁶. This underscores the necessity of rigorous experimental design for validating refined protein allocation models.

The previous model³⁹ also effectively captured temperature-dependent growth and protein allocation by explicitly incorporating chaperones. In contrast to our experimentally motivated assumption—that flux rates scale linearly with the mass of corresponding protein classes—their approach conceptualizes amino acid synthesis and translation as enzymatic kinetic processes. It is noteworthy that when the translation half-saturation constant is very low, and the amino acid synthesis efficiency is at most marginally higher than translation efficiency, the effective translation efficiency in their model approaches a constant. This result is consistent with our assumed formula (Eq. (8)). Additionally, incorporating the effect of substrate concentration on amino acid synthesis efficiency, the enzymatic kinetics related to amino acid synthesis become consistent with our assumption (Eq. (9)). As a result of the conceptual differences in defining protein availability, we employ a distinct functional form to characterize the temperature dependence of folding pressure. Consequently, while both models exhibit strong explanatory power for variations in growth rate, discrepancies emerge in their predictions regarding protein allocation, such as the variations in ϕ₂ and ϕ_r within the normal temperature range. Beyond this, our model provides additional explanatory insights into the temperature dependence of bacterial cell volume and constitutive β-galactosidase expression. We believe that this model and our model are complementary in nature.

In the previous analysis of bacterial physiology across temperatures, we assumed that protease activity was inefficient. Thus, the model predicts minimal expression of proteases across temperatures, and ϕ₃ = 0. This is consistent with quantitative proteomic data^36,64, which show that protease levels remain mainly at basal expression (Supplementary Fig. 5). Although protease upregulation is a documented part of the E. coli heat-shock response⁴, genome-wide data consistently show that ϕ₃ remains quantitatively small⁶⁴. In Supplementary Note 4, we explore scenarios in which this assumption is relaxed. The results show that as folding pressure increases, the fraction of expressed proteases rises in parallel with the fraction of chaperones. This pattern provides a plausible explanation for the observed upregulation of protease expression in minimal media at elevated temperatures^36,56,59,64.

Although a five-component protein allocation model was insufficient to fully capture E. coli’s temperature response, incorporating a chaperone sector significantly improves its ability to explain certain thermal adaptation phenomena. A more comprehensive model⁹¹ could provide a more complete representation. Moreover, the physiological interpretations provided by our model are based on the assumption that existing parameter values remain valid over an extended temperature range, even if some could be temperature-dependent. Consequently, predictions at extreme temperatures may be unreliable if these assumptions are not met. Given the scarcity of quantitative physiological parameters and the presence of conflicting experimental observations under these conditions, we anticipate that future work will elucidate the quantitative physiology of bacteria in such regimes. Such advances will substantially deepen our understanding of the fundamental principles governing bacterial growth in relation to temperature.

Methods

Optimization problem solving

Here we provide a concise description of solving the optimization problem (Eq. (14)). Although the flux rate of aberrant protein degradation by proteases, J_AP↦AA, is constrained by other flux rates according to J_AP↦PA = J_UP↦AP − J_AP↦AA≥0, we initially disregard this constraint because, under our model assumptions, the amount of protein aggregates does not affect the growth rate. To maximize growth, the two processes supplying amino acids, J_S↦AA = k₁ϕ₁ and J_AP↦AA = k₃ϕ₃, must collectively minimize the total proteomic cost required to deliver a given amino acid flux rate.

Assuming that the cost of acquiring amino acids from the external environment is always lower than that of obtaining them through aberrant protein degradation, i.e., k₁ > k₃, the optimal solution allocates a mass fraction of zero to the protease sector under all conditions, ϕ₃ = 0. This allocation automatically satisfies the constraint J_AP↦PA = J_UP↦AP − J_AP↦AA = J_UP↦AP≥0.

From the balance between amino acid supply and demand, J_S↦AA = J_AA↦UP, we obtain the relation ${\varphi }_1=\frac{k_r}{k_1}\left({\varphi }_r-{\varphi }_0\right)$. Given the constraint on the total allocation to these protein sectors, ϕ₁ + ϕ_r + ϕ₂ = 1 − ϕ_q, we obtain a direct relationship between ϕ_r and ϕ₂:

$$\left(1+\frac{k_r}{k_1}\right)\left({\varphi }_r-{\varphi }_0\right)+{\varphi }_2=1-{\varphi }_q-{\varphi }_0={\varphi }^{**}.$$

By substituting this relation into the expression for the growth rate μ, and differentiating with respect to either ϕ_r or ϕ₂, the optimal allocation of proteins (Eq. (16)) can be derived from the condition:

$$\frac{\mathrm{d}\mu }{\mathrm{d}{\varphi }_2}=0\mathrm{and}\frac{\mathrm{d}\mu }{\mathrm{d}{\varphi }_r}=0.$$

Classification for proteins class R, P₂

We adopt the protein classification scheme established by Proteomap⁹². Consistent with previous studies^28,93, the R-class proteins are defined as those encompassing functional modules involved in gene expression, specifically including “tRNA loading,” “translation factors,” “RNA polymerase,” and “ribosome.” The P₂-class proteins are defined as the cluster annotated “chaperones and folding catalysts.”

The quantitative proteomic dataset from prior studies⁶⁸ includes measurements for only 133 proteins. To enable direct comparison with model predictions, these data are linearly transformed based on a correlation derived from the proteomic datasets^36,64 (Supplementary Note 2).

Model fitting and protein allocation predicting

The model was fitted to temperature-dependent growth data^66,68—obtained under the same medium conditions—by the nonlinear least-squares regression using the scipy.optimize.curve_fit function in Python (see Supplementary Note 5 for the complete formula about μ). With the value of k_r and k₁ at 37 °C predetermined, the fitting procedure needs to determine the three remaining parameters, E_r, E₂, and N. The initial values were set as follows: E_r = 60 kJ/mol⁶⁸, N = 325 (the average protein length in E. coli⁴⁶), and E₂ = 0kJ/mol. In practice, the fitting results are robust to a wide range of initial values. When the value of k₁(T_n) for a given medium^71,72 is unknown, an initial value of k₁(T_n) = 20h⁻¹ is assigned for the fitting process.

We obtained the values of k₁, k_r, and ψ_f across temperature by fitting the model solely to growth rate data. Using the optimal solution of the model (Eq. (16)), we then predicted protein allocation across different temperatures.

Inferring k₁(T_n) for basic salts + succinate

The phenomenological parameter k₁ is not available for the medium “basic salts + succinate” used in ref. ⁷⁵ or other succinate-supplemented minimal media. Assuming that folding pressure and translational capacity remain constant at a given temperature, k₁ can be estimated using Eq. (21). If we further assume that the salt composition of the minimal medium has a negligible influence on k₁, then the reported k₁(T_n) value for M9 + glucose²⁹, together with the measured growth rate of E. coli in M9 + succinate and M9 + glucose³⁶, can be used to estimate k₁(T_n) ≈ 1.40 h⁻¹ for basic salts + succinate medium.

Author contributions

D.W., Q.Z. and H.S. conceived the study and developed the theory. D.W. performed the investigation and wrote the draft. D.W., Q.Z., and H.S. discussed, reviewed, and revised the paper.

Data Availability

The data used in this paper can be found or generated using the provided Python code in the GitHub repository https://github.com/DeyuWang-itp/protein_allocation.

Code availability

The code used in this paper can be found in the same GitHub repository.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41540-026-00693-4.