Quantifying the entropic cost of cellular growth control

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Quantifying the entropic cost of cellular growth control

Daniele De Martino, Fabrizio Capuani, and Andrea De Martino

Phys. Rev. E 96, 010401(R) – Published 10 July 2017

Abstract

Viewing the ways a living cell can organize its metabolism as the phase space of a physical system, regulation can be seen as the ability to reduce the entropy of that space by selecting specific cellular configurations that are, in some sense, optimal. Here we quantify the amount of regulation required to control a cell's growth rate by a maximum-entropy approach to the space of underlying metabolic phenotypes, where a configuration corresponds to a metabolic flux pattern as described by genome-scale models. We link the mean growth rate achieved by a population of cells to the minimal amount of metabolic regulation needed to achieve it through a phase diagram that highlights how growth suppression can be as costly (in regulatory terms) as growth enhancement. Moreover, we provide an interpretation of the inverse temperature $β$ controlling maximum-entropy distributions based on the underlying growth dynamics. Specifically, we show that the asymptotic value of $β$ for a cell population can be expected to depend on (i) the carrying capacity of the environment, (ii) the initial size of the colony, and (iii) the probability distribution from which the inoculum was sampled. Results obtained for E. coli and human cells are found to be remarkably consistent with empirical evidence.

Received 5 May 2017
Revised 19 June 2017

DOI:https://doi.org/10.1103/PhysRevE.96.010401

Physics Subject Headings (PhySH)

Emergent biological functions from complex interactions Nonequilibrium statistical mechanics Population dynamics

Biological networks Gene regulatory networks Populations

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Daniele De Martino¹, Fabrizio Capuani^2,*, and Andrea De Martino^2,3

¹Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
²Soft and Living Matter Laboratory, Institute of Nanotechnology, Consiglio Nazionale delle Ricerche, 00185 Rome, Italy
³Italian Institute for Genomic Medicine, 10126 Turin, Italy

^*Present address: Istituto Nazionale di Fisica Nucleare, Unità di Roma 1, Rome, Italy.

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Vol. 96, Iss. 1 — July 2017

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Images

Figure 1
Both the enhancement ( $β > 0$ ) and the suppression ( $β < 0$ ) of growth imply regulatory costs in terms of reduced metabolic phase-space accessibility. (a) The $〈 λ 〉 / λ_{max}$ versus $I$ trade-off curve computed from the E. coli iJR904 genome-scale metabolic network model assuming a glucose-limited minimal medium ( $λ_{max} = 1 / h$ ). The blue, green, and orange markers denote the values of $〈 λ 〉 / λ_{max}$ (i) found for $β = 0$ (i.e., for an unbiased sampling of the feasible space), (ii) estimated for E. coli in the human gut (roughly corresponding to a doubling time of 40 h), and (iii) computed in [6] for two sets of GR distributions, respectively, described by the values $〈 λ 〉 / λ_{max} ≃ 0.28$ and $≃ 0.45$ . (b) The $〈 λ 〉 / λ_{max}$ versus $I$ trade-off curve computed from the human catabolic core network with $λ_{max} ≃ 0.046 / h$ , corresponding to a fast GR for cancer cells (doubling time $≃ 15$ h). Blue and green markers represent, respectively, the estimated renewal rates of various healthy human tissues and the estimated GRs of different types of cancers (data were obtained from [10]). Note that reducing the mean GR by a given factor generically requires a smaller amount of regulation compared to speeding it up by the same factor.
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Figure 2
Different metabolic strategies control growth suppression and enhancement. Modulation of selected metabolic variables with $〈 λ 〉 / λ_{max}$ for maximum-entropy flux configurations of E. coli found under glucose-limited (maximum uptake 4.4 mmol/ $g_{DW}$ /h) aerobic conditions with $λ_{max} ≃ 0.4 / h$ . (a) Mean fluxes through acetolactate synthase (ACLS), length-2 futile cycles, and glucose import. (b) Fraction of carbon excreted as ${CO}_{2}$ (cyan) and as other carbon compounds (brown). All fluxes are normalized to the maximum glucose import flux. $β$ increases from $- \infty$ to $+ \infty$ as $〈 λ 〉 / λ_{max}$ grows. The blue triangle marks the value of $〈 λ 〉 / λ_{max}$ corresponding to $β = 0$ . Faster mean GRs can be achieved by reducing energy dissipation by futile cycles and improving yields via respiration. Slower rates appear to require the creation of bottlenecks in key pathways like the aminoacid biosynthesis route.
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Figure 3
The asymptotic value of $β$ controlling the phase-space organization of a cell population increases as the initial population size decreases. (a) $β^{★}$ versus $ln (K / N_{0})$ for the minimal model of population growth described in the text, obtained by solving Eq. (8) for $q (λ) \propto {(1 - λ / λ_{max})}^{171}$ . Note that $〈 λ 〉$ increases with $β$ . (b) Inverse of the measured doubling times of different cancer cell types (proxies for the mean GRs) as a function of the inverse inoculum size $1 / N_{0}$ . Data were obtained from [21].
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