Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 529
Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability; Waddington’s legacy revisited under the spirit of Einstein
KUNIHIKO KANEKO Department of Pure and Applied Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan and ERATO Complex Systems Biology Project, JST, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan (Email, [email protected])
Questions on possible relationship between phenotypic plasticity and evolvability, and that between robustness and evolution have been addressed over decades in the fi eld of evolution-development. Based on laboratory evolution experiments and numerical simulations of gene expression dynamics model with an evolving transcription network, we propose quantitative relationships on plasticity, phenotypic fl uctuations, and evolvability. By introducing an evolutionary stability assumption on the distribution of phenotype and genotype, the proportionality among phenotypic plasticity against environmental change, variances of phenotype fl uctuations of genetic and developmental origins, and evolution speed is obtained. The correlation between developmental robustness to noise and evolutionary robustness to mutation is analysed by simulations of the gene network model. These results provide quantitative formulation on canalization and genetic assimilation, in terms of fl uctuations of gene expression levels.
[Kaneko K 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability; Waddington’s legacy revisited under the spirit of Einstein; J. Biosci. 34 529–542]
1. Introduction 2005; Ciliberti et al. 2007; Kaneko 2007). In any biological system, these changes have two distinct origins: genetic 1.1 Questions on evolution and epigenetic. The former concerns structural robustness of the phenotype, i.e. rigidity of phenotype against genetic (i) How is evolvability characterized? Is it related with changes produced by mutations. On the other hand, the latter developmental process? Naively speaking, one may expect concerns robustness against the stochasticity that can arise in that if developmental process is rigid, phenotype is not environment or during developmental process, which includes easily changed, so that evolution does not progress so fast. fl uctuation in initial states and stochasticity occurring during Then one may expect that evolvability is tightly correlated developmental dynamics or in the external environment. with phenotypic plasticity. Here the plasticity refers to the Now, there are several questions associated with changeability of phenotype as a result of developmental robustness and evolution. Is robustness increased or dynamics (Callahan et al. 1997; West-Eberhard 2003; decreased through evolution? In fact, Schmalhausen (1949) Kirschner and Gerhart 2005; Pigliucci et al. 2006). If there proposed stabilizing evolution and Waddington (1942, is correlation between plasticity and evolvability (Ancel and 1957) proposed canalization. Both of them discussed the Fontana 2002), one may formulate relationship between possibility that robustness increases through evolution. On evolvability and development quantitatively. the other hand, since the robustness increases rigidity in (ii) Besides plasticity, an important issue in evolution is phenotype, the plasticity as well as the evolvability may be robustness. Robustness is defi ned as the ability to function decreased accordingly through evolution. against possible changes in the system (Barkai and Leibler Besides these robustness-evolution questions, another 1997; Wagner et al. 1997; Alon et al. 2000; Wagner 2000, issue to be addressed is the relationship between the two
Keywords. Evolvability; genetic assimilation; phenotypic fl uctuation; phenotypic plasticity
J. Biosci. 34(4), October 2009 J. Biosci. 34(4), October 2009, 529–542, © Indian Academy of Sciences 529 530 Kunihiko Kaneko types of robustness, genetic and epigenetic. Do the two types is due only to genetic heterogeneity. This variance is of robustness increase or decrease in correlation through denoted as Vg here. Then, the robustness to mutation is evolution? In fact, the epigenetic robustness concerns with estimated by this variance. If Vg is smaller, genetic change developmental stability. Higher epigenetic robustness is, gives little infl uence on the phenotype, and larger genetic (or more stable the phenotype in concern shaped by development mutational) robustness is implied. is. On the other hand, genetic robustness concerns with Phenotypic plasticity, on the other hand, represents the stability against mutational change, i.e. evolutionary the degree of change in phenotype against variation stability. Hence the question on a possible relationship in environment. Quantitatively, one can introduce the between genetic and epigenetic robustness concerns with following “response ratio” as a measure of plasticity. the issue on evolution-development relationship, that is an It is defi ned by the ratio of change in phenotype to important topic in biology. environmental change. For example, measure the (average) concentration (x) of a specifi c protein. We measure this concentration by (slightly) changing the concentration 1.2 Formulation of the problem in terms of (s) of an external signal molecule. Then, response ratio stochastic dynamical systems R is computed as the change in the protein concentration divided by the change in the signal concentration (R = dx/ Here we study the questions raised above, in terms of ds)1. This gives a quantitative measure for the phenotypic fl uctuations. Let us fi rst discuss epigenetic robustness, plasticity. In general, there can be a variety of ways to i.e. the degree of phenotype change as a result of change the environmental condition. In this case, one may developmental process. The developmental dynamics to also use the variance of such response ratios over a variety shape a phenotype generally involves stochasticity. As has of environmental changes, as a measure of phenotypic been recognized extensively, gene expression dynamics are plasticity. noisy due to molecular fl uctuation in chemical reactions. By using these variances V , V and plasticity the questions Accordingly, the phenotype, as well as the fi tness, of ip g we address are represented as follows (Gibson and Wagner isogenic individuals is distributed (Oosawa 1975; Spudich 2000; de Visser et al. 2000; Weinig 2000; Ancel and Fontana and Koshland 1976). Still, too large variability in the 2002; West-Eberhard 2003; Kirschner and Gerhart 2005): phenotype relevant to fi tness should be harmful to the survival of the organisms. Phenotype that is concerned (i) Does the evolution speed increase or decrease with fi tness is expected to keep some robustness against with robustness?— relationship between evolution such stochasticity in gene expression, i.e. robustness in speed with Vg or Vip. ‘developmental’ dynamics to noise. (ii) Are epigenetic and genetic robustness related?
When the phenotype is less robust to noise, this — relationship between Vg and Vip. distribution is broader. Hence, the variance of isogenic (iii) Does robustness in-(de-)crease through evolution? — change of V or V through evolution. phenotypic distribution, denoted as Vip here, gives an g ip index for robustness to noise in developmental dynamics (iv) How is phenotypic plasticity related with (Kaneko 2006, 2007). For example, distributions of protein robustness and evolution speed? — relationship abundances over isogenic individual cells have been between R and Vg or Vip. measured, by using fl uorescent proteins. The fl uorescence To answer these questions, we have carried out evolution level which gives an estimate of the protein concentration experiments by using bacteria (Sato et al. 2003) and is measured either by fl ow cytometry or at a single cell numerical simulations on catalytic reaction (Kaneko and microscopy (McAdams and Arkin 1997; Hasty et al. 2000; Furusawa 2006) and gene networks (Kaneko 2007, 2008), Elowitz et al. 2002; Bar-Even et al. 2004; Furusawa et al. while we have also developed a phenomenological theory 2005; Kaern et al.2005; Krishna et al. 2005). on the phenotype distribution. Here we briefl y review Genetic robustness is also measured in terms of these results and discuss their relevance to evolution- fl uctuation. Due to mutation to genes, the phenotype development. (fi tness) is distributed. Since even phenotype of isogenic individuals is distributed, the variance of phenotype 1As will be discussed in Section 2.3, it is often relevant to defi ne distribution of heterogenic population includes both the the phenotype variable after taking a logarithm. In this case a phe- contribution from phenotypic fl uctuation in isogenic notype variable is defi ned as x = log(X) with X as the original vari- individuals and that due to genetic variation. To distinguish able, say, the protein concentration. Likewise, external signal often the two, we fi rst obtain the average phenotype over isogenic works in a logarithmic scale, so that the external parameter s is also individuals, and then compute the variance of this average defi ned as log(S), with S as a signal concentration. In this case R = phenotype over heterogenic population. Then this variance dlog(X)/dlog(S), which, indeed, is often adopted in cell biology.
J. Biosci. 34(4), October 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 531
2. Macroscopic approach I: 2.2 Supports from numerical experiments Fluctuation-response relationship To confi rm this relationship between the evolution speed and One might suspect that isogenic phenotype fl uctuations isogenic phenotypic fl uctuation quantitatively, we have also
Vip are not related to evolution, since phenotypic change numerically studied a model of reproducing cells consisting without genetic change is not transferred to the next of catalytic reaction networks. Here the reaction networks of generation. However, the variance, a degree of fl uctuation mutant cells were slightly altered from the network of their itself, can be determined by the gene, and accordingly it is mother cells. Among the mutants, those networks with a heritable (Ito et al. 2009). Hence, there may be a relationship higher concentration of a given, specifi c chemical component between isogenic phenotypic fl uctuation and evolution. In were selected for the next generation. Again, the evolution fact, we have found evolutionary fl uctuation–response speed was computed by the increase of the concentration relationship in bacterial evolution in the laboratory, where at each generation, and the fl uctuation by the variance of the evolution speed is positively correlated with the variance the concentration over identical networks3. From extensive of the isogenic phenotypic fl uctuation (Sato et al. 2003). numerical experiments, the proportionality between the two This correlation or proportionality was confi rmed in a was clearly confi rmed (Kaneko and Furusawa 2006). simulation of the reaction network model (Kaneko and Furusawa 2006). The origin of proportionality between 2.3 Remark on Gaussian type distribution the isogenic phenotypic fl uctuation and genetic evolution has been discussed in light of the fl uctuation–response In the above experiment and numerical model, the relationship in statistical physics (Einstein 1926; Kubo fl uorescence or the concentration obeys approximately et al. 1985). In the present section we briefl y summarize the log-normal distribution, i.e. log(fl uorescence) or these recent studies. log(concentration) obeys nearly the Gaussian distribution. In the following analysis that we borrow from statistical physics, 2.1 Selection experiment by bacteria the distribution is assumed to be nearly Gaussian. If the distribution is of the log-normal type, we adopt such quantity We carried out a selection experiment to increase the taking a logarithm, so that the distribution of the phenotypic fl uorescence in bacteria and checked a possible relationship variable is nearly Gaussian, in order to apply the theoretical between evolution speed and isogenic phenotypic discussion. In fact, Haldane suggested that most phenotypic fl uctuation. First, by attaching a random sequence to the N variables should be defi ned after taking a logarithm. In the terminus of a wild-type Green Fluorescent Protein (GFP) above analyses between fl uctuation and response, all the gene, protein with low fl uorescence was generated. The quantities are defi ned and computed after taking a logarithm gene for this protein was introduced into Escherichia coli, log(fl uorescence) and log(concentration) are used to compute as the initial generation for the evolution. By applying the evolution speed and the variance. The linear relationship random mutagenesis only to the attached fragment in the between the two is confi rmed by using these variables. gene, a mutant pool with a diversity of cells was prepared. Then, cells with highest fl uorescence intensity are selected 2.4 Fluctuation-response relationship for the next generation. With this procedure, the (average) fl uorescence level of selected cells increases by generations. To understand the proportionality between the phenotypic The evolution speed at each generation is computed as fl uctuation and the evolution speed, we fi rst discuss a the difference of the fl uorescence levels between the two possible general relationship between fl uctuation and generations. Then, to observe the isogenic phenotypic response. Here, we refer to a measurable phenotype quantity fl uctuation, we use a distribution of clone cells of the selected (e.g. logarithm of the concentration of a protein) as a bacteria. The distribution of the fl uorescence intensity of the “variable” x of the system, while we assume the existence clones was measured with the help of fl ow cytometry, from of a “parameter” which controls the change of the variable. which the variance of fl uorescence was measured. The parameter corresponding to the variable x is indicated From these data we plotted the evolution speed divided as a. In the case of adaptation, this parameter is a quantity by the synonymous mutation rate versus the fl uorescence to specify the environmental condition, while, in the case variance2. The data support strong correlation between the of evolution, the parameter is an index of genotype that two, suggesting proportionality between the two (Sato et al. governs the corresponding phenotypic variable. 2003). 3Again, the phenotypic variable x is defi ned by log(concentration), 2To be precise, the phenotype variable x is defi ned by log (fl uores- and the variance is measured by the distribution of log cence), as will be discussed in Section 2.3. (concentration) (see Section 2.3).
J. Biosci. 34(4), October 2009 532 Kunihiko Kaneko
Even among individuals sharing the identical Remark: Although the above formula is formally gene (the parameter) a, the variables x are distributed. similar to that used in statistical physics, it is not The distribution P(x;a) of the variable x over cells grounded on any established theory. It is a proposal is defi ned, for a given parameter a (e.g. genotype). here. We set up a phenomenological description so Here the “average” and “variance” of x are defi ned that the formula is consistent with the data obtained by with regards to the distribution P(x;a) (see fi gure 1 experiments and numerical simulations on reaction network for schematic representation of the fl uctuation-response models. (However, such phenomenological description relationship). is the fi rst necessary step in science, as the history of Consider the change in the parameter value from a to a→ thermodynamics tells.) The choice of the distribution form a+δa. Then, the proposed fl uctuation-response relationship is not derived from the fi rst principle, but is based on several (Sato et al. 2003; Kaneko 2006) is give by assumptions. First, the distribution must be close to Gaussian, or the <>xx −<> aa+Δ a ∝<()δx 2 >, (1) variable has to be scaled properly so that the corresponding Δa distribution is nearly Gaussian. If the variance of the distribution of the variable in concern is not very 2 2 large, by suitable choice of variable transformation, where
Noting α = <(δx)2>, and neglecting deviation between (i) a — parameter to characterize the environmental α(a + Δa) and α(a ), we get condition 0 0 (ii) change in a — change in environment <>xx −<> (iii)
J. Biosci. 34(4), October 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 533
and phenotypic fl uctuation. It is the so-called fundamental theorem of natural selection by Fisher (1930) which states
that evolution speed is proportional to Vg, the variance of the phenotypic fl uctuation due to genetic variation. It is given by the variance of the average phenotype for each genotype over a heterogenic population. In contrast, the evolutionary fl uctuation – response relationship, proposed above, concerns phenotypic fl uctuation of isogenic individuals as
denoted by Vip. While Vip is defi ned as a variance over clones,
i.e., individuals with the same genes, Vg is a result of the heterogenic population distribution. Hence, the fl uctuation–
response relationship and the relationship concerning Vg by Fisher’s theorem are not identical.
If Vip and Vg are proportional through an evolutionary course, the two relationships are consistent. Such
proportionality, however, is not self-evident, as Vip is related
to variation against the developmental noise and Vg is against the mutation. The relationship between the two, if it exists, Figure 1. Schematic diagram on fl uctuation-response postulates a constraint on genotype–phenotype mapping relationship. The distribution function P(x;a) is shifted by the and may create a quantitative formulation of a relationship change of the parameter a. If the variance is larger the shift is between development and evolution. larger. Here we formulate a phenomenological theory so that the evolutionary fl uctuation-response relationship and 2.6 Evolutionary fl uctuation-response relationship Fisher’s theorem is consistent (see fi gure 2). Recall that originally the phenotype is given as a function of gene, so This fl uctuation-response relationship can be applied to that we considered the conditional distribution function evolution by making the following correspondence: P(x;a). However, the genotype distribution is infl uenced (i) a — a parameter that specifi es the genotype by phenotype through the selection process, as the fi tness (ii) change in a — genetic mutation for selection is a function of phenotype. Now, consider a (iii) change in
− 2 ()xX0 3. Macroscopic approach II: Stability theory for Pxa(),= N exp[ − + 2α()a genotype-phenotype mapping (5) 1 Cx()())()]−−−−, X a a a a 2 00μ 0 The evolutionary fl uctuation-response relationship men- 2 ˆ tioned above casts another mystery to be solved. There with N as a normalization constant. The Gaussian distri- exp(−−1 (aa )2 ) is an established relationship between evolutionary speed bution 2 μ 0 represents the distribution of geno-
J. Biosci. 34(4), October 2009 534 Kunihiko Kaneko
types at around a = a0, whose variance is (in a suitable by recalling that Vip (a) = α(a). Thus we get the proportionality unit) the mutation rate μ. The coupling term C(x–X0) (a–a0) between Vip and Vg. represents the change of the phenotype x by the change of Note that if we choose the distribution consisting of genotype a. Recalling that the above distribution (5) can be heterogenic individuals taking a given phenotype value 2 rewritten as x only, the variance <(a–a0) >fi x e d – x is given simply by μ. The variance of average phenotype over such population −− −α 2 (()())xX00 Caa a Pxa(),= N exp[ − sharing the phenotype value x is defi ned as Vig (Kaneko and 2α()a Furusawa 2008). Then, by using eq. (7) we get V = (αC)2 < (6) ig Ca2α() 1 (a–a )2 > = μ (αC)2, instead of eq.(9). Accordingly, the +−−( )(aa )2 ], 0 fi x e d – x 2 2μ 0 inequality (8) is rewritten as the average phenotype value for given genotype a Vip ≥ Vig, (11) μ satisfi es VV= while ip μ ig is also obtained without assuming μ « max μ . If the evolution process successively increases the x a ≡,=+−.∫ xP() x a dx X C ( a a )()α a (7) max 00 selected value of x by generations, the distribution at each generation is centered at around a given phenotype. In such Now we postulate evolutionary stability, i.e. at each stage of case, V , instead of V , may work as a relevant measure for the evolutionary course, the distribution has a single peak in ig g the phenotypic variance due to genetic change. Then the (x, a) space. This assumption is rather natural for evolution inequality (11) can work as a measure for stability. to progress robustly and gradually. When a certain region of Note again that the above relationships (10) and (11) are phenotype, say x > x , is selected, the evolution to increase thr not derived from the fi rst principle, but a phenomenological x works if the distribution is concentrated. On the other description to be consistent with numerical experiments. hand, if the peak is lost and the distribution is fl attened, then There are several assumptions to obtain the relationships. selection to choose a fraction of population to increase x no First, whether genotype is represented by a continuous longer works, as the distribution is extended to very low- parameter a is not a trivial question, as gene is coded fi tness values (see also a remark at the end of this section). by a genetic sequence. Possible change by mutation to In the above form, this stability condition is given by αC 2 1 the sequence can be represented by a walk along a high- −≤0,..ie 2 2μ dimensional hyper-cubic lattice, and whether one can introduce a scalar parameter a instead is not self-evident. A candidate for such scalar parameter is Hamming distance μ ≤≡.1 μ (8) αC 2 max from the fi ttest sequence, or a projection according to the corresponding phenotype (Sato and Kaneko 2007). We This means that the mutation rate has an upper bound note that such scalar genetic parameter is often adopted in
μmax beyond which the distribution does not have a peak population genetics also. in the genotype–phenotype space. In the above form, the Second, description by using a two-variable distribution distribution becomes fl at at μ = μmax, so that mutants with in genotype and phenotype P(x,a) is not trivial, either. As low fi tness rate appear, as discussed as error catastrophe the gene controls the phenotype, use of the conditional by Eigen (Eigen and Schuster 1979) (see also the remark distribution P(x|a), i.e. the distribution of phenotype for 2 at the end of this section). Recalling Vg = <(x(a)–X0) > and given genotype a would be natural, to compute P(x,a) as 2 2 2 eq.(6), Vg is given by (Cα) < (δa) >. Here, (δa) is computed p(a)P(x|a). In contrast, the eq. (5) takes symmetric form by the average over P(x,a), so that it is obtained by with regards to x and a. Here the form is chosen after taking
2 C 2 1 2 into account of selection process based on phenotype x. This Naa∫ ()exp[()()]−−−μ aada . 0 2α()a 2 0 selection process introduces a feedback from phenotype to From this formula we obtain the genotype distribution as in fi gure 2. Considering quasi- static evolutionary process to select phenotype x, the form 22 μ μαC μ = = α max . (5) is chosen for P(x,a). Vg 2 μ (9) − μα − μ 11C max Third, the stability assumption is introduced for robust and gradual evolution, postulating that P(x,a) has a single If the mutation rate μ is small enough to satisfy μ« μ , we max peak in (x,a) space at each generation. Fourth, existence get of threshold mutation rate μmax is implicitly assumed, μ beyond which the stability condition is not satisfi ed. In VV∼ , g μ ip (10) other words, existence of error catastrophe is implicitly max assumed. Such catastrophe does not exist for every coupling
J. Biosci. 34(4), October 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 535 form in eq. (5). For example, if we adopted the distribution with stochasticity in genotype-phenotype mapping,
−−((xcaa − ))(2 aa − )2 as demonstrated by Vip > 0, which is a result of noise −−0 0 , exp()22αμ encountered during the developmental dynamics. In fact, in the example to be discussed in the next section, the instead, the stability condition would always be satisfi ed, catastrophe occurs as a result of the decrease in the noise as long as α > 0 and μ > 0. In other words, we assumed strength in developmental dynamics. In this sense, the error existence of such loss of stability with the increase of the catastrophe here is not identical with that by Eigen. mutation rate, to choose the coupling form in eq. (5). (see also a remark below). 4. Microscopic approach: Gene regulation network Fifth, to get the proportionality (10), additional model assumption is made on the smallness of the mutation rate μ, as well as the constancy of μ and μ through the max As the above discussion on phenotypic variances on genetic evolutionary course. and epigenetic origins is based on several assumptions, Note again we have used the expansion up to the second it is important to study the evolution of robustness order around a0 and X0 in eq. (5), i.e. the Gaussian-like and phenotypic fl uctuations, by using a model with distribution as well as the coupling term C(x–X )(a–a ). 0 0 “developmental” dynamics of phenotype. To introduce In fact, this specifi c form is not necessary. By taking a such dynamical systems model, one needs to consider the generalized from, the single-peaked distribution assumption following structure of evolutionary processes. leads to the Hessian condition for logP(x,a) around the peak at a = a and x = X . This leads to the inequality (11)4. For (i) There is a population of organisms with a distribution 0 0 of genotypes. example, when we take the coupling C′(x–X0)(a–a0) / α in eq. 2 (ii) Phenotype is determined by genotype, through (5), the inequality (8) is replaced by μ < αC′ =μmax, whereas 2 2 “developmental” dynamics. the equations on Vg is replaced by Vg = μ C′ / (1–μC′ /α), so that the fi nal form in eq. (9) is identical. (iii) The fi tness for selection, i.e. the offspring number, is given by the phenotype. Remark: At μ = μmax, the distribution (5) becomes fl at. In fact, such loss of stability of a fi tness distribution (iv) The distribution of genotypes for the next roughly corresponds to error catastrophe by Eigen and generation changes according to the change of Schuster (1979), where non-fi tted mutants accumulate with genotype by mutation, and selection process the increase of mutation rate, due to their combinatorial according to the fi tness. explosion. When fi tted genotypes are rare, they are not During evolution in silico, procedures (i), (iii), and (iv) are sustained against the increase of the mutation rate. adopted as genetic algorithms. Here, however, it is important In our formulation, however, we cannot tell anything that the phenotype is determined only after (complex) about the distribution when the inequality (8) (or equivalently ‘developmental’ dynamics (ii), which are stochastic due to eq. [11]) is not satisfi ed. In the formulation with Gaussian the noise therein. distribution, the distribution of phenotype is symmetric around We previously studied a protocell model with catalytic the peak. However, near this catastrophic point, the variance reaction network in which cells that have a higher increases so that the expansion up to the second order in eq. (5) concentration of a specifi c chemical are selected among will no longer be valid. In general, the genetic sequence giving the networks having different paths. Extensive numerical rise to fi tted phenotype is rare, and thus the distribution at simulations confi rm the inequality Vip > Vig, error catastrophe at Vip ~ Vig, and the proportionality between Vip and Vg. Note μ ~ μmax is expected to be extended mostly to the lower fi tness side, due to possible higher order terms in the expansion of that in the numerical evolution of this model, none of the fi ve assumptions in the last section are adopted. a–a0 and x–X0. (In other words, this asymmetry to the side of lower fi tness values in the distribution is also expected As another example, we have studied gene expression from the projection to a one-dimensional scalar variable a dynamics that are governed by regulatory networks (Glass from a genetic sequence.) Accordingly, it is expected that and Kauffman 1973; Mjolsness et al. 1991; Salzar-Ciudad the evolution to select a higher fi tness phenotype no longer et al. 2000). The developmental process (ii) is given by this works when the inequality is not satisfi ed, as discussed by gene expression dynamics. It shapes the fi nal gene expression Eigen as error catastrophe. profi le that determines the evolutionary fi tness. Mutation A remark should be made here: In the discussion of at each generation alters the network slightly. Among the Eigen, phenotype is uniquely determined by the genotype. mutated networks, we select those with higher fi tness values. In contrast, the present error catastrophe is associated To be specifi c, the dynamics of a given gene expression level xi is described by: 4Note V in (Kaneko and Furusawa 2006) needs to be interpreted M g dx/= dtγση(( f∑ J x ) − x ) + () t , as V here, as remarked in Kaneko and Furusawa (2008). i ij j i i (12) ig j
J. Biosci. 34(4), October 2009 536 Kunihiko Kaneko
Figure 2. Schematic representation of genotype-phenotype relationship. The genotype a is distributed by p(a), whereas there is a stochastic mapping from the genotype to phenotype as a result of developmental dynamics with noise. Thus the phenotype x is distributed even for isogenic individuals, so that the distribution P(x,a) is defi ned. Mutation-selection process works on phenotype x, so that a feedback from P(x,a) to the genotype distribution P(a) is generated. where Jij = –1,1,0, and ηi(t) is Gaussian white noise given by As the model contains a noise term, whose strength is
<ηi(t)ηj(t′)> = δi,jδ(t–t′). M is the total number of genes. The given by σ, the fi tness can fl uctuate among individuals sharing function f represents gene expression dynamics following the same genotype Jij. Hence the fi tness F is distributed. This Hill function, or described by a certain threshold dynamics leads to the variance of the isogenic phenotypic fl uctuation
(Kaneko 2007, 2008). denoted by Vip ({J}) for a given genotype (network) {J}, i.e. The value of σ represents the strength of noise encountered in gene expression dynamics, originated in molecular =;−,∫ ˆ 2 (13) Vip () {J} dFP()(()) F {J} F F {J} fl uctuations in chemical reactions. We prefi x the initial condition for this “developmental” dynamical system (say, at the condition that none of the genes are expressed). The where Pˆ (F;{J}) is the fi tness distribution over isogenic ¯ ˆ phenotype is a pattern of {xi} developed after a certain time. species sharing the same network Jij, and F ({J}) = ∫FP
The fi tness F is determined as a function of xi. Sometimes, (F; {J})dF is the average fi tness. To check the relationship not all genes contribute to the fi tness. It is given by a between Vip and Vg, we either use the Vip for {J} that gives function of only a set of “output” gene expressions. In the ¯ the peak value of the fi tness distribution or the average Vip paper (Kaneko 2007), we adopted the fi tness as the number = ∫d {J}p ({J}) V ({J}), where p({J}) is the population … ip of expressed genes among the output genes j = 1, 2, , k < distribution of the genotype {J}. In both cases, the same result M, i.e. x greater than a certain threshold. Selection is applied j on the relationship between V and V is obtained. Here we after the introduction of mutation at each generation in the ip g briefl y describe the result with V¯ . At each generation, there transcriptional regulation network, i.e. the matrix J . ip ij exists a population of individuals with different genotypes Among the mutated networks, we select a certain fraction {J}. On the other hand, the phenotypic variance by genetic of networks that has higher fi tness values. Because the distribution is given by network Jij determines the ‘rule’ of the dynamics, it is natural to treat J as a measure of genotype. Individuals with different ij =−<>,∫ 2 (14) Vig d{J}p()(() {J} F J F ) genotypes have different sets of Jij.
J. Biosci. 34(4), October 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 537 where
course after a few generations. (Here as Vig << Vip, eliminate ruggedness in developmental potential is possible
the difference between Vig and Vg is negligible). only for a suffi ciently high noise level. Thus, the results are consistent with those given by the We expect these relationships (i)–(iii) as well as the phenomenological theory in the last section. Here, however, existence of the error catastrophe to be generally valid for we should note that there is not a direct correspondence systems satisfying the following conditions. with the phenomenological distribution theory and the gene (A) Fitness is determined through developmental expression network model. So far no direct derivation of dynamics. the former from the latter is available: The genotype here (B) Developmental dynamics is complex so that it may is given by the matrix {J}, and is not a priori represented fail to produce the target phenotype that gives a by a scalar parameter a. The accumulation of low-fi tness high fi tness value, when perturbed by noise. mutants with the decrease of noise level corresponds to (C) There is effective equivalence between mutation the error catastrophe at V ~ V but existence of such error ip ig to gene and noise in the developmental dynamics, catastrophe itself is not a priori assumed in the model. with regards to phenotypic change. The proportionality of the two variances is not evident in the model, either. Indeed, as mentioned above, the Condition (A) is generally satisfi ed for any biological proportionality is observed only after some generations, system. Even in a unicellular organism, phenotype is shaped when evolution progresses steadily. This might correspond through gene expression dynamics. The condition (B) is to the assumption for the quasi-static evolution of P(x,a). the assumption on the existence of ’error catastrophe’. A Why does the system not maintain the highest fi tness state phenotypic state with a higher function is rather rare, and it under a small phenotypic noise level with σ < σc? Indeed, the is not easy to reach such a phenotype through the dynamics dynamics of the top-fi tness networks that evolved under involving many degrees of freedom. In our model, the such low noise level have distinguishable dynamics from condition (B) is provided by the complex gene expression those that evolved under a high noise level. It was found dynamics to reach a target expression pattern. In a system that, for networks evolved under σ < σc, a large portion of with (A)–(B), slight differences in genotype may lead the initial conditions reach attractors that give the highest to a large differences in phenotype, as the phenotype is
fi tness values, while for those evolved under σ < σc, only a determined after temporal integration of the developmental
J. Biosci. 34(4), October 2009 538 Kunihiko Kaneko dynamics where slight differences in genotype (i.e. in the to the evolution speed according to Fisher’s theorem, the rule that governs the dynamics) are accumulated through phenotypic plasticity and evolvability are correlated through the course of development and amplifi ed. This introduces the phenotypic fl uctuation. Although such relationship was “error catastrophe” as a result of noise in developmental discussed qualitatively over decades, quantitative conclusion dynamics. has not been drawn yet. Here we have proposed such Postulate (C) also seems to be satisfi ed generally; by noise, quantitative relationship by phenomenological distribution developmental dynamics are modulated to change a path in theory, to be consistent with the data from numerical and in the developmental course, whereas changes introduced by vivo experiments. mutation into the equations governing the developmental Waddington proposed genetic assimilation in which dynamics can have a similar effect. In the present model, the phenotypic change due to the environmental change is phenotypic fl uctuation to increase synthesis of a specifi c gene later embedded into genes (Waddington 1953, 1957). The product and mutation to add another term to increase such proportionality among phenotypic plasticity, Vip, and Vg production can contribute to the increase in the expression is regarded as a quantitative expression of such genetic level of a given gene in a similar manner. assimilation, since Vg gives an index for the evolution speed. Evolvability may depend on species. Some species called living fossil preserve their phenotype over much longer 5. Discussion generations than others. An origin of such difference in evolvability could be provided by the degree of rigidness Our theory has several implications in development and in developmental process. If the phenotype generated evolution. We briefl y discuss a few topics. by developmental process is rigid, then the phenotypic fl uctuation as well as phenotypic plasticity against 5.1 Evolvability and its relationship with fl uctuations: environmental change is smaller. Then according to the answer to question 1 theory here, the evolution speed will be smaller. Evolvability depends not only on species but on each gene In section 2, we have proposed the proportionality between in a given species. Some proteins could evolve more rapidly phenotypic plasticity and phenotypic fl uctuation Vip as a than others. If we apply the fl uctuation-response relationship natural consequence of fl uctuation-response relationship. in section 2 the expression level of each gene, correlation Then we have proposed the proportionality between this between the level of fl uctuation of each gene expression and epigenetic phenotypic fl uctuation Vip and the fl uctuation its evolution speed is expected to exist not only through the due to genetic variation, Vg. Since the latter is proportional evolutionary course, but also for different genes in a given
1 σ =.1 σ=.06 200 0.1 σ=.01 180 Vg=Vip 160 140 0.01 120 100 80 0.001 60 Vg 40 20 0 1e-04
1e-05
1e-06 1e-05 1e-04 0.001 0.01 0.1 1 Vip
¯ Figure 3. The relationship between Vg (Vig) and V ip. Vg is computed from the distribution of fi tness over different {J} at each generation ¯ (eq. [14] , and V ip by averaging the variance of isogenic phenotype fl uctuations over all existing individuals (eq. [13]). Computed by using the model eq. [12] (see [Kaneko 2007] for detailed setup). Points are plotted over 200 generations, with gradual colour change as displayed.
σ = 0.01 (○), 0.06 (×), and 0.1 (■). For σ > σc ≈ 0.02, both decrease with successive generations.
J. Biosci. 34(4), October 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 539
Figure 4. Schematic representation of the basin structure, represented as a process of descending a potential landscape. In general, developmental dynamics can have many attractors, and depending on initial conditions, they may reach different phenotypes, so that the developmental landscape is rather rugged. After evolution, global and smooth attraction to the target phenotype is shaped by developmental dynamics, if gene expression dynamics are suffi ciently noisy. generation. To discuss such dependence on each protein, we through the evolution under a fi xed fi tness condition, if there have studied the model discussed in section 4, consisting exits a certain level of noise in the development. Robustness of expression levels of a variety of genes, and found the evolves, as discussed by Schmalhausen’s stabilizing proportionality between Vg and Vip over expressions of a selection (Schmalhausen 1949) and Waddington’s variety of genes. Indeed, we have measured the variances canalization (Waddington 1942, 1957). On the other hand, for expression levels of each gene i, and defi ned their genetic the proportionality between the two variances implies and epigenetic variances Vg (i) and Vip (i) for each. From the correlation between developmental and mutational several simulations, we have confi rmed the proportionality robustness. Note that this evolution of robustness is possible between Vg (i) and Vip (i) over many genes i (Kaneko 2008). only under a suffi cient level of noise in development. Although direct experimental support is not available yet, Robustness to noise in development brings about robustness recent data on isogenic phenotypic variance and mutational to mutation. variance over proteins in yeast may suggest the existence Our result demonstrates the role of noise during of such correlation between the two (Landry et al. 2007; phenotypic expression dynamics to evolution of robustness. Lehner 2008). Despite recent quantitative observations on phenotypic fl uctuation, noise is often thought to be an obstacle in 5.2 Evolution of robustness and relevance of noise: tuning a system to achieve and maintain a state with a Answer to the question 2 higher function, because the phenotype may be deviated from a state with a higher function, perturbed by such As mentioned in the introduction, biological robustness noise. Indeed, questions most often asked are how a given is the insensitivity of fi tness or phenotype to system’s biological function can be maintained in spite of the changes. Thus, the developmental robustness to noise is existence of phenotypic noise (Ueda et al. 2001; Kaern et al. characterized by the inverse of Vip, while the mutational 2005). In contrast, we have revealed a positive role of noise robustness is characterized by the inverse of Vg. In Section 4, to evolution quantitatively. We also note that relevance of it is shown that the two variances decrease in proportion noise to adaptation (Kashiwagi et al. 2006; Furusawa and
J. Biosci. 34(4), October 2009 540 Kunihiko Kaneko
Kaneko 2008) and to cell differentiation (Kaneko and Yomo gene expression pattern in the model of Section 1999) is recently discussed. 4 after some generation. To be specifi c, some of In our theory, robustness in genotype and phenotype are the target genes should be “off” instead of “on”. represented by the single-peaked-ness in the distribution P(x After this alteration in the fi tness condition,
= phenotype, a = genotype). If we write the distribution in a both the fl uctuations Vip and Vg increase over “potential form” as some generations, and then decrease to fi x the expression of target genes. It is interesting that P(x,a) = exp (–U(x,a)), (15) both the expression variances increase keeping the stability condition means existence of global minimum proportionality. At any rate, the restoration of of the potential U(x,a). With the increase in the slope of the fl uctuation emerges as a result of the change in attraction in the potential, the robustness increases. The environmental condition. After this change, both the smooth attraction in the developmental potential shown in variances increase to adapt the new environmental Section 4 is a representation of canalization in the epigenetic condition (K Kaneko, unpublished). Hence, the landscape by Waddington, whereas the correlation in the variances again work as a measure of plasticity. It attractions in phenotypic (x) and genetic (a) directions is one will also be important to incorporate environmental representation of genetic assimilation. fl uctuation into our theoretical formulation. (3) Interaction: Another source to sustain the level of fl uctuation is interaction among individuals, which 5.3 Future issues may introduce a change in the ‘environmental condition’ that each individual faces with. Here as The proportionality between the phenotypic variances of the variation in phenotypes is larger, the variation in isogenic individuals and that due to genetic variation in the interaction will be increased. Due to the variation Section 2 and Section 3 is not a result of derivation from the in the interaction term, the phenotypic variances will fi rst principle, but is a result of phenomenological description be increased. Then, mutual reinforcement between based on several assumptions. Although numerical results on diversity in interaction and phenotypic plasticity gene expression network and catalytic reaction network may be expected, which will result in both the models support the relationship, there remains a large gap genetic and phenotypic diversity. Co-evolution for between the macroscopic phenomenological theory and such reinforcement between phenotypic plasticity microscopic simulations. It should be important to establish a and genetic diversity will be an important issue to statistical physics theory to fi ll the gap between the two (see be studied in future. also Sakata et al. [2009] for a study towards this direction). (4) Speciation: So far, we have considered gradual All of the plasticity, fl uctuation, and evolvability decrease evolution keeping a single peak in phenotype in our model simulations through the course of evolution, distribution. On the other hand, the bimodal under selection by a fi xed fi tness condition. Considering distribution should appear at the speciation. that none of the phenotypic plasticity, fl uctuation, and Previously we have shown that robust sympatric evolvability vanishes in the present organisms, some process speciation can occur as a mutual reinforcement for restoration or maintenance of plasticity or fl uctuation of bifurcation of phenotypes as a result of should also exist. To consider such process, the followings developmental dynamics and due to genetic are necessary: variation (Kaneko and Yomo 2000; Kaneko 2002). (1) Fitness with several conditions: In our model study It will be important to reformulate the problem in (and also in experiments with artifi cial selection), the terms of the present distribution theory. fi tness condition is imposed by a single condition. In (5) Sexual reproduction: Our simulation in the present nature, the fi tness may consist in several conditions. paper was based on asexual haploid population. On In this case, we may need more variables in the the other hand, the phenomenological theory on the characterization of the distribution function P, say distribution could be applicable also to a population P(x,y,a,b) etc. In such multi-dimensional theory, with sexual reproduction. In the sexual reproduction, interference among different phenotypic variable x however, recombination is a major source of genetic and y may contribute to the increase or maintenance variation, rather than the mutation. Then, instead of of fl uctuations. the mutation rate, the recombination rate could (2) Environmental variation: If the environmental control the error catastrophe. Here, the degree of condition changes in time, the fl uctuation level genetic heterogeneity induced by recombination may be sustained. To mimic the environmental strongly depends on the genotype distribution at the change, we have altered the condition of target moment. (For example, for an isogenic population,
J. Biosci. 34(4), October 2009 Relationship among phenotypic plasticity, phenotypic fl uctuations, robustness, and evolvability 541
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