Entropy production of a steady-growth cell with catalytic reactions
EEntropy production of a steady-growth cell with catalyticreactions
Yusuke Himeoka and Kunihiko Kaneko
Department of Basic Science, University of Tokyo,Komaba, Meguro-ku, Tokyo 153-8902, Japan
Abstract
Cells generally convert external nutrient resources to support metabolism and growth. Un-derstanding the thermodynamic efficiency of this conversion is essential to determine the generalcharacteristics of cellular growth. Using a simple protocell model with catalytic reaction dynamicsto synthesize the necessary enzyme and membrane components from nutrients, the entropy pro-duction per unit cell-volume growth is calculated analytically and numerically based on the rateequation for chemical kinetics and linear non-equilibrium thermodynamics. The minimal entropyproduction per unit cell growth is found to be achieved at a non-zero nutrient uptake rate, ratherthan at a quasi-static limit as in the standard Carnot engine. This difference appears becausethe equilibration mediated by the enzyme exists only within cells that grow through enzyme andmembrane synthesis. Optimal nutrient uptake is also confirmed by protocell models with manychemical components synthesized through a catalytic reaction network. The possible relevance ofthe identified optimal uptake to optimal yield for cellular growth is also discussed. a r X i v : . [ q - b i o . S C ] M a r . INTRODUCTION A cell is a system that transforms nutrients into substrates for growth and division. Byassuming that the nutrient flow from the outside of a cell is an energy and material source, thecell can be regarded as a system to transform energy and matter into cellular reproduction.It is important to thermodynamically study the efficiency of this transformation[1–5].Regarding material transformation, the yield is defined as the molar concentration ofnutrients (carbon sources) needed to synthesize a molar unit of biomass (cell content) andhas been measured in several microbes [6–10]. As the conversion of nutrients to cell contentis not perfect and material loss to the outside of a cell occurs as waste, the yield is generallylower than unity. The yield also changes with nutrient conditions, and measurements inseveral microbes show that the yield is maximized at a certain finite nutrient flow rate. Thebasic logic underlying the optimization of yield at a finite nutrient flow rate rather than ata quasi-static limit is not fully understood.A cell can also be regarded as a type of thermodynamic engine to transform nutrientenergy into cell contents. In this case, it is necessary to study the thermodynamic efficiencyor entropy production during the process of cell reproduction. The thermodynamic efficiencyof metabolism has been measured in several microbes under several nutrient conditions [9,11–15], and Westerhoff and others computed it by applying the phenomenological flow-forcerelationship of the linear thermodynamics to catabolism and anabolism [4, 16] to show thatthe efficiency is optimal at a finite nutrient flow. Although such a phenomenological approachis important for technological application, a physiochemical approach is also necessary tohighlight difference between cellular machinery and the Carnot engine by characterizingthe basic thermodynamic properties in a simple protocell model. Indeed, when viewed asa thermodynamic engine, a cell has remarkable differences from the standard Carnot-cycleengine. The cell sits in a single reservoir, without a need to switch contacts between differentbaths. The cell grows autonomously to reproduce. To consider the nature of such a system,it is necessary to establish the following three points distinguishing the cell from the standardCarnot engine[17].First, cells contain catalysts (enzymes). The enzyme exists only within a compartmen-talized cell encapsulated by a membrane and thus enables reactions to convert resourcesto intracellular components to occur within a reasonable time scale within a cell but not2utside the cell. Without the catalyst, extensive time is required for the reaction. Thus,the reaction is regarded to occur only in the presence of the catalyst. This leads to anintriguing non-equilibrium situation: Let us consider the reaction R + C ↔ P + C with R asthe resource, P as the product, and C as the catalyst. Then, under the existence of C , thesystem approaches an equilibrium concentration ratio with [ R ] / [ P ] = exp( − β ( µ R − µ P )) and µ R and µ P as the standard chemical potential of the resource and product, respectively, andwith β as the inverse temperature. In contrast, outside the cell, R and P are disconnectedby reactions within the normal time scale; therefore, their concentration ratio can take onany value. In this sense, the external environment is non-equilibrium in nature, in contrastto the intracellular environment. This leads to a remarkable difference from the standardCarnot engine.Second, while considering the dynamical process, it is important to note that the catalystsare synthesized within the cell as a result of catalytic reactions. The time scale to approachequilibrium can depend on the abundance of the catalyst, which depends on the reactiondynamics themselves. Based on the first and second points mentioned above, the approachto equilibrium in the intracellular environment depends on catalyst abundance, which alsodepends on the flow rate of nutrients from outside the cell. Hence, the thermodynamicefficiency could show non-trivial dependence upon the nutrient flow.Third, cell volume growth results from membrane synthesis from nutrient components,facilitated by the catalyst, whereas the concentrations of catalyst and nutrient are dilutedby cell growth, which results in a non-standard factor for thermodynamic characteristics.These three issues, which are fundamental to cell reproduction, are mutually connectedand thus inherent to a self-reproducing, or autopoietic, system. In contrast to dynamicalsystems studies for self-reproduction in catalytic reaction networks [18–21], however, thethermodynamic characteristics for such systems have not been fully explored.In the present study, we determine these characteristics using simple reaction dynamicsconsisting of the nutrient, catalyst, and membrane. In Sec. II, we consider a simple protocellmodel consisting of a membrane precursor and catalyst under a given nutrient flow. Theentropy production by chemical production per unit cell volume growth is shown to beminimized at a certain finite nutrient flow. The mechanism underlying this optimizationis discussed in relation to the abovementioned three characteristics of a cell. The entropyproduction by material flow is discussed in Sec. II.A and basically does not change the3onclusion described above. A protocell model consisting of a variety of catalysts thatform a network, together with nutrients and membrane precursors, has been investigated toconfirm that the conclusion described above is not altered. The biological relevance of ourresults is discussed in Sec. III. II. ENTROPY PRODUCTION OF AN AUTOPOIETIC CELLA. Two-component model
First, we study the entropy production σ resulting from the intracellular reaction forthe minimal protocell model consisting only of the synthesis of the enzyme and membraneprecursor from the nutrient, which then leads to cellular growth [8, 22–24](see FIG.1 forschematic representation). The model consists of nutrient, membrane precursor, and en- FIG. 1. Schematic representation of our three-component protocell model. N, MP, and Edenote nutrient, membrane precursor, and enzyme, respectively. The nutrient is taken up from theextracellular nutrient pool by diffusion, indicated by a blue arrow. All chemical reactions, indicatedby black solid arrows, are reversible and catalyzed by the enzyme, as indicated by dashed arrows.Membrane precursors are transformed to the membrane as indicated by the green ring with someleaks. The membrane growth results in an increase in cell volume. zyme, where the enzyme and membrane precursor are synthesized from the nutrient undercatalysis by the enzyme. Moreover, by assuming that the diffusion constant of the nutrient issufficiently large, the internal nutrient concentration is regarded to be equal to the externalnutrient concentration. Based on the rate equation for chemical kinetics, our model is given4y the following two-component ordinary differential equation dxdt = κ x x ( kX − x ) − xλ,dydt = κ y x ( lX − y ) − φy − yλ. (1)where the variables x and y denote the concentrations of the enzyme and membrane pre-cursor, respectively, whereas λ ≡ V dVdt denotes the cell volume growth rate to be determined.Here, the notation of parameters is as follows: • X : nutrient concentration. • k = e − β ( µ x − µ nut ) , l = e − β ( µ y − µ nut ) , with µ nut , µ x and µ y as the standard chemical poten-tial of nutrient, x , and y , respectively. • κ i : catalytic capacity of the enzyme for i component ( i = x, y ). • φ : consumption rate of the membrane precursor to produce the membrane, such thatthe volume growth rate λ is given by λ = γφy , where γ is the conversion rate frommembrane molecules to cell volume.In the stationary state, λ takes a positive constant value of y > X > σ . In computing σ , spatial inhomogeneity isnot considered through the assumption of local homogeneous equilibrium. Thus, the entropyproduced during the doubling in the protocell volume is given by S = σ (cid:90) T V e λt dt = σλ V , where V is the initial cell volume and T is doubling time of the protocell volume.We denote η ≡ σ/λ as the entropy production per unit cell-volume growth. Generally, if η is smaller, the thermodynamic efficiency for a cell growth is higher. For larger η , moreenergetic loss occurs in the reaction process. Hereafter, we study the dependence of η onthe nutrient condition and the growth rate λ .In this subsection, we consider only the entropy production by the chemical reaction; theentropy production by the flow of chemicals from the outside of the cell will be consideredin the next section. The calculation of entropy production among different componentsis performed by virtually introducing chemical baths for different components that are5utually in disequilibrium and then applying linear non-equilibrium thermodynamics forcalculation. This may result in stringent requisites; however, this step is adopted to addressthe thermodynamic efficiency of a cell with growth, as general steady-state thermodynamicsare not established currently. Then, the entropy production by the reactions is given by σ = (cid:80) i J i A i T , where J i is the chemical flow and A i is the affinity for each reaction. Here weset T = 1 without losing generality.For calculation, we assume that κ x and κ y are identical for simplicity, denoted as κ . Then,by rescaling the variables as ˜ x = xγ, ˜ y = yγ, ˜ X = lXγ, τ = tφ. (2)Eq.(1) is written as d ˜ xdτ = ˜ κ ˜ x (˜ k ˜ X − ˜ x ) − ˜ x ˜ y,d ˜ ydτ = ˜ κ ˜ x ( ˜ X − ˜ y ) − ˜ y − ˜ y , (3)where ˜ κ = κφγ and ˜ k = k/l . The stationary solution of the equation for ˜ κ = 1 is given by˜ x = ˜ k ˜ X (1 + ˜ k ˜ X )1 + ˜ X + ˜ k ˜ X , ˜ y = ˜ k ˜ X X + ˜ k ˜ X .
Following this assumption, the entropy production by chemical reaction σ at the stationarystate is calculated as σ = σ x + σ y with σ i = J i A i T for the enzymatic reaction i = x and for themembrane reaction i = y . Here, the flows are given by ˜ J x = ˜ κ ˜ x (˜ k ˜ X − ˜ x ) and ˜ J y = ˜ κ ˜ x ( ˜ X − ˜ y ),whereas the affinities are given by A x = T ln(˜ k ˜ X/ ˜ x ) and A y = T ln( ˜ X/ ˜ y ). We omit the tildefor affinities because the affinities are not affected by scale transformation. Therefore, weobtain ˜ σ = ˜ κ ˜ x (˜ k ˜ X − ˜ x ) ln(˜ k ˜ X/ ˜ x ) + ˜ κ ˜ x ( ˜ X − ˜ y ) ln( ˜ X/ ˜ y ) . The dependence of ˜ η ≡ ˜ σ/ ˜ y = γη upon ˜ k and ˜ X , thus obtained, is plotted in FIG.2 for ˜ κ = 1.As shown, the entropy production rate per unit growth shows a non-monotonic dependenceon the nutrient concentration and is minimized at a non-zero nutrient concentration. Becausenutrient uptake rate is a monotonic function of nutrient concentration, this result means thatthe entropy production rate per unit growth η is minimal at a finite nutrient uptake rate.6his result is in strong contrast with the thermal engine, where the entropy production isminimal at a quasi-static limit. FIG. 2. The logarithm of ˜ η plotted as a function of nutrient concentration and ˜ k , with the colorcode given in the side bar. It is calculated from the solutions of Eq.(3). The parameter ˜ κ is chosento be 1 .
0. For given ˜ k , there is an optimal nutrient concentration that gives the minimum η . (Tildeis omitted in the figure.) FIG.3(a),(b) shows the entropy production rate per unit growth σ x /λ, σ y /λ for each re-action which produces component x and y , respectively. This shows that the non-monotonicdependence on the nutrient in FIG.2 is attributable to σ y /λ . (a) (b) FIG. 3. The logarithm of the entropy production per unit growth rate σ x /λ and σ y /λ for theenzyme and membrane precursor synthesis reactions, respectively, plotted as a function of thenutrient concentration ˜ X and the rate constant ˜ k , computed by Eq.(3). (a). σ x /λ for the enzymeproducing reaction and (b) σ y /λ for the membrane precursor producing reaction.
7s mentioned above, an important characteristic of cells is that intracellular reactionsare facilitated by enzymes that are autonomously synthesized. Thus, the equilibrium dis-tribution of chemicals in the presence of enzymes is different from the external chemicaldistribution. The decrease in η under low nutrient concentrations is explained accordingly:The extracellular concentrations of the nutrient and of the membrane precursor are far fromequilibrium in the presence of catalysts. Therefore, their intracellular concentrations underconditions of low nutrient uptake remain far from equilibrium and still similar to the ex-ternal concentrations because of insufficiency of the enzyme. However, when the amountof nutrient uptake increases, the amount of enzyme increases and the system approachesintracellular equilibrium; therefore, the entropy production rate per unit growth decreases.In contrast, with further increases in nutrient uptake, the entropy production rate increasesas a result of the increase in cellular growth; entropy production σ = (cid:80) i J i A i T by the reactionincreases linearly with the reaction speed J i . In the steady state, the reaction speed J i isroughly estimated by λx , with x as the concentration of the product of the reaction. Forexample, the dynamics of the enzyme concentration are given by dxdt = x ( kX − x ) − λx .At steady state, the enzyme production rate x ( kX − x ) is balanced with λx according toEq.(1). Thus, σ x increases with λx . In summary, for a cell with a high growth rate, increasedenzyme abundance is needed, which, however, leads to higher entropy production [26][27].In contrast, if the enzyme concentration is fixed externally, the entropy production rateper unit growth η is minimized at the zero limit of nutrient concentration. In this case, thereaction dynamics Eq.(1) are reduced to dydt = c ( lX − y ) − φy − φy . (4)where c is a constant representing the concentration of the enzyme. In this case, the sta-tionary solution is given by y = [ − (1 + c/φ ) + (cid:112) (1 + ( cφ ) ) + 4 clX/φ ], and accordingly η − = (1+ y ) ln( lX/y ). There is no optimal nutrient concentration in this expression because ∂η − ∂X is always positive for any X, l >
0. This is consistent with the explanation mentionedabove for Eq.(3). If the enzyme abundance is fixed to be independent of the nutrient uptake,the speed of approaching equilibrium is not altered by the nutrient condition; therefore, theentropy production just increases monotonically because of the cell volume growth.8 . Additional entropy production by material flow
Thus far, we considered only entropy production by chemical reactions. In addition, thematerial flow also contributes to entropy production, which is taken into account here.To discuss the flow of nutrients, the dynamics of the nutrient concentration cannot beneglected. By including the temporal evolution of the nutrient concentration, the dynamicsof the cellular state are given by dsdt = − κ x x ( ks − x ) − κ y x ( ls − y ) − sλ + D ( s ext − s ) ,dxdt = κ x x ( ks − x ) − xλ, (5) dydt = κ y x ( ls − y ) − φy − yλ. where x, y and s are the enzyme, membrane precursor, and nutrient concentration, respec-tively, and λ = V dVdt = γφy . The rate constants k and l are determined by the standardchemical potential of each chemical. Additionally, the nutrient is taken up with rate D fromthe extracellular environment with a concentration s ext .Entropy production by chemical flow is derived from nutrient uptake and membrane con-sumption, which (again by assuming linear nonequilibrium thermodynamics) are given by (cid:126)J s · ∇ ( − µ s /T ) and (cid:126)J y · ∇ ( − µ y /T ), respectively, where (cid:126)J i is the material flow of component i and µ is the chemical potential. Integration of the terms over a narrow layer having aspatial gradient results in D ( s ext − s ) s ext − ss /T and φy/T . We neglect the entropy productionof the solvent with the assumption that intra- and extracellular solvent concentrations areidentical[28]. The contribution of dilution of the nutrient resulting from cellular growthis approximated as σ s ≈ sλ by using the formula of entropy change resulting from theisothermal expansion of an ideal solution[29]; for other species, we use the same formula.We choose that κ x , κ y , D, γ and φ are equal to unity and that l = k , for the sake ofsimplicity. Indeed, the characteristic behavior of η is independent of this choice. Then,the fixed-point solutions of Eq.(5) are obtained against two parameters k and s ext . Fromthe solution, the entropy production per unit growth is computed, as shown in FIG.5(a).We note that here again the minimal η is achieved for a finite nutrient uptake, i.e., undernonequilibrium chemical flow. In FIG.5(b), we plotted η flow , the entropy production exclud-9ng that derived from the chemical reaction. It increases monotonically with the externalnutrient concentration. Entropy production is primarily derived from chemical reactions;therefore, the conclusion of subsection A is unchanged.Note that the so-called thermodynamic efficiency is defined as η th = − J a ∆ G a J c ∆ G c where J c and J a are the rates of catabolism and anabolism, and ∆ G c and ∆ G a are the affinities ofcatabolism and anabolism [4, 11]. Here, the optimality with regard to entropy production η also leads to the optimal thermodynamic efficiency, which, in the present case, is computedby η th = J y µ y /J s µ s where J s = D ( s ext − s ) and J y = φy are the absolute values of the uptake (and consump-tion) flow of chemical species s (and y ), and µ i is the chemical potential of the i th chemicalspecies. It is computed by using the chemical potential of nutrient µ s = µ + T ln( s/s ) with µ as the standard chemical potential for the nutrient and s as its standard concentration(The chemical potential for x and y are computed in the same way). This thermodynamicefficiency also takes a local maximum value at a non-zero nutrient uptake rate (see FIG.4). FIG. 4. The thermodynamic efficiency for the model Eq.(5) plotted as a function of the externalnutrient concentration s ext and the rate constant k . The parameters were set as µ s = 0 . D = 1 . φ = 1 . γ = 1 . κ x = κ y = 1 .
0. The standard concentrations were chosen to be 10 − . a) (b) FIG. 5. The entropy production plotted as a function of the external nutrient concentration s ext and the rate constant k , calculated from the fixed-point solution of Eq.(5); (a) the logarithm oftotal entropy production per unit cell growth, η ; and (b) the logarithm of the entropy productionper unit growth by material flow and dilution only. The parameters are chosen to be κ x = 1 . , κ y =1 . , D = 1 . , φ = 1 . , γ = 1 .
0, and l = k . III. EXTENSION TO A MULTI-COMPONENT MODEL
It is worthwhile to check the generality of our result for a system with a large number ofchemical species as in the present cell. For this purpose, we introduce a model given by dx dt = N (cid:88) j=1 N − (cid:88) k=2 ( C (1 , j ; k ) k j x j − C ( j, k ) k j x ) x k + ( X − x ) − x λ,dx i dt = N (cid:88) j=1 N − (cid:88) k=2 ( C ( i, j ; k ) k ij x j − C ( j, i ; k ) k ji x i ) x k − x i λ, (1 < i < N − , (6) dx N dt = N (cid:88) j=1 N − (cid:88) k=2 ( C ( N, j ; k ) k Nj x j − C ( j, N ; k ) k jN x N ) x k − φx N − x N λ,λ = x N . where the variables x , x N , and x i (1 < i < N) denote the concentrations of the nutrient,membrane precursor, and enzymes, respectively, and X is the external concentration ofthe nutrient. Each element of the reaction tensor C ( i, j ; k ) is unity if the reaction of j to i catalyzed by k exists; otherwise, it is set to zero. Here, the nutrient and the membrane11recursor cannot catalyze any reaction, whereas the other components i = 1 , ..N − C ( i, j ; k ) is equal to unity if and only if C ( j, i ; k ) equals unity. For the sake ofsimplicity, we assume that catalytic capacity, nutrient uptake rate, membrane precursor con-sumption rate, and the conversion rate from membrane molecule to cell volume are unity.The standard chemical potential µ i for each chemical species is assigned by uniform randomnumbers within [0 , k ij is given by min { , exp( − β ( µ i − µ j )) } accordingly [31].Numerical simulations reveal that there again exists an optimal point of η for each randomlygenerated reaction network of N = 100. The dependence of η on the nutrient concentrationis plotted in FIG.6(a), overlaid for different networks. Although the nutrient concentrationto give the optimal value is network-dependent, it always exists at a finite nutrient concen-tration; therefore, the entropy production is minimized at a non-zero nutrient concentration. (a) (b) FIG. 6. The entropy production and deviation from equilibrium calculated from the steady-statesolution of the multi-component model Eq.(6), plotted as a function of the external nutrient con-centration. The results of 10 randomly generated networks are overlaid. (a). η ; and (b). Kullback-Leibler divergence of the steady-state distribution from the Boltzmann distribution. The numberof chemical species is set as 100, whereas the parameter φ is chosen to be unity, and the ratio ofthe number of reactions to the number of chemical species is set to 3. To determine a possible relationship with the optimality of η and equilibrium in thepresence of a catalyst We also computed the Kullback-Leibler (KL) divergence of the steadystate distribution from the equilibrium Boltzmann distribution as a function of the external12utrient concentration, expressed as D KL ( p || q ) = N (cid:88) i =1 p i ln p i q i , (cid:16) with p i = e − µ i (cid:80) j e − µ j , q i = x sti (cid:80) j x stj (cid:17) , where x sti is the concentration of the i th chemical species in the steady state. The KL diver-gence for each network shows non-monotonic behavior, as shown in FIG.6(b). Although theoptimal nutrient concentration does not agree with the optimum for η , each KL divergencedecreases in the region where η is reduced. In this sense, it is suggested that the reductionof η in our model Eq.(6) is related to the equilibration process of abundant enzymes syn-thesized as a result of a relatively high rate of nutrient uptake as discussed for Eq.(1) andEq.(5). IV. SUMMARY AND DISCUSSION
To discuss the thermodynamic nature of a reproducing cell, we have studied simple proto-cell models in which nutrients are diffused from the extracellular environment and necessaryenzymes for the intracellular reactions are synthesized to facilitate chemical reactions, in-cluding the synthesis of membrane components, which leads to the growth of cell volume. Inthe models, cell growth is achieved through nutrient consumption by the reactions describedabove. We computed η , which is the entropy production per unit cell volume growth andfound that the value was minimized at a certain nutrient uptake rate. This optimizationstems from the constraint that cells have to synthesize enzymes to facilitate chemical reac-tions, i.e., the autopoietic nature of cells. In general, the concentrations of nutrients andmembrane components in extracellular environments are different from those in equilibriumachieved in the presence of enzymes, and the intracellular state moves towards equilibriumby synthesizing enzymes to increase the speed of chemical reactions. The equilibrationreduces the entropy per unit chemical reaction. However, faster cell volume growth leads toa higher dilution of chemicals; therefore, faster chemical reactions are required to maintainthe steady-state concentration of chemicals. Because entropy production by the reactionincreases (roughly linearly) with the frequency of net chemical reactions, η then increases fora higher growth range. Thus, the existence of an optimal nutrient content is explained by13he requirement for reproduction mentioned in the introduction, i.e., equilibration of non-equilibrium environmental conditions facilitated by the enzyme, autocatalytic processes tosynthesize the enzyme, and cell-volume increase resulting from membrane synthesis.In the present model, all chemical components thus synthesized are not decomposed; theyare only diluted. However, each component generally has a specific decomposition time ordeactivation time as a catalyst. We can include these decomposition rates, which can alsobe regarded as diffusion to the extracellular environment with a null concentration. Then,the equilibration effect is clearer, although the results regarding optimal nutrient uptake areunchanged.In the present study we focused on the entropy production that corresponds to dissipatedenergy per unit growth. In microbial biology, however, material loss is discussed as bio-logical yield, as mentioned in the introduction, and it is thus reported that the optimalyield is achieved at a certain finite nutrient flow. Material loss is not directly includedin the present model; therefore, we cannot discuss the yield derived directly from entropyproduction. However, it may be possible to assume that energy dissipation is correlatedwith material dissipation.For example, the stoichiometry of metabolism is suggested to depend on dissipated energy[33]. Here, metabolism consists of two distinct parts: catabolism and anabolism. Forcatabolism, the energy is transported through energy currency molecules such as ATP,NADPH, and GTP, which are synthesized from the nutrient molecule. In this process,molecular decomposition also occurs, leading to the loss of nutrient molecules. In addition,the abundance of energy-currency molecules and the utilized energy are correlated. Hence,for both catabolism and anabolism, the energy dissipation and material loss are expectedto be correlated. Indeed, a linear relationship between the yield and the inverse of ther-modynamic loss (i.e., quantity similar to 1 /η here) is suggested from microbial experiments[33, 34].Considering the correlation between energy and matter, the minimal entropy production ata finite nutrient flow that we have shown here may provide an explanation for the findingof optimal yield at a finite nutrient flow. Future studies should examine the relationshipbetween minimal entropy production and optimal yield in the future by choosing an appro-priate model that includes ATP synthesis and waste products in a cell. Currently, althoughour models are too simple to capture such complex biochemistry in a cell, they should14nitiate discussion regarding the thermodynamics of cellular growth. ACKNOWLEDGMENTS
The authors would like to thank A. Kamimura, N. Takeuchi, Y. Kondo, T. Hatakeyama,A. Awazu, Y. Izumida, T. Sagawa, and T. Yomo for the useful discussion. The presentwork was partially supported by the Platform for Dynamic Approaches to Living Systemfrom the Ministry of Education, Culture, Sports, Science, and Technology of Japan and theDynamical Micro-scale Reaction Environment Project of the Japan Science and TechnologyAgency. [1] E. Schr¨odinger.
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