Direction and Constraint in Phenotypic Evolution: Dimension Reduction and Global Proportionality in Phenotype Fluctuation and Responses
aa r X i v : . [ q - b i o . P E ] F e b Chapter 1
Direction and Constraint in PhenotypicEvolution: Dimension Reduction and GlobalProportionality in Phenotype Fluctuation andResponses
Kunihiko Kaneko 1) and Chikara Furusawa 2)
Abstract
A macroscopic theory for describing cellular states during steady-growthis presented, which is based on the consistency between cellular growth and molecu-lar replication, as well as the robustness of phenotypes against perturbations. Adap-tive changes in high-dimensional phenotypes were shown to be restricted within alow-dimensional slow manifold, from which a macroscopic law for cellular stateswas derived, which was confirmed by adaptation experiments on bacteria understress. Next, the theory was extended to phenotypic evolution, leading to propor-tionality between phenotypic responses against genetic evolution and environmentaladaptation. The link between robustness to noise and mutation, as a result of robust-ness in developmental dynamics to perturbations, showed proportionality betweenphenotypic plasticity by genetic changes and by environmental noise. Accordingly,directionality and constraint in phenotypic evolution was quantitatively formulatedin terms of phenotypic fluctuation and the response against environmental change.The evolutionary relevance of slow modes in controlling high-dimensional pheno-types is discussed.
In a chapter in the previous volume of Evolutionary Systems Biology [34], we dis-cussed the evolutionary fluctuation-response relationship, which states that if phe-notypic variance due to noise is high, then evolution rapidly occurs. This suggestsa correlation between short-term phenotypic dynamics and long-term evolutionaryresponses. This, in some sense, is a quantitative expression of Waddington’s geneticassimilation [72].
1) Research Center for Complex Systems Biology, Universal Biology Institute, University ofTokyo, 3-8-1 Komaba, Tokyo 153-8902, Japan 2) Center for Biosystems Dynamics Research,RIKEN, 6-2-3 Furuedai, Suita, Osaka 565-0874;and Universal Biology Institute, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan 1 Kunihiko Kaneko 1) and Chikara Furusawa 2)
Can we push this viewpoint forward to determine the direction of phenotypicevolution in a high-dimensional phenotypic space (i.e., with a large degree of free-dom components)? Can one predict which traits are likely to evolve among manycomponents before evolution progresses?To answer the question, we first investigate the characteristics of responses ofphenotypes with a large degree of freedom. We review the general relation in phe-notypic responses of cells over many components, demonstrating that global propor-tionality exists among all logarithmic changes in concentrations against adaptationto different environmental conditions. We first discuss this proportionality as a gen-eral consequence of steady exponential growth cell, following [37]. We show thatwhen a cell grows and divides while maintaining its composition, the abundancesof each component increases at the same rate; this constraint supports the globalproportional relationship.However, the growth-rate constraint is not enough to explain the experimental ob-servations. Global proportional changes across all components are confirmed evenacross many different environmental conditions. This result cannot be explainedby the constraint of steady-growth alone. We will see that another important con-straint, imposed by the robustness in a phenotypic state shaped throughout evolu-tion, is essential. From this evolutary robustness, phenotypes can change mostlyalong only one or in a few dimensions, although the original phenotypic space ishigh-dimensional as a consequence of the huge diversity in the components of cells.Based on [21], We demonstrate this evolutionary dimension-reduction both throughtheory and simulations, whereas its consequence is consistent with experimentalobservations.This constraint in phenotypic changes is extended to changes that occur in evolu-tion. We demonstrate that long-term phenotypic changes via evolution and short-term changes via adaptation are highly correlated. Global proportionality in thephenotypic changes by environmentally induced adaptation and those by geneti-cally induced evolution is confirmed across all components, both in simulations andlaboratory evolution experiments [20].In contrast, response and fluctuation are two sidesof thesamecoin, as has beendemonstrated by statistical mechanics (see also the first volume of EvolutionarySystems Biology [34]). Hence, a similar correlation in concentration fluctuationsis expected across all components. Indeed, we demonstrate a proportional relation-ship between fluctuations by gene mutation and those by noise over the concen-tration of all components. Recall that the variances in each trait (phenotype) dueto genetic variation are proportional to the evolution rate of the trait according tothe fundamental theorem of natural selection by Fisher [14]. Hence, the evolutionrate of each trait is correlated with its variance by noise. This variance is predeter-mined before mutation and selection. This means that the evolutionary potential ofeach trait is determined in advance by the phenotype changeability which is affectedby environmental variation or noise before genetic changes occur. This enables theprediction of phenotypic evolution. Among the high-dimensional phenotypic space,evolution progresses along the direction in which the variation by the noise or en-vironmental response is larger, which is predetermined before mutation. Although itle Suppressed Due to Excessive Length 3 genetic variation itself is random and undirected, phenotype evolution tends to showdirectionality.
To describe changes in the cellular state in response to environmental changes, weintroduce a simple theory by assuming that cells undergo steady-growth. When acell grows and reproduces in this steady state, all components, e.g., expressed pro-teins, must be approximately doubled [32, 17].Consider a cell consisting of M chemical components. In the cellular state understeady-growth conditions, the cell number increases exponentially over time, as doesthe cell volume V , as given by dV / dt = µ V . In a steady-growth cell, the abundanceof all components increases at the same rate, preserving the concentration of eachcomponent during the cell cycle.To formulate the constraint for steady-growth, let us denote the concentration x i ( > ) for each component i = , · · · , M . The cellular state is represented as apoint in an M -dimensional state space. Here, each component i is synthesized ordecomposed relative to other components at a rate f i ( { x j } ) , for instance, by therate-equation in chemical kinetics. Additionally, all concentrations are diluted bythe rate ( / V )( dV / dt ) = µ , so that the time-change of a concentration is given by dx i / dt = f i ( { x j } ) − µ x i . (1.1)For convenience, let us denote X i = log x i , and f i = x i F i . Then, Eq. (1) can be writtenas dX i / dt = F i ( { X j } ) − µ , which assumes that x i =
0, i.e., all components exist.Then, the stationary state is given by the fixed-point solution F i ( { X ∗ j } ) = µ for all i .In response to environmental changes, the term F i ( { X j } ) and growth rate µ change, as does each concentration x ∗ i ; however, the M − F = F = · · · = F M must be satisfied. Thus, a cell must follow a 1-dimensional curve in the M -dimensional space (see Fig. 1) under a given change in the environmental condi-tions (e.g., against changes in stress strength). Now, consider intracellular changesin response to environmental changes. Here each environmental change given by atype of stress a is parametrized by a single continuous parameter E a (such as thetemperature, degree of nutrient limitation, etc.). Using this parameterization E a , thesteady-growth condition leads to F i ( { X ∗ j ( E a ) } , E a ) = µ ( E a ) . We consider the parameter change from E to E , where each X ∗ j changes from X ∗ j at E , to X ∗ j + δ X j , which is accompanied by a change from µ to µ + δµ . Assuming agradual change in the dynamics x j , we introduce a partial derivative of F i ( { X ∗ j ( E ) } ) by X j at E = E , which gives the Jacobi matrix J i j . Assuming that the environmentalchange is small and that phenotypic changes are sufficiently small and follow onlythe linear term in δ X j , we obtain Kunihiko Kaneko 1) and Chikara Furusawa 2) X Iso-μline for E a Iso-μline for E b X X E Fig. 1.1
Schematic representation of our theoretical analysis: Changes in gene expression in ahigh-dimensional state space under different perturbations E a and E b are presented. Upon a givenenvironmental change, the phenotypic changes in component concentrations follow a curve satis-fying the constraint showing that the growth rates of all components are identical, i.e., an iso- µ line F = F = · · · = F M , in an M -dimensional state space. For a different environmental vector, thelocus in the state space follows a different iso- µ line. ∑ j J i j δ X j ( E ) + γ i δ E = δµ ( E ) (1.2)with γ i ≡ ∂ F i ∂ E . Under our linear conditions, δµ ∝ δ E , so that δµ = αδ E holdsfor a constant α . Accordingly, we obtain ∑ j J i j δ X j ( E ) = δµ ( E )( − γ i / α ) . Hence, ∑ j J ij δ X j ( E ) δµ ( E ) = ∑ j J ij δ X j ( E ′ ) δµ ( E ′ ) so that δ X j ( E ) δ X j ( E ′ ) = δµ ( E ) δµ ( E ′ ) (1.3)is obtained overall j . The formula can be compared with experimental observations.Note that it can be applied to any component. For example, one can use the concen-tration of either mRNA or protein, depending on the available experimental data. To explore the relationship between changes in global gene expression and growthrate, we analyzed transcriptome data of Escherichia coli obtained under three en-vironmental conditions: osmotic stress, starvation, and heat stress, as presented in[48]. In the experiments, the cells were initially cultured under minimal mediumat 37 ◦ C. After the transient response to the introduction of a given stress, the cellswere harvested when the growth rate reached a constant value. The data were takenonly from an exponentially growing steady state. For each stress condition, three itle Suppressed Due to Excessive Length 5 levels of stresses (s = high, medium, low) were used, so that the absolute expressionlevels, represented by x j for j -th gene, were measured over 3 × X j = log x j ), that is δ X j ( E ) = X j ( E ) − X Oj (i.e.,log ( x j ( E ) / x Oj ) ) for genes j , where E represents a given environmental condition and X Oj represents the log-transformed gene expression level under the original condi-tion [37].To examine the validity of the theory for global changes in expression inducedby the environmental stresses, we plotted the relationship between the differencesin expression ( δ X j ( E as ) , δ X j ( E as )) in Fig. 2(a)-(c) for s = low and s = medium,where a is either osmotic, heat, or starvation stress. A common proportionality wasobserved in concentration changes across most mRNA species, which is consistentwith the theory [37].According to our theory, the proportion coefficient in the expression level shouldagree with the growth rate. Here, for each condition, the change in the growth rate δµ ( E as ) was also measured ( a is either osmotic, heat, or starvation stress). The slopefitted from the data agrees well with the common ratio δ X j ( E as ) / δ X j ( E as ) . d X (E ) j mediumheat d X ( E ) j l o w hea t -2-1 0 1 2 -2 -1 0 1 2 -2-1 0 1 2 -2 -1 0 1 2 -2-1 0 1 2 -2 -1 0 1 2 d X (E ) j mediumosmo d X ( E ) j l o w o s m o d X (E ) j mediumstrv d X ( E ) j l o w s t r v (a) (b) (c) Fig. 1.2
Examples of the relationship between changes in gene expression δ X j ( E as ) and δ X j ( E as ) for genes in E. coli. δ X j represents the difference in the logarithmic expression level of a gene j between non-stressed and stressed conditions, where s and s represent two different stressstrengths, i.e., low and medium. (a), (b), and (c) show the plot for a = osmotic stress, heat stress,and starvation, respectively. The fitted line indeed agrees well with that expected from the growth-rate change in Eq.(3). Reproduced from [37]. In this respect, the theory based on the steady-growth state and linearization ofchanges in stress works well for analyzing transcriptome changes in bacteria. In-deed, relevance of growth-rate to global trend in transcriptome changes was notedin several experiments [57, 52, 2, 39], and the formulation in the last section canprovide a step to understand such global trend. Here, however, the steady-growththeory is not sufficient. First, the global proportionality is satisfied even under a
Kunihiko Kaneko 1) and Chikara Furusawa 2) stress condition that reduces the growth rate to below 20% or as compared to thestandard. Such expansion of the linear regime is beyond the simple theory. Theother important point missed by this steady-growth theory will be discussed in thenext section.
Until now, we have compared the responses against a given type of environmen-tal condition with different strengths. In general, possible environmental changesare described by a vector as E . In the study, the changes are given by E = λ a e a with different strengths λ a by fixing e a . However, one can also compare expressionchanges across different types of stress conditions. In this case, the environmentalchange E is no longer represented by a scalar variable, and the responses against theenvironmental changes given by different vectors e a and e b must be compared.However, the above theory cannot predict the common proportionality. This isbecause each one-dimensional curve upon a given type of environmental change isgenerally located along a different direction in the state space (see Fig. 1). This canalso be understood using Eqs. (1)-(3) in Sec. 2. To compare the responses againstdifferent types of environmental changes using Eqs. (2) and (3), one needs γ ai ≡ ∂ F i ∂λ a ,which depends on the type of environmental (stress) condition a . Hence, rather thanEq. (3), we obtain δ X j ( E a ) δ X j ( E b ) = δµ ( E a ) δµ ( E b ) . ∑ i L ji ( − γ ai / α a ) ∑ i L ji ( − γ bi / α b ) (1.4)Then, because γ ai = γ bi , in general, the proportionality cannot be determined as inEq. (3). Although the theory cannot predict the simple proportional relationship,one can plot the experimental data as Fig.2, even across different conditions for os-motic pressure, heat, and starvation. An example of such plot is given in Fig. 3(a)(see also [37]). To further support the proportionality, we plotted δ X j ( E ) for thechanges in concentrations of thousands of proteins (rather than mRNA) under dif-ferent conditions by using proteome analysis [63] in Fig. 3(b). Interestingly, in bothcases, a strong correlation between δ X j ( E a ) and δ X j ( E b ) was still observed for allcomponents j , even under different environmental conditions. Although more genesdeviated from the common proportionality, lowering the correlation coefficients ascompared with those for the same type of stress, the global proportionality still heldfor most genes (Note that Fig. 3(a) and (b) show more than 1000 points, and sothat a single line fits most of these points). Thus, the global proportionality is stillvalid. Further, as shown in Fig. 3(c), the slope approximately agrees with the rate ofgrowth change, as in Eq. (3). Additionally, other data suggested such a correlationin mRNA abundance under different environmental conditions [39, 68]. itle Suppressed Due to Excessive Length 7 -3-2-1 0 1 2 3-3 -2 -1 0 1 2 3 d X ( A c e t a t e ) j d X (Mannose) j (b) (c) dm (E ) / dm (E ) a s l o p e i n ( d X ( E ) , d X ( E ) ) jj d X (Osmo) j d X ( S t a r v a t i o n ) j (a) b ab Fig. 1.3
Common proportionality in
E. coli gene expression changes. (a) shows an example ofthe relation between mRNA expression changes δ X j ( E a ) and δ X j ( E b ) , where a and b are osmoticstress and starvation, respectively. (b) shows an example of the relation between protein expressionchanges in the mannose carbon source and acetate carbon source conditions. The blue lines in (a)and (b) represent the slope calculated by the ratio of observed growth rate changes. (c) Relationbetween the slope of the change in protein expression and the change in the growth rate. Theabscissa represents δµ ( E a ) / δµ ( E b ) , whereas the ordinate is the slope in δ X j ( E a ) / δ X j ( E b ) . Theslope was obtained by fitting the protein expression data. Reproduced from [21]. Because gene expression dynamics are very high-dimensional, this correlationsuggests that a strong constraint exists in adaptive changes in expression dynamicsthat cannot be explained by the simple theory assuming only steady-growth. Theglobal proportionality is beyond the scope of the simple theory presented in Eqs. (2)and (3). Thus, this proportionality is not generic in any dynamical systems satisfyingonly steady-growth.In summary, two questions remain: how to evaluate a broad range of linearityregime and evaluating proportionality under different environmental conditions. Toanswer these questions, some factor other than steady-growth must be evaluated.Of course, cells are not only constrained by steady-growth but also are a prod-uct of evolution. Through evolution, cells can efficiently and robustly reproducethemselves under external conditions. Therefore, the above two features may resultfrom evolution. In the next section, we examine the validity of the hypothesis thatevolutionary robustness constrains intracellular dynamics to exhibit global propor-tionality in the adaptive changes of many components.
The above hypothesis regarding the consequence of evolutionary robustness is dif-ficult to evaluate experimentally, as the experimentally available data are only fromorganisms that currently exist as a result of evolution: One cannot be compare them
Kunihiko Kaneko 1) and Chikara Furusawa 2) with the data before evolution. Hence, we used numerical evolution for some mod-els.To this end, we utilize simple cell models consisting of a large number of com-ponents and numerically evolve them under a given fitness condition to determinehow the phenotypes of many components evolve. Two models are adopted in whichphenotypes are generated by dynamical process for intracellular components. One isa catalytic reaction network model [17, 19] in which catalysts are synthesized withthe aid of other catalysts so that the concentrations of a set of catalytic moleculesconstitute the phenotypic state space. The other model adopts a gene regulation net-work [22, 49, 8, 33] in which proteins are expressed as a result of mutual activationor inhibition from other proteins. Both dynamics involve a large number of com-ponents, i.e., chemical concentrations or protein expression levels, which determinethe phenotypes. The growth rate or fitness is determined by these phenotypes.In these models, genes govern the network structure and parameters for the re-actions that establish the rules for such dynamical systems. The phenotype of eachorganism, as well as the growth rate (fitness) of a cell, is determined by such reactiondynamics, whereas the evolutionary process consists of selection according to theassociated fitness and genetic change in the reaction network (i.e., rewiring of thepathway). The global proportionality in concentration changes across componentsis confirmed in both the models after evolution.Here, we explain the results of analysis using the catalytic network. Despite itssimplicity, this model captures the basic characteristic of cells such as the power-law abundance and log-normal fluctuations of cellular components, adaptation withfold-change detection, among other factors [17, 18, 19, 36].In the model, the cellular state is represented by the numbers of k chemicalspecies, i.e., ( N , N , · · · , N k ) , whereas their concentrations are given by x i = N i / V with the volume of the cell V [17, 19]. There are m ( < k ) resource chemicals S , S , · · · , S m whose concentrations in the environment and within a cell are givenby s , · · · , s m and x , · · · , x m , respectively. Each reaction leading from one chemi-cal i to another chemical j was assumed to be catalyzed by a third chemical ℓ , i.e., i + ℓ → j + ℓ . The resource chemicals are transported into the cell with the aid ofother chemical components named as “transporters.” We assumed that the uptakeflux of nutrient i from the environment is proportional to Ds i x t i , where chemical t i acts as the transporter of nutrient i , and D is a transport constant. For each nutrient,there is one corresponding transporter, represented by t i = m + i . The other k − m chemical species are catalysts synthesized from other components via the catalyticreactions mentioned above. The catalytic reactions result in nutrient transformationinto cell-component chemicals. With the uptake of nutrient chemicals from the envi-ronment, the total number of chemicals N = ∑ i N i in a cell increases. A cell, then, isdivided into two cells when the total number of molecules exceeds a given threshold.Here, to achieve a higher growth rate, the synthesis of the cell components mustprogress concurrently with nutrient uptake. Hence, the cellular growth rate dependson the catalytic network, which is determined by genes. With evolution this growthrate, i.e., the fitness can be increased. itle Suppressed Due to Excessive Length 9 Because of this fitness, the evolutionary procedure is carried out as follows. First,we prepared n parent cells with slightly different reaction networks, randomly gen-erated with a given connection rate. We applied stochastic reaction simulation ofthe above model and selected n / L cells with high growth rates. From each of the n / L parent cells, L mutant cells were generated by replacing a certain fraction ofreaction paths, whose rate is determined by the mutation rate. Next, we obtained n cells of the next generation, which contained slightly different reaction paths. Werepeated the same procedure to obtain the next generation population, and so on.The simulation of evolutionary dynamics was performed under a constant envi-ronmental condition o = { s o , · · · , s om } . Under the original condition, the concentra-tions were set at s o = s o = · · · = s om = / m , at which evolution progresses so that thecell growth rate, i.e., inverse of the average division time, is increased. Using the model described in the last subsection, we analyzed the response ofthe component concentrations to the environmental change from the original con-dition. Here, the environmental condition is given by the external concentration { s , · · · , s m } . We then changed the condition to s ( ε , E ) j = ( − ε ) s oj + ε s E j , where ε is the intensity of the stress and E = { s E , · · · , s E m } denotes the vector of the new,stressed environment, in which the values of component s E , · · · , s E m were determinedrandomly to satisfy ∑ j s E j =
1. For each environment, we computed the reaction dy-namics of the cell to obtain the concentration x ( ε , E ) j in the steady-growth state, fromwhich the logarithmic change in the concentration δ X ( ε , E ) j = log ( x ( ε , E ) j / x o j ) was ob-tained; the change in growth rate µ , designated as δµ ( ε , E ) , was also computed.We next examined whether changes in δ X ( ε , E ) j satisfy the common proportion-ality across all components under a variety of environmental changes for the ranges0 ≤ ε ≤
1. We examined the degree of proportionality both for the random networksbefore evolution and those after evolution under the given environmental conditions.We also tested whether the proportion coefficient is consistent with δµ ( ε , E ) .First, we computed the response of expression to the same type of stress, i.e.,the same vector E with different intensities ε . As shown in Fig. 4(a), we examinedthe correlation between changes in component concentrations δ X ( ε , E ) j and δ X ( ε , E ) j caused by different magnitudes of environmental change ( ε = ε + ε , at ε > ε = . ε = . δµ ( ε , E ) / δµ ( ε , E ) and fitted slope in ( δ X ( ε , E ) j , δ X ( ε , E ) j ) across all compo-nents. Fig. 4(b) shows the ratio of the slope to δµ ( ε , E ) / δµ ( ε , E ) (which turns tobe unity when Eq. (3) is satisfied) as a function of the magnitude of environmen- tal change ε . These results demonstrated that for the evolved network, Eq. (3) ismaintained under large environmental changes, whereas it holds only against smallchanges for random networks. These results confirmed the expansion of the linearregime by evolution. (a) (b) -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 e =0.02 e =0.08random netevolved net c o rr e l a t i o n r a t i o strength of stress e Fig. 1.4
Common proportionality in concentration changes in response to the same type of stressin the catalytic-reaction network model. (a) Coefficient of correlation between the changes in com-ponent concentrations δ X ( ε , E ) j and δ X ( ε , E ) j with ε = ε + ε . For the random and evolved net-works, the correlation coefficients with a small ( ε = .
02) and large ( ε = .
08) environmentalchange are plotted, which were obtained using 100 randomly chosen environmental vectors E . (b)Ratio of the slope in the concentration changes to the growth rate change as a function of the in-tensity of stress ε . The ratio in the ordinate becomes unity when Eq. (3) is satisfied. Reproducedfrom [21]. Next, we examined the correlation of concentration changes under different typesof environmental stressors. Fig. 5(a) shows examples of ( δ X ( ε , E a ) j , δ X ( ε , E b ) j ) ob-tained by three networks from different generations. For the initial random net-works, there was no correlation, whereas a modest correlation emerged in the 10thgeneration. Later, over evolution, common proportionality was observed; for in-stance, in the 150th generation, the proportionality reached more than two digits.To demonstrate the generality of the proportionality over a variety of environmen-tal variations, we computed the coefficients of correlation between δ X ( ε , E a ) j and δ X ( ε , E b ) j for a random choice of different vectors E a and E b . Fig. 5(b) shows thedistributions of the correlation coefficients obtained by the random network andevolved network (150th generation). Remarkably, global proportionality was ob-served even under different environmental conditions that had not been experiencedthrough the course of evolution.The relationship between δµ ( ε , E a ) / δµ ( ε , E b ) and the slope in ( δ X ( ε , E a ) j , δ X ( ε , E b ) j ) is presented in Fig. 5(c), whereas the ratio of the slope in ( δ X ( ε , E a ) j , δ X ( ε , E b ) j ) to δµ ( ε , E a ) / δµ ( ε , E b ) is shown in Fig. 5(d) as a function of ε . The slope of δ X agreesrather well with the growth rate change as given by Eq. (3) for the evolved networksas compared to the random networks. itle Suppressed Due to Excessive Length 11 (a) (d) dm / dm s l o p e i n ( , ) r a t i o strength of stress e
0 0.2 0.4 0.6 0.8 1 0.1 1 10 -2-1.5-1-0.5 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2gen=150gen=10gen=0 d X j ( e , E ) a (c) correlation coefficient f r e q u e n c y random netevolved net (b) d X j ( e , E ) b d X j ( e , E ) a d X j ( e , E ) b ( e , E ) a1 ( e , E ) b2 Fig. 1.5
Common proportionality in concentration changes in response to different types of stressin the catalytic-reaction network model. (a) Concentration changes across different types of envi-ronmental stressors. For three different networks from different generations, ( δ X j ( E a ) , δ X j ( E b )) are plotted by means of different randomly chosen vectors E a and E b . (b) Distributions of coef-ficients of correlation between the changes in component concentrations δ X ( ε , E a ) j and δ X ( ε , E b ) j .Red and green curves represent the distributions of a random network and evolved network (150thgeneration) obtained from 1000 pairs of E a and E b , respectively. The magnitude of environmentalchange ε is fixed at 0 .
8. (c) Relation between the ratio of growth rate change δµ ( ε , E a ) / δµ ( ε , E b ) and the slope in ( δ X ( ε , E a ) j , δ X ( ε , E b ) j ) . (d) Ratio of the slope in the relation between concentrationchanges and growth rate change as a function of the intensity of stress ε in the case of differenttypes of stress. Reproduced from [21]. The global proportionality over all components across various environmentalconditions suggests that changes δ X ( ε , E ) j across different environmental conditionsare constrained mainly along a one-dimensional manifold after evolution has pro-gressed (even) under a single environmental condition. To verify the existence ofsuch constraints, we carried out the principal component (PC) analysis of the dataof δ X ( ε , E ) j across different environmental changes E and ε . As shown in Fig. 6(a),we plotted the data in the space with the first three PC axes. In the evolved network,high-dimensional data from X ( ε , E ) j were located along a one-dimensional curve.Note that the contribution of the first PC reached 74% in the data. In contrast, thedata from the random network were scattered, and no clear structure was visible, asshown in Fig. 6(b). Furthermore, for the evolved network, the value of the first PCagrees rather well with the growth rate [21]. (a) (b) PC1PC2PC3
PC1PC2PC3
Fig. 1.6
The change in X ( ε , E ) j with environmental changes in principal-component space. Compo-nent concentrations X ( ε , E ) j at randomly chosen various E and ε values are presented for (a) evolvedand (b) random networks. In (a), the contributions of the first, second, and third components were74%, 8%, and 5%, respectively. Reproduced from [21]. We then examined the evolutionary course of the phenotype projected on thesame principal-component space, as depicted in Fig. 6(a). As shown in Fig. 7(a),the points from { X j } generated by random mutations in the reaction network areagain located along the same one-dimensional curve. Furthermore, those obtainedby environmental variation or noise in the reaction dynamics also lie on this one-dimensional curve, as shown in Fig. 7(b). Thus, the phenotypic changes are highlyrestricted, both genetically and non-genetically, within an identical one-dimensionalcurve. As shown in Figs. 6 and 7, variation in the concentration due to perturbationswas much larger along the first PC than along the other components. This suggeststhat relaxation is much slower in the direction of the first component than in theother directions.In summary, we observed emergent global proportionality which is far beyondthe trivial linearity in response to tiny perturbations. After evolution, the linearityregion expanded to a level with an order-of-magnitude change in the growth rate.Additionally, the proportionality over different components across different environ-mental conditions was enhanced through evolution. In this global proportionality,evolutionary dimension reduction in phenotypic dynamics underlies changes in thehigh-dimensional phenotype space across a variety of environmental conditions, ge-netic variations, and noise, which are confined to a common one-dimensional man-ifold. The first principal-component mode corresponding to the one-dimensionalmanifold is highly correlated with the growth rate.Notably, this dimension reduction by evolution has also been observed in someother models. First, even when the fitness for selection does not affect the growthrate but rather some other quantity (such as the concentration of a component),the phenotypic change is mainly constrained along a one-dimensional manifold.Second, even when the environmental condition (e.g., concentrations of externalnutrients) is not fixed but rather fluctuates over generations, restriction within theone-dimensional manifold is observed [62]. Third, there are preliminary reports that itle Suppressed Due to Excessive Length 13 the evolution of gene regulation networks and spin-glass models [59] also showdimension reduction of phenotypic changes, as observed in our results. (a) (b) PC1PC2PC3
PC1PC2PC3
Fig. 1.7
Change in component concentration X j because of (a) mutations and (b) noise in reactiondynamics. In (a), mutations were added to the evolved reaction network by randomly replacing0.5% of the reaction paths. The red dots show the concentrations of components after mutations,which are projected onto the same principal-component space, as depicted in Fig. 6(a). The graydots represent the concentration changes caused by environmental changes for the reference, whichare identical to those shown in Fig. 6(a). The red dots in (b) represent the concentration changesobserved at each cell division, which are caused by the stochastic nature of reaction dynamics.Reproduced from [21]. Following the observation of global proportionality and dimension reduction fromthe high-dimensional phenotypic space in the last section, we proposed the follow-ing hypothesis:Phenotypic dynamics involve a large number of variables, and their state spaceis generally high-dimensional. However, phenotypic changes induced by environ-mental perturbations are constrained mainly along a low-dimensional (often one-dimensional)manifold(majoraxis).Alongthismanifold,thephenotypicdynamicsare much slower than those across the manifold. Further, phenotypic changes in-duced by evolutionary changes (i.e., due to genetic changes governing phenotypicdynamics)also progressmainly alongthismanifold.Thefitnessgraduallychangesalongthemanifold,whereasitrapidlydecreasesacrossthemanifold.Indeed, the results described in the last section support the hypothesis, where thedominance of the first principal mode in a phenotypic change emerges after evolu-tion, and the phenotypic changes as a result of mutation are constrained along theprincipal-mode axis (see Fig. 7). Moreover, expression data from bacterial evolutionstudies support this hypothesis, as described later.This hypothesis is plausible considering the evolutionary robustness of pheno-types: First, in most cases, phenotypes (e.g., concentrations of chemicals) are shaped as a result of complex dynamics involving a large degree of freedom. For exam-ples, these dynamics can be determined by catalytic-reaction or gene-regulatory net-works, which are determined by genes. In general, networks with higher fitness arerare, and thus a mutation-selection process is needed to achieve higher fitness. Be-cause of the complexity of the dynamics, stochastic perturbations may influence thefinal phenotype in general, unless the networks are evolved to reduce the influenceof perturbations [34].As evolution progresses robustness of the state against perturbations will be ac-quired. Otherwise, because of inevitable noise during the dynamics, a rare fitter stateis not sustained. Increased robustness to perturbations is expected to result from evo-lution (see also [8, 33]). Accordingly, in the state space, the dynamics provide flowto the selected (fitter) phenotype against perturbations for most directions, as shownschematically in Fig. 8. Strong contraction to the attracted state is shaped by evolu-tion. However, there is (at least) one exceptional direction that does not possess sucha strong contraction. This is the direction along which evolution to increase fitnessprogresses. Along this direction, phenotype states can be changed rather easily byperturbations. Otherwise, it is difficult for evolution to progress. Hence, as schemat-ically shown in Fig. 8, only along the direction in which evolutionary changes pro-ceed, the relaxation is slow, whereas for other orthogonal directions, the change ismuch faster. X X X X X k X Major mode W
Fig. 1.8
Schematic representation of the dimension-reduction hypothesis. In the state space of X ,dominant changes are constrained along the mode W following the major axis and its connectedmanifold, whereas the attraction to this manifold is much faster. Now, let us consider the relaxation dynamics to the original state (attractor) at agiven generation, as represented by Eq. (2). The relaxation dynamics are representedby a combination of eigenmodes with negative eigenvalues. The magnitude of (neg-ative) eigenvalues will be large, except for one (or a few) eigenvectors, whereas thatalong the direction of evolution will be much closer to zero; that is, relaxation alongthe direction of evolution is slower. Hence, variance along the largest principal com-ponent will be dominant, as demonstrated numerically in Fig. 6. itle Suppressed Due to Excessive Length 15
Indeed, in a recent study [62], the eigenvalues of the Jacobi matrix in Eq. (2)were numerically computed by using the catalytic-reaction dynamics adopted in thelast subsection. The results confirmed that one eigenvalue was close to zero and allother (negative) eigenvalues had much larger magnitudes. This separation of oneeigenvalue occurred because of evolution.This hypothesis indicates that only one mode dominates in Eq.(2). Althoughthe original dynamics are high-dimensional, most changes occur along the one-dimensional manifold W , corresponding to the eigenvector for the smallest eigen-value (or its nonlinear extension). Let us denote this direction as w . From thisdominance of the single dominant mode W , the global proportionality in Eq. (3) isnaturally derived across different conditions E . Because changes in all components X j , δ X j s are constrained along the dominant mode W , they are given by the projec-tion of the change in W onto each X j axis, which is in turn given by cos θ j , where θ j is the angle between W and each axis X j . This angle is determined by the givenphenotype state only and is independent of the type of environmental perturbation.Thus, the proportionality observed over different strengths of an identical environ-mental condition is valid across different types of environmental perturbations.Note that the change along W is parametrized by the growth rate δµ because itshows tight one-to-one correspondence with the principal coordinate. Then δ W isrepresented as a function of δµ . Given that δ W ∝ δµ ( E ) for small δµ , Then, Eq.(3) can be extended to different environmental vectors as follows: δ X j ( E ) δ X j ( E ′ ) = δµ ( E ) δµ ( E ′ ) . (1.5)Indeed, the above argument can be formulated explicitly by using linear algebrafor the relaxation dynamics to the original state (attractor) at a given generation,as represented by Eq. (2). By using L = J − , it follows that δ X = L ( δµ I − γδ E ) , with I is an unit vector ( , , , .. ) T . The matrix L is represented by ∑ k λ k w k v Tk ,with their eigenvalues λ k , and the corresponding right (left) eigenvectors w k ( v k ),respectively. Noting that the hypothesis in the present section postulates that themagnitude of the smallest eigenvalue of J (denoted as k =
0) is much smaller thanthat of the others. In other words, the absolute eigenvalue of λ for L = J − ismuch greater than others. Then, the major response to environmental changes, isgiven only by the projection to this mode for λ . In other words, the major axis forthe change δ X is given by w . Using this reduction to this mode w , and throughstraightforward calculation(see [21] for details), we obtain δ X = λ w ( δµ ( v · I )) .Accordingly, the global proportionality relationship Eq. (5) is reproduced.In summary, we can explain two basic features observed in experiments and sim-ulations using the above theoretical formulation:(1) Overall proportionality in expression level changes across most componentsand across various environmental conditions:
This is because high-dimensionalchanges are constrained to changes along the major axis, i.e., eigenvector w .(2) Extended region of global proportionality:
Because the range in variationalong w is large, the change in the phenotype is constrained to points near this eigenvector, causing the proportionality range of phenotypic change to extend viaevolution. Furthermore, as long as the changes are nearly confined to the manifoldalong the major axis, global proportionality reaches the regime nonlinear to δε . According to the hypothesis in the last section, the change due to genetic variationwould also be constrained along with this major axis w , as the most changeabledirection w is the direction in which evolution has progressed and will progress.Indeed, in the simulation described in Sec. 5, the phenotypic changes caused by themutational change are constrained along the manifold spanned by the first principalmode in the environmentally induced phenotypic changes (see Fig.6). Accordingly,we expect to observe global proportionality between the concentration changes in-duced by a given environmental condition (stress) ( δ X j ( Env ) ) and those derivedfrom evolution with genetic changes ( δ X j ( Gen ) ), as given by δ X j ( Gen ) δ X j ( Env ) = δµ ( Gen ) δµ ( Env ) . (1.6)For example, when cells are subjected to a stressed condition Env , the growthrate is reduced so that δµ ( Env ) <
0, whereas the expression levels change δ X j ( Env ) accordingly. Next, the cells evolve under this given stressed condition over severalgenerations along with genetic changes. After n generations under genetic evolution,the growth rate recovers to some degree so that the growth rate shows a difference of δµ ( Gen ) from the original (no-stressed) state, satisfying 0 ≥ δµ ( Gen ) ≥ δµ ( Env ) .The accompanied expression change, denoted by δ X j ( Gen ) , is then expected tosatisfy δ X j ( Gen ) δ X j ( Env ) = δµ ( Gen ) δµ ( Env ) ≤ j in a similar manner as in Eq. (5). Because | δµ ( Gen ) | isreduced with the progression of evolution, changes in the components introduced bythe environmental change are reduced. Thus, there is an evolutionary tendency thatthe original expression pattern is recovered. This is reminiscent of the Le Chatelierprinciple in thermodynamics.We next examined if the above relationship would hold in numerical simulationand bacterial evolution experiments. itle Suppressed Due to Excessive Length 17 We employed the catalytic reaction network model in Sec. 5. After evolving the cellas described in the section under the given environmental condition, we switchedthe nutrient condition at a given generation. This caused the growth rate to decrease,which was later recovered through genetic evolution over generations. We computedthe phenotypic changes induced by the environmental and evolutionary changes toexamine the validity of the above relationship.After altering the nutrient conditions, the abundances of all the componentswere changed. The average change of these abundances was denoted by: δ X Envj ≡h X j ( ) i − h X j ( ) i = log h N j ( ) ih N j ( ) i , where generation 1 refers to the time point immedi-ately after the environmental change, and generation 0 denotes the generation rightbefore this nutrient change. Similarly, we defined the response by genetic evolutionafter m generations by δ X Genj ( m ) = h X j ( m ) i − h X j ( ) i . Figs. 9(a) and (b) show theplot of δ X Envj versus δ X Genj ( m ) for m = r ( m ) for δ X Genj ( m ) δ X Envj across compo-nents j . According to Eq. (6), this agrees with the growth rate change given bythe ratio of δµ Gen ( m ) = µ ( m ) − µ ( )( ≤ ) to δµ Env = µ ( ) − µ ( )( < ) at eachgeneration m . As shown in Fig. 9(c), the proportion coefficient r ( m ) was plottedagainst this growth rate recovery δµ Gen ( m ) / δµ Env . The agreement between thetwo is clearly discernible. This proportion coefficient r ( m ) is initially close to 1(i.e., m ∼ m , the value decreased towards zero, inconjunction with recovery of the growth rate δµ Gen ( m ) / δµ Env . Thus, as stated inLe Chatelier principle mentioned in the previous section, evolution shows a commontendency to reduce changes in components introduced by environmental change. -3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 d X j G e n ( ) d X jEnv (a) -3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 d X j G e n ( ) d X jEnv (b) p r o p o r t i o n c o e ffi c i e n t r dm Gen / dm Env (c)
Fig. 1.9
Response to environmental change versus response by evolution. Relationship betweenthe environmental response δ X Envj and genetic response δ X Genj ( m ) . (a) and (b) show the plots for m =
5, and 50, respectively. The black solid lines are y = x for reference. (c) Relationship be-tween growth recovery rate δµ Gen ( m ) / δµ Env and the proportion coefficient r ( m ) . The proportioncoefficient r ( m ) was obtained by using the least-squares method for the relationship of δ X Envj and δ X Genj ( m ) for m = ∼ y = x for reference. Reproduced from [20].8 Kunihiko Kaneko 1) and Chikara Furusawa 2) To verify the relationship given by Eq.(7), we analyzed time-series transcriptomedata obtained in an experimental evolution study of E. coli under conditions ofethanol stress [30, 31]. In this experiment, after cultivation of approximately 1,000generations (2,500 h) under 5% ethanol stress, 6 independent ethanol-tolerant strainswere obtained, which exhibited an approximately 2-fold increase in specific growthrates compared with the ancestor. For all independent culture series, mRNA sampleswere extracted from approximately 10 cells at 6 different time points, and the ab-solute expression levels were quantified by microarray analysis. All mRNA sampleswere obtained from cells in an exponential growth phase, (see [31] for details).Using the expression data taken at several generations through adaptive evolu-tion, we analyzed the common proportionality in expression changes. The environ-mental response of the j -th gene δ X Envj is defined by the log-transformed ratio ofthe expression level of the j -th. Similarly, the evolutionary response at n hours afterthe exposure to stress δ X Genj ( n ) is defined by the log-transformed ratio of the ex-pression level at n hours to that of the non-stress condition. We found a commontrend between the environmental and genetic responses over all genes [20].Furthermore, as shown in Fig. 10(a), the agreement between r ( n ) and the growthrecovery ratio δµ Gen ( n ) / δµ Env , as predicted by Eq.(7), was discernible, where δµ Gen ( n ) and δµ Env are the growth rate differences at n h and 24 hours after theexposure to stress, respectively. These results demonstrate that the evolutionary dy-namics with growth recovery were accompanied by gene expression changes, whichwere reduced from those introduced by the new environment.How does { X j } change in the state space of a few thousand dimensions? As itis hard to see a high-dimensional state space, we used the first, second, and thirdprincipal components determined from the data for each generation of E. coli geneexpression. (Approximately 31% of the change in each data set can be explainedby the 1st component, whereas 15% is explained by the 2nd component). When thedata points were plotted in the space of each principal component P i -axis, they weredistributed mainly along the P -axis direction, whereas the spread in the P and P axes was limited. Furthermore, the value of this first component was approximatelyproportional to the growth rate. This is consistent with the finding that the growthrate is a major factor in determining the change in expression of each gene. Now,Fig.10(b) shows the cell-state changes by projecting the expression state X j at eachgeneration onto these 3 component axes. Here, six independent data are superim-posed, which were obtained by repeating the same experiment. Mutations occurredat different sites by each experiment; each of the six strains has a different geneticsequence. Nevertheless, all experimental samples changed with the same curve. Thephenotypic changes to increase fitness are rather deterministic as compared to ran-dom changes in genetic sequences. Shaping the relevant phenotypic change is apriority in evolution, whereas several possibilities exist to achieve such changes ge-netically. itle Suppressed Due to Excessive Length 19 p r o p o r t i o n c o e ffi c i e n t r dm Gen / dm Env strain Astrain B strain Cstrain Dstrain Estrain F parentstrainPC1 (31.0%) P C ( . % ) P C ( . % ) (a) (b) Fig. 1.10
Response to environmental change versus response by evolution in E. coli adaption toethanol stress. (a) Relationship between growth recovery rate δµ Gen ( n ) / δµ Env and the proportioncoefficient r ( n ) . The proportion coefficient r ( n ) was obtained by using the least-squares methodfor the relationship of δ X Envj and δ X Genj ( n ) for n = δµ Gen ( n ) / δµ Env was calculated based on the experimental measurements(see [31] for details). Among the 6 independent culture lines in [31], the results of 5 culture lineswithout genome duplication are plotted. The black line is y = x for reference. (b) Changes in PCAscores during adaptive evolution. Starting from the parent strain, changes in the expression profilesduring adaptive evolution are plotted as orbits in the three-dimensional PCA plane. Reproducedfrom [20]. In the previous section, we discussed the relationship between environmental andevolutionary responses. According to statistical physics, response and fluctuationare proportional. In evolution, analogously, we previously proposed an evolutionaryfluctuation-response relationship: the phenotypic response throughout the evolution-ary course is proportional to the phenotypic variance induced by noise [61, 34].Consider a system characterized by a gene parameter a and phenotypic trait X . Wecan then evaluate the change in X against that in the gene parameter value from a to a + ∆ a . Then, the proposed fluctuation-response relationship is given by h X i a + ∆ a − h X i a ∆ a ∝ h ( δ X ) i , (1.8)where h X i a and h ( δ X ) i = h ( X − h X i ) i a are the average and variance of the phe-notypic trait X for a given system parameterized by a , respectively.If a is assigned as a parameter specifying the genotype (e.g., number of substi-tutions in the DNA sequence), this relationship implies that the evolutionary rate,i.e., change in average phenotype per generation, is proportional to the varianceof the isogenic (clonal) phenotypic distribution, denoted by V ip = h ( δ X ) i . In fact,several model simulations and some experiments support this type of evolutionaryfluctuation-response relationship [61, 36].This evolutionary fluctuation-response relationship is associated with the phe-notypic variance V ip of isogenic individuals, which is caused by noise. In standard population genetics, in contrast, the phenotypic variance due to genetic variation,named as genetic variance V g is considered. In fact, the so-called fundamental theo-rem of natural selection proposed by Fisher [14] states that the evolutionary rate isproportional to V g . Thus, for both the evolutionary fluctuation-response relationshipand Fisher’s theorem to be valid, V ip and V g must remain proportional throughout theevolutionary course. Indeed, such a relationship between these two variances wasconfirmed through evolutionary simulations of a catalytic reaction network modeland gene regulation network [36, 33].The origin of such a relationship can be explained as follows. In general, develop-mental dynamics to shape the phenotype are quite complex, and the final state maybe diverted by perturbations in the initial condition or by those that occur duringthe dynamics. Even if the fit phenotype is shaped by developmental dynamics, theperturbations due to noise during the dynamics may result in different non-fit states.Thus, the phenotype may be rather sensitive to noise. After evolution progresses, therobustness of the fitted state to noise is increased. In this case, the global attractionto the target phenotype is shaped. This agrees with the hypothesis in Sec. 6.Genetic changes, in contrast, can also cause perturbations to such dynamics. Asthe robustness to noise is shaped, the robustness against genetic changes is also ex-pected to increase. Through evolution, as the dynamics become increasingly robustto noise, they also become more robust to genetic changes, resulting in a correla-tion between the two types of robustness. As the robustness to noise is increased,the phenotypic variance V ip will decrease; similarly, an increase in the robustnessto genetic mutation leads to a decrease in V g . Hence, throughout evolution, both V ip and V g decrease in correlation (or in proportion). Thus, proportionality between V ip and V g is expected, as observed in the evolution simulations [36, 33]. Furthermore,this proportionality of the two variances is extended to that between the phenotypicvariance X i for each component i by noise (written as V ip ( i ) ) and that by geneticvariation (denoted by V g ( i ) ) [20, 35, 34]. This, indeed, is expected by assuming thatthe relaxation of phenotypic changes is much slower along the dominant mode W ,phenotypic fluctuations due to noise are nearly confined to this axis. In summary, we demonstrated the 2-by-2 global proportionality of phenotypicchanges occurring between responses and fluctuations and between the perturba-tions due to environmental (noise) and genetic changes. This proportionality is ex-plained by evolutionary dimension reduction, which states that phenotypic changesdue to environmental changes and genetic variation are constrained along a uniquelow-dimensional manifold, as observed in bacterial and numerical evolution exper-iments. Furthermore, expression changes induced by environmental stress, for mostgenes, are reduced through evolution to recover the growth, which is analogous tothe Le Chatelier principle in thermodynamics [6, 43]. itle Suppressed Due to Excessive Length 21
We demonstrated in numerical evolution that high-dimensional phenotypic changeis mainly constrained along with the mode w , the eigenvector corresponding to theeigenvalue of the relaxation dynamics closest to zero. The change in the phenotypicstate is larger along the direction of w , and the variable W along this directionslowly returns to the steady state value. The time scale of this mode is distinctivelyslower than others, as confirmed directly by evolution simulation of the catalyticreaction and gene regulation networks [62]. This separation of the timescale of theslowest mode from others is theoretically expected in order to make the robustnessof the fitted state and plasticity along the evolutionary course compatible.Formation of one (or few) slow mode as given by W separated from other modesis significant in evolutionary biology. It may be possible that this type of mode isstraightforwardly given by the expression of some specific genes that changes moreslowly than others. However, it may be more natural that this W is expressed as acollective change in the expressions of several genes rather than as a single protein.Because the slow mode is expressed by the first principal component, determinationof genes whose expression levels contribute more to the first principal componentwill improve the understanding of how plasticity and robustness are compatible.The slow, dominant mode W emerges from evolution, but accelerates evolution.When faster and slower variables coexist and interact, the slower mode generallyfunctions as a control variable for the faster variables. Accordingly, if the slowermode is modified by a genetic change, most faster variables will be influenced si-multaneously. Furthermore, because the mode W can influence the fitness µ , thephenotypic evolution will be feasible simply by the change in this slow mode W .In contrast, if many variables change in a similar time scale, the genetic changesintroduced to each will influence each other, and make directional phenotypicchanges difficult to progress. This situation is reminiscent of the proverb “Too manycooks spoil the broth.” The emergence of slow modes governing the others has alsobeen observed in the evolution of pattern formation [42].The correlation between evolutionary and environmental responses raises a ques-tion regarding how the two processes with quite different time scales are correlated.The presence of the slow mode W suggests a possible answer to this question. Adap-tive dynamics, which originally show a much faster timescale than the evolutionarychange, will be slowed along the mode W , whereas the evolutionary change, whichoriginally has a much slower time scale is fastest, along the direction of the mode W . Thus, along the dominant mode W , the timescales of phenotypic adaptation andevolution can approach each other.Of course, further studies are needed to establish a phenomenological theoryfor phenotypic evolution. The generality of the evolutionary dimension reductionand resultant constraint in phenotypic evolution must be explored. The conditionrequired for the emergence of dimensional reduction should also be determined.The models we studied satisfy the following conditions: (i) phenotypes with higherfitness are shaped by complex high-dimensional dynamical systems and (ii) the frac-tion of such fit states is rare in the state space and in the genetic-rule space. Thesetwo features are also consistent with our theoretical argument. The evolution of astatistical-physics model on interacting spins that preliminarily supports the dimen- sion reduction also satisfies these conditions [59, 60]. Studies of statistical physicand high-dimensional dynamical systems are needed to reveal the condition for evo-lutionary dimension reduction. Further experimental confirmation of the dimensionreduction, as well as the directionality and constraint in phenotypic evolution, isneeded, including in multicellular organisms.Note that dimension reduction, or separation of a slow eigenmode from otherfaster modes, has been discussed in several other topics. They include protein dy-namics [70, 69], laboratory ecological evolution [16], learning in brain [44], andneural networks [64], among others. As a possible relationship, the sloppy param-eter hypothesis by Sethna [10] is proposed, which suggests that many parametersemployed in biological models are irrelevant. Further studies are necessary to ex-plain the universality of such evolutionary dimension reduction.There are some limitations to dimension reduction. In the studies described here,we assumed steady-growth state, i.e., exponential phase. Under nutrient-limitedconditions, however, there occurs a transition from such exponential growth to thestationary phase with suppressed growth, as has been investigated both experimen-tally [23, 51] and theoretically [28, 47]. As such phase with suppressed growth maynot be selected as a robust fitted state, whether the dimension reduction is valid to itrequires further analysis. Acknowledgment
The authors would like to thank Takuya U Sato and Tetsuhiro Hatakeyama forstimulating discussions. This research was partially supported by a Grant-in-Aid forScientific Research (S) (15H05746) from the Japanese Society for the Promotionof Science (JSPS) and Grant-in-Aid for Scientific Research on Innovative Areas(17H06386 and 17H06389) from the Ministry of Education, Culture, Sports, Sci-ence and Technology (MEXT) of Japan.