Are Microbes Thermodynamically Optimised Self-Reproducing Machines?

Are Microbes Thermodynamically Optimised Self-Reproducing Machines? Nima P. Saadat 1,† , Tim Nies 1,† , Yvan Rousset 1,† and Oliver Ebenhöh 1,2,∗ 1 Institute for Quantitative and Theoretical Biology, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, DE 2 Cluster of Excellence on Plant Sciences (CEPLAS), Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, DE * Correspondence: oliver.ebenhoeh@hhu.de † These authors contributed equally to this work.


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Life is certainly one of the greatest wonders on earth. The ability to grow and reproduce 28 Since ω f is the maximum value of the yield that is consistent with the laws of thermodynamics, 145 Roels defined the thermodynamic efficiency of a growth process as η th = Y sx /ω f , where Y sx is the 146 yield of biomass (x) given a certain substrate (s). By calculating the values of η th for different organic 147 compounds as energy source, he observed that highly reduced as well as highly oxidized substrates 148 lead to a rather low efficiency. Roels concluded that substrates with a degree of reduction above 4.2 149 (the degree of reduction of biomass) contain enough energy to allow that all carbon could in principle 150 be converted into biomass. In contrast to this, for compounds with a degree of reduction below 4.2, 151 this is not possible due to energetic reasons.

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How to incorporate these results into a mathematical model of bacterial growth is extensively 153 discussed by Esener in his report "Theory and Application of unstructured Models: Kinetic and 154 Energetic Aspects" [13]. Together with his paper published in 1982, in which Esener et al. describe 155 how to include aspects of varying biomass composition under a changing environment in a formal 156 description of bacterial growth [29], his work is a good example of an early attempt to develop 157 a consistent model explaining bacterial growth. These approaches are clearly related to modern 158 approaches to explain bacterial growth laws (see [30,31]). For a very well written discussion comparing 159 most of the efficiency measures discussed above (and much more), see [22,32]. 160 In combination, these investigations and concepts of thermodynamic efficiency in bacterial 161 growth provided the basis for further developments, including methods to estimate the energy and 162 entropy of formation for biomass, which represents a parameter of fundamental importance for 163 energetic calculations concerning life [27,28,33,34]. Particularly valuable were Battley's contributions 164 for estimating the Gibbs free energy of formation of biomass, ∆ f G b (-65.10 kJ/C-mol, γ b = 4.998, N 2 as 165 nitrogen source), and the enthalpy of formation for biomass, ∆ f H b . By using the well known relation of 166 entropy, enthalpy and Gibbs free energy, ∆G = ∆H − T∆S, he even attempted to estimate the entropy 167 of formation of biomass, but concluded that this method is too prone for errors, because it highly 168 depends on the quality of the approximation of enthalpy and Gibbs free energy [28].

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The introduction of novel experimental techniques, often referred to as high-throughput 'omics' investigate microbial growth rates, using flux balance analysis (FBA) as a method to predict metabolic fluxes in optimised growth scenarios [39]. Well curated genome-scale metabolic models can provide 179 valuable information about flux distributions of cells growing on different media. In combination with 180 experimental measurements, this approach has successfully been applied to investigate uptake rates of 181 carbon sources and byproduct secretion rates for several microbial organisms [40][41][42][43]. However, there 182 are a number of phenomena associated with microbial growth which cannot be explained by these 183 type of models. For example, they do not explain the observation of fermentative metabolism and only 184 partial oxidation of organic substrates in aerobic conditions and high substrate availability (known as 185 the Crabtree effect in yeast or the Warburg effect in cancer cells) [44]. In general, flux balance analysis 186 is very limited when investigating fundamental principles of microbial growth.

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Thermodynamic approaches are typically used in genome-scale metabolic models to avoid infeasible flux distributions in the solution space. Thermodynamics gives information about the correct direction of biochemical reactions [45]. Essentially any enzymatic reaction can be reversible, and the Gibbs free energy of reaction (∆ R G) defines the direction in which a reaction proceeds (always in the direction of negative ∆ R G). The Gibbs free energy of reaction depends on the metabolite concentrations by where n i is the stoichiometric coefficient of metabolite i, c i its concentration, and ∆ f G  195 In recent years, resource allocation models and models including molecular crowding have  In the 1990s, the development of the so-called fluctuation theorems [56-59] constituted a major advance in non-equilibrium statistical physics. Basically, they represent a generalisation of the second law of thermodynamics. They link the probability to observe an entropy increase σ during a time τ to the probability to observe an entropy decrease by the same amount in the same time. This class of theorems can be generally expressed by where P (+σ) and P (−σ) denote the probabilities to observe an entropy increase or decrease by σ during time τ, respectively. More recently, J. L. England [60] has proposed to apply this approach to self-replicating systems. Obviously, replication is a highly irreversible biological process and can be described in the language of statistical physics as a system that goes from a macrostate I (a single cell and the substrates in the surrounding medium), to a macrostate II (two daughter cells and the substrates), where each macrostate corresponds to an extremely high number of microstates. England's reasoning starts from the fact that particles obey classical mechanics at the microscopic scale, and therefore follow a reversible dynamics. This allows quantifying the reversibility of a microscopic transition by the associated change in entropy. Applying these microscopic considerations to the macroscopic scale, the author obtains a generalization of the second law of thermodynamics for macroscopic irreversible biological processes. While the classical second law of thermodynamics states that the increase of entropy of a closed system is always positive and obeys the inequality where ∆Q ex is the amount of heat exchanged with the environment and ∆S int the internal entropy increase of the system. England's derivation adds a new term to this relation, where π(I → II) (respectively π(II → I)) stands for the probability that the system evolves from 229 macrostate I to macrostate II (respectively from II to I). When a macroscopic transition is irreversible 230 (π(II → I) π(I → II)), the logarithm becomes negative, increasing the lower bounds for heat 231 dissipation and entropy increase.

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Equation (4) allows to have a closer look at self-replication. England applies this relation to a population of exponentially growing cells. He denotes the growth rate as g and the reverse rate (highly unlikely to happen) as δ. It allows to express π(I → II) as gdt and π(II → I) as δdt. Hence, equation (4) becomes ∆q T + ∆s int ≥ ln (g/δ), where ∆q and ∆s int are the respective intensive quantities. From this relation, one can see that the maximum duplication rate g max is obtained when the right and left terms are equal. Therefore, Because the net growth rate is (g − δ), this last relation is one of particular interest. The right hand 233 term shows the dependency on three quantities. The maximum growth rate will increase with heat 234 dissipation, ∆q, internal entropy change, ∆s int , and the rate at which the reverse process would occur, 235 δ. The author emphasizes an interesting property: for identical entropy changes and decay rates, a 236 replicator that dissipates more heat compared to another will have a higher maximum growth rate. A 237 second particularly interesting aspect is the dependency on δ and ∆s int . Low degree of organization 238 and low stability make a replicator more competitive.

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Application of equation (4) to any self-replicator needs an estimation of the rates g and δ. England RNA. Therefore, such a ligation for DNA is thermodynamically not possible.

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The same approach is applied to bacterial cell division. England considers a system with a single 249 bacterium at constant temperature T in a rich medium. The system is initially in macrostate I (a single 250 bacterium and the substrate), and will evolve through cell division to macrostate II (two bacteria and 251 the substrate). Application of equation (4) to this system provides a lower limit on heat dissipation 252 for cell replication. This amount is six times lower than what was experimentally observed for E. coli, 253 which is, according to England, surprisingly close to the thermodynamic limit.  Employing now the formula to calculate the degree of reduction when N 2 is assumed to be the nitrogen source, in combination with the relation between the energy of combustion for an organic compound and its 274 degree of reduction (Battley assumes -107.90 kJ/av e − ), he was able to estimate the energy of formation 275 for biomass as -65.10 kJ/C-mol (not including ions). The n X in equation (7)    Another possibility for a straight-forward approach to combine thermodynamic concepts from 292 black box models with genome-scale models is a separate analysis of anabolism and catabolism.

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In order to calculate properties of anabolism, such as those predicted by Battley in 1993 [28], from 294 genome-scale models, we pursue the following approach: Genome-scale models from the BIGG 295 database were modified in two steps. First, all reactions that can produce ATP are disabled by 296 introducing a dummy compound representing "unusable" ATP. Second, two strictly coupled reactions 297 are introduced that import ATP into the cytosol, and export ADP and orthophosphate with the same 298 rate. The strict coupling of import and export ensures that only energy but no matter is introduced into 299 the system. Thus, the modified model is unable to produce ATP from any carbon source and instead 300 must use the imported ATP as energy source. Therefore, this modification separates anabolism from 301 catabolism by simulating an external "ATP battery" providing the organism with external chemical 302 energy, replacing the usual catabolic pathways.

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These modified models were used to simulate anabolism separate from catabolism. In particular,

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we calculated the minimum amount of ATP required to incorporate one carbon atom from the nutrients 305 into biomass and the minimum number of CO 2 molecules that are released in this process. For this, 306 the biomass production rate was fixed to the value obtained from the original model (in all cases 307 the objective was maximisation of the biomass production rate) and subsequently minimizing all database are for E. coli, and use the same biomass definition. For all E. coli models except the "core model", the required ATP per biomass carbon is between 2 and 3.5. Interestingly, the anabolism versus 316 metabolism ratio of released carbon dioxide for the E. coli models is very close to the ratio predicted by 317 Battley [28] (indicated by the dotted black line).

Figure 2.
Anabolic properties of genome-scale models of the BIGG database. The y-axis indicates the minimum required amount of ATP per biomass carbon. The x-axis displays the ratio of carbon dioxide released by anabolism to carbon dioxide released by overall metabolism (including anabolism and catabolism).

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To increase fundamental understanding of microbial self-replication and its limitations, between two macroscopic states has been derived. This allowed calculating a lower bound for the 355 produced heat during self-replication as a function of the internal entropy, growth and decaying rates.

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A consequence of these calculations is that a self-replicating microbe that dissipates heat with a rate 357 close to the thermodynamic minimum is optimal in the sense that energy loss is minimised. However, 358 the maximal rate of self-replication increases with increased heat dissipation. The finding that the 359 heat dissipation of E. coli is not far from the calculated minimum needed for self-replication hints at 360 evolution towards thermodynamic efficiency. However, the calculations imply that replication rates 361 are increased with higher internal entropy and an increased rate of spontaneous self-decay. Both

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properties are not commonly found in microbial organisms. These properties may be beneficial to 363 increase growth rate from a thermodynamic perspective, but are probably disadvantageous regarding 364 other evolutionary pressures.

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Evidently, there is no simple and unique answer to the question whether microbes are 366 thermodynamically optimised self-replicating machines. Although all three concepts described here 367 are concerned with the same phenomenon, each represents a different perception and viewpoint on 368 thermodynamic optimality of microbial growth. In our opinion these three concepts, as different as 369 they may be, host an enormous potential to complement each other into an extended understanding of 370 thermodynamic limitations and optimality of microbial growth.

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The minimum amount of heat dissipation, and the upper limit of Gibbs free energy dissipation 372 define fundamental thermodynamic limitations of microbial growth. The lower bound is a consequence 373 of the extended second law of thermodynamics: It is impossible to replicate with less dissipated heat.

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The upper bound is an empiric observation which so far has not experienced a theoretical explanation.

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It implies that there is a principle upper limit for microbial growth rates. In addition, black box models Temperature in Joule, k B set to 1 P (±σ) Probability to observe an entropy production of ±σ π(II → I) Probability to observe a transition from macrostate II to I π(I → II) Probability to observe a transition from macrostate I to II g Duplication rate