Antonine Nicoglou
email: antoninenico@gmail.com

Bradshaw

The evolution of phenotypic plasticity: Genealogy of a debate in genetics

Keywords: phenotypic plasticity, evolution, Bradshaw, quantitative genetics, adaptive plasticity, extended synthesis

The evolution of phenotypic plasticity

Antonine Nicoglou

Introduction

Since the end of 1970s, some biologists (e.g., [START_REF] Gould | Ontogeny and phylogeny[END_REF][START_REF] Hamburger | Embryology and the Modern Synthesis in Evolutionary Theory[END_REF][START_REF] Coleman | Morphology in the Evolutionary Synthesis[END_REF][START_REF] Lauder | Introduction[END_REF][START_REF] Wallace | Can embryologists contribute to an understanding of evolutionary mechanisms?[END_REF] have started to question the adequacy of the genocentric conception of the Modern Synthesis -which brought together Mendelian genetics, and evolution through population genetics [START_REF] Huxley | Evolution: The Modern Synthesis: The Definitive Edition[END_REF][START_REF] Huxley | Evolution: The Modern Synthesis: The Definitive Edition[END_REF] -in explaining the evolution of phenotypic traits, suggesting that developmental issues should also be included in the synthesis [START_REF] Gilbert | Resynthesizing Evolutionary and Developmental Biology[END_REF]. More recently, some authors have stressed that phenotypic plasticity should be seen as one important element in an extended synthesis of evolution including these developmental issues (see, Pigliucci & Müller, 2010). The present paper comes back on a precise controversy in the history of phenotypic plasticity in biology, which is a debate in the 1980s between two representative views (among others) -these of Via & Lande and Scheiner & Lyman -concerning the evolution of phenotypic plasticity. The aim of the paper is to lay the groundwork for a genealogy of phenotypic plasticity and to show that the notion was defined and discussed since a long time before the 2010s, and that it adopted a specific meaning in the emerging field of quantitative genetics. Through a clarification of the origins and the basis of a specific debate concerning the evolution of phenotypic plasticity that occurred in the 1980s, the purpose here is to enlighten some of the implicit ideas, which used to be associated with the notion of plasticity at that time, and to show the reasons why it remains difficult to associate the notion with something different from the Modern Synthesis.

In 1965, Anthony D. Bradshaw (1926Bradshaw ( -2008) ) published an article entitled The Evolutionary Significance of Phenotypic Plasticity in Plants. In the article, he proposed for the first time a model intended to explain the evolution of norms of reaction mainly based environments. And in such a perspective, they concluded that the evolution of plasticity should be seen as a by-product of evolution, assuming that selection would not act directly upon plasticity, which Via did not consider to be a distinct trait with its own genetic etiology. On the other side, Samuel Scheiner and Richard Lyman wanted a tool by which they could model adaptive evolution of reaction norms. And in their perspective, they considered that plasticity had its own genetic basis, independent from the one of the plastic trait, and, therefore, that it had its own model of evolution [START_REF] Via | Genotype-environment interaction and the evolution of phenotypic plasticity[END_REF], 1987;[START_REF] Scheiner | The genetics of phenotypic plasticity. II. Response to selection[END_REF].

Since both authors refer to Bradshaw's seminal article of 1965, one purpose of the present paper is to analyze in what way Bradshaw's understanding of phenotypic plasticity might have influenced the works of the main protagonists of the debate and so how it might have perpetuated in the current literature. In other words, the purpose is both to understand whether those divergent approaches concerning the evolution of plasticity might have a common ground and whether the differences between them could help to enlighten the current understanding biologists have of phenotypic plasticity. There are two interrelated problems here. The first one concerns their common reference to Bradshaw and the relationship of their models with Bradshaw's view on plasticity. The second one concerns the disagreement, our precise understanding of it, and the potential consequences for the current understanding of phenotypic plasticity in biology. However, in order to answer the second problem -that is to say: what was the disagreement about? -, it seems necessary to provide some main clues concerning the first one -that is to say: what was Bradshaw's influence on the protagonists of the debate? With this being said, I shall then come back to my main goal here and conclude on the reasons why the disagreement seems to have disappeared in the current understanding biologists have of the evolution of phenotypic plasticity. But let us first start with Bradshaw's view on phenotypic plasticity.

Bradshaw and the notion of "phenotypic plasticity"

In the article of 1965, British ecologist and geneticist Anthony Bradshaw proposed a model to explain the evolution of what was previously considered as the "shape of reaction norms" and that he will call "plasticity." In the article, Bradshaw pointed out the importance of the environmental effects on organisms: "We are becoming increasingly aware that the individual cannot be considered out of the context of its environment" (Bradshaw, 1965, p. 115). Before him, the "instability" produced by the environment was mostly conceived as a source of perturbation for geneticists [START_REF] Falconer | The Problem of Environment and Selection[END_REF]. And as Bradshaw claimed: "Any modifications induced by the environment during the course of an experiment are usually considered only an embarrassment" (Bradshaw, 1965, p. 148).

Therefore, they mostly tried to eliminate it in their studies.

Another example, which is somehow less expected, is that of Conrad Hall Waddington (1905Waddington ( -1975)), who was interested in the issue of "stability" through the process of canalization. His most quoted book The Strategy of the Genes [START_REF] Waddington | The strategy of the genes: a discussion of some aspects of theoretical biology[END_REF] has been often, and justifiably, used by biologists and theorists to argue that he was one of the first evolutionary biologists to make an effort to bring genetics and development together, and, therefore, to pay a specific attention to environment. However, in most other books where he dealt with the question of "stability" (or "canalization"), he did not refer explicitly to the environment. In Principles of Embryology [START_REF] Waddington | Principles of Embryology[END_REF], in which he developed and explained the process of "canalization," the term "environment" and its derivatives (e.g., "environmental") appeared only 16 times in a book of 528 pages.

In the book How Animals Develop, (Waddington, 1962 3 , [1935]) the term "environment" and its derivatives appeared only 6 times in a book of 148 pages. Waddington considered the process of canalization as the expression of some robustness to genetic variability and not to environmental variability. It appears that Waddington, like many geneticists of his time, mostly depicted -with the notable exception of The Strategy of the Genes -the environment, even if influent, as a source of perturbation that should be removed from the analysis of phenogenesis for experimental purposes. He also explicitly considered that environmental effects were minimal during early development of organisms (Waddington, 1962, p. 122). Despite this major rejection of the "unstable" factor, the problem remained, nevertheless, to understand how the individual could maintain some stability in an unstable environment. After World War II, the question became one of the main issues among geneticists (e.g., [START_REF] Dobzhansky | The Genetics of Homeostasis in Drosophila[END_REF][START_REF] Mather | Genetical control of stability in development[END_REF][START_REF] Jinks | Stability in Development of Heterozygotes and Homozygotes[END_REF][START_REF] Kimura | Stochastic Processes and Distribution of Gene Frequencies Under Natural Selection[END_REF][START_REF] Lerner | Genetic homeostasis[END_REF][START_REF] Lewontin | The Adaptations of Populations to Varying Environments[END_REF][START_REF] Waddington | Canalization of development and genetic assimilation of acquired characters[END_REF][START_REF] Levins | Theory of fitness in a heterogeneous environment. II. Developmental flexibility and niche selection[END_REF].

Unlike most of his colleagues and even though he was a geneticist, Bradshaw did not see the environment as a disruptive force. Far from it, he explicitly incorporated the environment in his genetic analysis (see [START_REF] Fitter | Anthony David Bradshaw[END_REF]. Therefore, in the article of 1965, Bradshaw was striving to demonstrate: "first, that environmental effects on the phenotype were as important as genetic effects (rather than simply inconvenient error), and second, that these effects were themselves under genetic control and could therefore evolve" (Fitter, 2010, p. 31). Moreover, Bradshaw undermined, at least partly, some of his predecessor's contribution by downplaying the importance of development for understanding plasticity (i.e., [START_REF] Waddington | The strategy of the genes: a discussion of some aspects of theoretical biology[END_REF]. In Bradshaw's view, if plasticity could be related to the general pattern of development of a character, it should not be seen as properly explained by these developmental pathways:

It has been argued […] that at the present, owing to the complexities of the developmental pathways concerned, the interactions between pathways and environment will be so complex that it is unlikely that much progress in understanding can be made until more detailed work on the pathways has been carried out. Considerations of basic causes, however, have led to the argument that the degree of plasticity shown by a character can be related to the basic pattern of its developmental pathway. (Bradshaw, 1965, p. 117) While basic developmental pathways are important, it does not seem possible for them [START_REF] Stebbins | Variation and Evolution in Plants[END_REF][START_REF] Harper | Approaches to the study of plant competition[END_REF] to provide an explanation of all observable difference in plasticity. (Bradshaw, 1965, p. 118) In other words, Bradshaw's purpose was not only to emphasize the importance of plasticity for the understanding of the evolution of phenotypic traits, it was also a way, for him, to establish a strong position in the overall theoretical debate, by downplaying some of his protagonists assumptions. Furthermore, Bradshaw depicted "plasticity" -as he chose, after Nilsson-Ehle (1873-1949) to call it -as having, in many circumstances, an adaptive value, which should be studied per se. For Bradshaw, Plasticity is therefore shown by a genotype when its expression is able to be altered by environmental influences. The change that occurs can be termed the response. Since all changes in the characters of an organism which are not genetic are environmental, plasticity is applicable to all intragenotypic variability. (Bradshaw, 1965, p. 116) In Bradshaw's view, the "response" to environmental variation can be understood as an adaptation. Plasticity embodies phenotypic variability as a consequence of the genotypic response to environmental variation. Indeed, Bradshaw considered that plasticity is "genetically determined" (Bradshaw, 1965, p. 145). This conception lies squarely in the lines of the genotype/phenotype distinction's definition offered by [START_REF] Johannsen | The Genotype Conception of Heredity[END_REF] Such a definition had already influenced many biologists interested in phenogenesis, such as [START_REF] Timoféeff-Ressovsky | Über das phänotypische Manifestation des Genotypes. II. Über idio-somatische Variationsgruppen bei Drosophila funebris[END_REF] or [START_REF] Dobzhansky | Genetics and the Origin of Species[END_REF][START_REF] Dobzhansky | A Review of Some Fundamental Concepts and Problems of Population Genetics[END_REF]. Dobzhansky had precisely suggested that in the study of phenogenesis one should focus on adaptation, such as any other traits genotypically determined, it also became a possible target for natural selection. When Dobzhansky disseminated the notion of "adaptive norm" to the West and particularly to the United States, he did it trying to demonstrate that a mutation does not alter a particular morphological character but that it introduces a change in the reaction norm [START_REF] Dobzhansky | A Review of Some Fundamental Concepts and Problems of Population Genetics[END_REF]. Gradually, the notion of "norm of reaction" was replaced, in the literature, by the notion of "reaction range" (Sinnott, Dunn & Dobzhansky, 1950, p. 22). 4 It was a way to indicate that phenotypic variability depended largely on interactions between inherited (or genetic) factors and the environment inside a precise range. However, the stated objective was to put a new emphasis on variability (as opposed to a strong genetic determinism, which was completely leaving aside the environment in the explanation of the evolution of traits). Yet, the use of the notion of "norm of reaction" by Dobzhansky led to its conceptual change: it became in 1955 essentially a problem of genetics since the norm of reaction itself was interpreted as a unit of Mendelian inheritance. With Bradshaw, the conceptual shift associated with the reaction norm initiated by Dobzhansky (from a developmental/phenogenesis perspective with Woltereck, to a population genetic/evolutionary perspective with Dobzhansky) received a new notion -phenotypic plasticity -to escape confusions with the controverted concept of "norm of reaction" and its potential derivatives (e.g. norm of reactivity, reaction range, Reaktionsnorm, etc.), and "plasticity" became mainly a tool of population genetics/evolution. Therefore, even if Bradshaw followed Herman Nilsson-Ehle's (the Swedish geneticists who first referred explicitly to the notion of "plasticity" to describe the effect of the environment on the phenotype) conclusions5 concerning the adaptive value of plasticity, his interpretation concerning the genetically determined character of plasticity depicts a major difference between the two authors. Nilsson-Ehle used to understand plasticity as something "purely adaptive" (as opposed to "evolutionary adaptive" in Darwinian terms): he considered it from being part of self-regulation mechanisms (mechanisms by which an organism responds to environmental changes by phenotypic changes without necessarily a genetic basis). Therefore, Nilsson-Ehle, contrary to all expectation (since he realized his works in the context of the emerging Mendelian genetics), did not define plasticity as "a property of a single genotype" such as [START_REF] Johannsen | The Genotype Conception of Heredity[END_REF] who understood the "norm of reaction" as the range of reaction of a single genotype to changing environments. The fact that Nilsson-Ehle did not refer to "plasticity" in accordance with theories developed by Johannsen -one of the most influential scientists of the rising discipline of genetics at that time -explains maybe why geneticists, before Bradshaw's seminal study in the mid-1960s, would not use so much the term "plasticity." Bradshaw, on his side, considered "adaptive plasticity" in an evolutionary perspective. As such, he thought that it might be submitted to a genetic determination of some sort. If the formulation of the notion of "norm of reaction," in the beginning of the 20 th century could be understood as a key-notion in the story of Mendelian genetics, involving a mitigation of genetic determinism, with Bradshaw the notion of "plasticity" became part of the story of quantitative genetics. The review he offered of it opened the possibility for a precise analysis of its evolution.

However, if Bradshaw opened a new field of study -the study of the evolution of plasticity with optimality models -that will be further developed in the 1980s, in his review of 1965, he was mostly discussing his predecessor's positions. It seems that some of these discussions might have introduced confusions for quantitative geneticists who started being interested on plasticity in 1980's. For instance, Bradshaw distinguished what he called "physiological plasticity" (which corresponds to all different forms of plasticity)6 from "morphological plasticity" (which corresponds to a particular manifestation of physiological plasticity, that refers to developmental changes). Such a distinction should be linked to the fact that in the 1940s, the Austrian embryologist and geneticist Richard Goldschmidt (1878Goldschmidt ( -1958) ) had made a clear distinction between "norm of reaction" and "norm of reactivity" (Goldschmidt , 1940, p. 250). He used the distinction to stress the genetically determined nature of the "norm or reaction" compared to the "norm of reactivity" (that would be applied to any physiological reactivity of the organism), while Bradshaw equated "physiological plasticity" and "morphological plasticity" and considered that both of them were genetically determined. According to Bradshaw, in both cases, one was dealing with "adaptive plasticity." Therefore, the only reason, for him, to keep drawing a distinction between the two notions -morphological plasticity versus physiological plasticity -was to make clear that the kind of plasticity biologists were mainly dealing with, in their work, was "morphological plasticity," since this type of plasticity was most easy identified in nature. However, since physiological plasticity would start to be broadly studied in quantitative genetics -but with a different meaning from that of Bradshaw 7 -such a distinction (between physiological and morphological plasticity) might appear nowadays as a source of confusion for biologists who are dealing with plasticity.

In order to refine his definition, Bradshaw also made a distinction between plasticity and another phenomena: "phenotypic flexibility." This was likewise used by [START_REF] Thoday | Components of fitness[END_REF] to describe the ability of an organism to function in a range of environments. In Bradshaw's view, he difference between "plasticity" and "flexibility" is that "plasticity" depicts the variability of the norm of reaction while "flexibility" may also include stable responses to environmental variation. For Bradshaw, "flexibility" had nothing to do with "norm of reaction," since there was no actual phenotypic change.

Therefore, he did not consider heat resistance as a plasticity phenomenon but only as phenotypic flexibility. However, it seems that his position concerning such a distinction remains somehow unclear to his reader, since he kept pointing out the fact that phenotypic flexibility itself could occasionally involve both stable and plastic responses (Bradshaw, 1965, p. 117). This position would also be a source of confusion in the interpretation Bradshaw's readers did of his conception of plasticity. They would either consider plasticity as something definite, either as something that should, itself, be related to an adaptive trait.

After the publication of Bradshaw's article, in 1965, this difficulty will precisely become the subject of many controversies regarding the definition of "phenotypic plasticity." In this respect, the debate between the two pair of opponents Lande and Via on the one side, and Sheiner and Lyman on the other side -and more specially the precise opposition between Sara Via and Samuel Scheiner8 -can partly be seen as a symptom of a tacit confusion, introduced by Bradshaw, concerning his definitions of plasticity. In order to understand more precisely some of the reasons and expressions of the controversy, I will now turn to a sharper analysis of Bradshaw's evolutionary significance of phenotypic plasticity.

Since, for Bradshaw, plasticity was genetically determined, and since he saw it as an adaptation, he was, therefore, convinced that the general assumption, which associates the plasticity of a trait to its developmental pattern, was not exhaustive enough. And as a plant biologist, Bradshaw mainly argued against plant biologists on this topic. His argument was the following. First, Bradshaw related the fact that, [START_REF] Stebbins | Variation and Evolution in Plants[END_REF] has argued that characters formed by long periods of meristematic [the meristem is the tissue in most plants that contains undifferentiated cells and that one founds in zones of the plant where growth can take place] activity (such as over-all size, leaf number, etc.) will be more subject to environmental influences and are likely to be more plastic than characters formed rapidly (such as reproductive structures) or than characters whose pattern is impressed on primordia at an early stage of development (such as bud scales, leaves, etc.). This argument can be supported by evidence of the differences in plasticity shown by different characters in Achillea and Potentilla in the experiments of Clausen andassociates (1940, 1948). The plasticity shown by the characters of these http://dx.doi.org/10.1016/j.shpsc.2015.01.003 14 species can be related to their general pattern of development, i.e., its duration, complexity, the number of interacting processes involved, etc. (Bradshaw, 1965, p. 117-118) However, Bradshaw referred to a deductive theoretical reasoning to reject this argument:

If it was assumed that the degree of plasticity shown by a character was the outcome of the basic pattern of its developmental pathway, certain deductions would follow. First, since general pathways of characters cannot be changed readily, it should not be possible for plasticity to change readily. Second, since the same organ, e.g., leaf, usually has the same basic developmental pathway in different species, the organ should show the same plasticity in different species. (Bradshaw, 1965, p. 119) Bradshaw argued that those assumptions were false by referring to a number of studies, which showed that plasticity differs among different species that exhibit the same type of development. 9 Therefore, he considered that the plasticity is not specifically linked to the development of the plastic trait. For similar reasons, he also rejected the idea that plasticity is a property of the entire genome. Finally, he based his conception of plasticity on three main points:

(1) Plasticity is a property specific to individual characters in relation to specific environmental influences.

(2) Since plasticity of a trait changes among the different species of one genus and among the different varieties of the same species, Bradshaw concluded to a genetic independence of the plastic trait with the plasticity of a trait. He stressed that "such differences are difficult to explain unless it is assumed that the plasticity of a trait is an 9 Most of the studies refer to a conspicuous type of plasticity: the heterophylly shown by certain water plants. Heterophylly is rarely characteristic of a whole genus; more commonly, it is found only in particular species. For this reason, this type of plasticity is particularly handy to Bradshaw's argument. independent property of that trait and is under its own specific genetic control" (Bradshaw, 1965, p. 119).

(3) Finally, he quoted Waddington's works on canalization and genetic assimilation (1953b) and argued that a large measure of control on the degree of modification possible is afforded by the developmental pathway, the epigenetic landscape, which has its own genetic determinism as much as plasticity which offer the possibility to switch from one developmental pathway to another.

Taken together, these three points allowed Bradshaw to conclude that plasticitygenetically determined -is necessarily submitted to natural selection as any other trait in nature. Consequently, Bradshaw's approach was to introduce 'more' genetics rather than more development or environment when trying to make sense of plasticity.

In 2006, two years before his death, Bradshaw was invited to speak at a symposium organized by the Journal The New Phytologist and to comment his article of 1965. In the resulting article (his last published article), Bradshaw described the conference and offered a fascinating perspective on the origin of his ideas in the field [START_REF] Bradshaw | Unravelling phenotypic plasticity -why should we bother?[END_REF]. The article reveals, among other things, that his position has ultimately not much changed since the 1965 article, except, at least, that its has been further informed by recent works in the field. Such a determination also demonstrates a fixation of Bradshaw's views of the concept of "phenotypic plasticity," after he gave it its first literature review in 1965.

Bradshaw viewed his legacy correctly; the concept of "phenotypic plasticity" became widely used in biological literature after the publication of his review (see next section).

"Phenotypic plasticity" became an operative concept in biology where, in the past, and especially in embryology, the adjective "plastic" was mainly used with a metaphorical and an architectonic connotation linked to its use in natural philosophy from the 17 th century onwards. 10 Because of this scientific meaning, Bradshaw also permanently bound plasticity to the field of genetics (and even quantitative genetics), since the term "plasticity" became widely used in biology after Bradshaw's definition in the field of quantitative genetics.

It can also be noted that various past uses of the term are overshadowed by Bradshaw's definition of plasticity (Nilsson-Ehle's use but also the embryologists and zoologists' uses before him, such as Gavin de Beer's use (see, de Beer, 1940)) mainly because of the rise of genetics at this time and its importance for evolutionary studies. This settles a disciplinary division in which the geneticists are taking control of the terms of the debate, the scope of the conceptions, and above all are defining the parameters of evolutionary biology, versus the embryologists, the zoologists and even the botanists whose conception of plasticity is left out. It is in this precise context that was settled the debate, which arose in the 1980s concerning the evolution of phenotypic plasticity.

Therefore, before detailing both views concerning the evolution of phenotypic plasticity, I shall first detail the general context and conditions of the debate.

The debate between Lande & Via and Scheiner & Lyman

Following Bradshaw, and in the context of the Modern Synthesis in evolutionary biology, population geneticists interested in plasticity would mainly study its evolution.

More precisely, they sought to understand to what extent natural selection could explain the evolution of plasticity. Therefore, their attention focused on what they called "adaptive plasticity." 11 Like Bradshaw, these authors also downplayed the importance of developmental patterns in their understanding of plasticity. However, at this time -in 10 For more about the use of the concept of « plasticity » in the philosophy of nature in the 17 th and 18 th centuries, see Nicoglou (2012).

11 It should be noted that the different authors of the debate did not have exactly the same view on what "adaptive" means. However, since they used the same term "adaptive," they were under the impression that they referred to the same thing.

1980s -long after Bradshaw, when the discipline of quantitative genetics was already settled, it was not a claim for these authors but rather a matter of fact linked to the disciplinary constraints confining them to certain questions, such as: how to understand with genetic tools the evolution of phenotypic plasticity?

When first quantitative genetics models were established in order to describe the evolution of "phenotypic plasticity," the notion of plasticity was broadly defined in the same way Bradshaw used to define it: to characterize the phenotypic response to the environmental variations. At the beginning of the synthetic theory of evolution (in the 1930s-1940s), quantitative genetics became a common tool used in biological studies of evolution (e.g., Fischer, 1930;[START_REF] Wright | The genetical structure of populations[END_REF]. Later, it never stopped drawing attention in the analysis of phenotypic evolution (e.g., [START_REF] Lande | Genetic Variation and Phenotypic Evolution During Allopatric Speciation[END_REF][START_REF] Falconer | Introduction to quantitative genetics[END_REF][START_REF] Cheverud | Quantitative genetics of development: genetic correlations among age-specific trait values and the evolution of ontogeny[END_REF][START_REF] Lande | The measurement of selection on correlated characters[END_REF][START_REF] Slatkin | Quantitative Genetics of Heterochrony[END_REF][START_REF] Barton | Evolutionary Quantitative Genetics: How Little Do We Know[END_REF][START_REF] Shaw | Changes in Genetic Variances and Covariances: G Whiz![END_REF][START_REF] Roff | Evolutionary Quantitative Genetics[END_REF]. It became therefore natural that these techniques were also used to assess the evolution of phenotypic plasticity in works that heralded a new line of research on plasticity (e.g., [START_REF] Falconer | The Problem of Environment and Selection[END_REF]Via, 1984a[START_REF] Via | The Quantitative Genetics of Polyphagy in an Insect Herbivore. II. Genetic Correlations in Larval Performance Within and Among Host Plants[END_REF][START_REF] Via | Genotype-environment interaction and the evolution of phenotypic plasticity[END_REF]. At this point, I would like to suggest that even if people dealing with studies in evolution were mainly dealing with genetic factors, the problem of the evolution of plasticity was not only related to the question of knowing in which way genes were involved in the determination of plasticity. It was also related to the definition of plasticity.

In an article of 1986, Carl Schlichting, with an early interpretation, suggested that Via and Lande's type of study illustrated a renewed interest in the study of plasticity, after a period of twenty years, from Bradshaw, when such studies had been limited. At the same time, he admitted failing to understand the precise reasons for such a long eclipse and could only offer few suggestions:

Until 1980, theoretical work on plasticity was limited; and empirical research, with the notable exception of Subodh Jain's efforts [START_REF] Marshall | Phenotypic Placticity of Avena fatua and A. barbata[END_REF][START_REF] Jain | Inheritance of phenotypic plasticity in soft chess, Bromus mollis L.(Gramineae)[END_REF][START_REF] Jain | Adaptive strategies: polymorphism, plasticity, and homeostasis[END_REF], was largely unfocused. The reasons for such neglect are puzzling, especially considering the clarity of Bradshaw's review. Surely part of the problem was the growing fascination with the detection and measurement of "genetic" variation, of which plasticity must have seemed the antithesis. Another problem was that environmentally induced variability in an experiment is typically avoided at all costs. Experimental complexity and the problem of measuring plastic responses also retarded progress. (Schlichting, 1986, p. 669) Schlichting is certainly true from a disciplinary point of view, in this interpretation. However, he does not highlight the fact that at this time, when biologists were facing the problem of the evolution of plasticity, they had to do it mainly through a genetic approach, since their tools of analysis for phenotypic evolution were quantitative genetics. Like Bradshaw, they did not specifically distinguish different possible types of plasticity with different definitions (such as "histogenetic plasticity," "phylogenetic plasticity" or even "genetic plasticity" itself). 12 Most of them naturally focused on what they knew the best: "genetic plasticity," which had handily been defined by Bradshaw. Schlichting suggested, then, that the observed neglect between 1965 and 1980, concerning plasticity, was probably the result of an effect of myopia due to an exclusive focus on the genetic basis of variation rather than on its "plasticity." However, I argue that if one adopts a larger scope than those of the discipline of quantitative genetics, the causes of this perceived absence are not only the result of a focus on stability/genetics (rather than unstability/plasticity), but they are also linked to some remaining difficulties in defining plasticity. Indeed, as I have shown previously, Bradshaw's review was sometimes source of confusions concerning his understanding of the notion (e.g., concerning "adaptive plasticity"). Furthermore, when one looks at the uses of the term "plasticity" throughout the different disciplines of biology, one realizes that its study has, ultimately, still remained active between 1965 and 1980 (even if it was not specifically in genetics). Indeed, between 1965 and 1980 many ecologists, evolutionary biologists and botanists were interested in issues related to plasticity, both from a morphological and a physiological point of view (e.g., [START_REF] Harper | A Darwinian approach to plant ecology[END_REF]Rehfeld, 1979;[START_REF] Baker | Race[END_REF]. Their understanding of plasticity in biology relied on dynamic phenomena analysis such as homeostasis (e.g., [START_REF] Ashby | Homeostasis. In Cybernetics: Circular Causal and Feedback Mechanisms in Biological and Social Systems[END_REF][START_REF] Thoday | Components of fitness[END_REF][START_REF] Thoday | Balance, heterozygosity and developmental stability[END_REF][START_REF] Thoday | Homeostasis in a selection experiment[END_REF][START_REF] Hyde | Genetic homeostasis and behavior: Analysis, data, and theory[END_REF] or canalization (e.g., Waddington 1953a[START_REF] Waddington | The strategy of the genes: a discussion of some aspects of theoretical biology[END_REF]. Although all these studies did not explicitly and technically referred to the term "plasticity," it remains clear that they were all dealing with "plastic phenomena." In other words, in the 1980s, the term is still not fixed in biology, but is a malleable term about to be variously deployed and defined. Thus, its meaning and construction was still under debate (exposing the assumption about the local processes of that construction). However, in 1985, when the field of quantitative genetics was already settled, the idea of plasticity in biology became mostly associated with the notion of "phenotypic plasticity" -previously defined by Bradshaw -and was mostly seen as a notion of (quantitative) genetics (as opposed to a more intuitive or philosophical notion of plasticity). 13 With the article of 1965, Bradshaw had already contributed to the rejection of an "indirect" analysis of the evolution of plasticity, through the analysis, for instance, of the evolution of stability or through the analysis of its different forms of expression (i.e., homeostasis or canalization). He believed that the observation of a certain "lack of stability" should be considered as something different from "plasticity" (Bradshaw, 1965, p. 117). Therefore, the study of plasticity has been relatively neglected for a number of years after Bradshaw especially if one has in mind the study of evolution of plasticity. But http://dx.doi.org/10.1016/j.shpsc.2015.01.003 20 the key point is that, during all these years, some noticeable studies continued to address the issue of plasticity and even question its definition (e.g., [START_REF] Shapiro | Seasonal Polyphenism[END_REF][START_REF] Cavalli-Sforza | The Role of Plasticity in Biological and Cultural Evolution[END_REF][START_REF] Caswell | Phenotypic Plasticity in Life-History Traits: Demographic Effects and Evolutionary Consequences[END_REF][START_REF] Smith-Gill | Developmental Plasticity: Developmental Conversion versus Phenotypic Modulation[END_REF]. This illustrates the fact that, even after Bradshaw, two related problems remained that cannot be totally separated: with what model should one describe the evolution of plasticity? And what general definition could be given to the notion? If the first problem has mostly been worked out (despite the fact, or because, it provided a debate in 1980s), the second problem was not acknowledged, from the beginning, and led to a disputed genealogy of phenotypic plasticity after the debate (that could, or could not, be linked to development/phenogenesis, depending on the conception each members of the debate had of plasticity). I will now look more precisely at those conceptions.

Sara Via and Russell Lande's conception of the evolution of plasticity

Sara Via met Russ Lande while she was in graduate school and became one of her disciples, before working with him as a post-graduate student. Before working with Lande, Via was interested in how populations adapt to heterogeneous environments, and more precisely in population divergence and differentiation. Her question was about the status of individuals among the species: are the species that looks like generalist only like that because the individuals are placed in a specific environment; if they were placed in different environments would they act differently? In other words, her question was: in the generalist species, are individuals generalists or specialists? From this question, she then got into genotype-environment interaction by figuring out that she could study this question, which concerned generalist species, by using genotype-environment interaction and by testing populations in two environments (in her case, two host plants for leaf miners insects, her animal study model) (Via, 1984a;[START_REF] Via | The Quantitative Genetics of Polyphagy in an Insect Herbivore. II. Genetic Correlations in Larval Performance Within and Among Host Plants[END_REF]. From this first study, her interests turned into how specialization could evolve and she started working with Russ Lande on the measure of -not specialization -but variation in response to the environment. 14 The variation in response to the environment had been described by Bradshaw, who focused on plasticity shown by a genotype when its expression is able to be altered by environmental influences. Quantitative genetics focused on variation in multiple environments. For this reason, Via left the context of adaptation to heterogeneous environment, genotypeenvironment interaction, and went straight to the study of the evolution of plasticity. Therefore, while Lande and Via's explanatory model for the evolution of plasticity required quantitative genetics tools, it did not strictly speaking rely on a "quantitative genetic" definition of plasticity. Via was interested in plasticity because of the question of adaptation to heterogeneous environment, rather than because of the question of variation in multiple environment (the focus of quantitative genetics). Furthermore, she did not see the genotype as synonymous with "the reaction norm," and had come to plasticity because of her interest in genotype-environment interaction. Indeed, Via and Lande defined phenotypic plasticity by referring to Bradshaw's definition, but also to Georgy Gause (1910Gause ( -1986) ) and Ivan Schmalhausen's (1884Schmalhausen's ( -1963) )

conceptions of plasticity:

Environmental modification of the phenotype is common in the quantitative (polygenic) characters of organisms that inhabit heterogeneous environments. The profile of phenotypes produced by a genotype across environments is the 'norm of reaction' [START_REF] Schmalhausen | Factors of evolution: the theory of stabilizing selection[END_REF]; the extent to which the environment modifies the phenotype is termed 'phenotypic plasticity' [START_REF] Gause | Problems of evolution[END_REF][START_REF] Bradshaw | Evolutionary Significance of Phenotypic Plasticity in Plants[END_REF]. Because phenotypic response to environmental change may facilitate the exploitation of some environments and provide protection from others, the level of plasticity in a given trait is thought to be molded by selection [START_REF] Gause | Problems of evolution[END_REF][START_REF] Schmalhausen | Factors of evolution: the theory of stabilizing selection[END_REF][START_REF] Bradshaw | Evolutionary Significance of Phenotypic Plasticity in Plants[END_REF]. [START_REF] Via | Genotype-environment interaction and the evolution of phenotypic plasticity[END_REF], p.505) 14 Via thought that the moment you analyze two environments, it implies variation whether it is about specialization or not and that it implies genotype-environment interaction. From Sara Via's interview by E. Peirson on Monday, May 13, 2013.

From the start, Via and Lande's definition of plasticity provided a more nuanced picture, compared to the definition proposed by Bradshaw, which was: "Plasticity is […] shown by a genotype when its [phenotypic] expression is able to be altered by environmental influences" (Bradshaw, 1965, p. 116). Obviously, while the definition introduced by Bradshaw had tended to focus on the genotype, Via and Lande had in mind the conceptual controversies based on Soviet authors' views embracing genotype-environment interactions more fully (i.e., [START_REF] Gause | Problems of evolution[END_REF][START_REF] Schmalhausen | Factors of evolution: the theory of stabilizing selection[END_REF], and which were still relevant in their opinion in 1985.

Via's interview by [START_REF] Peirson | Sara Via's interview[END_REF] seems to confirm that her conception of plasticity was not only influenced by Bradshaw's definition but also by those Russian authors. 15 For Via, plasticity was "different phenotypes in different environments." She was mainly thinking "in terms of genetic correlations and what causes a genetic correlation." She thought of plasticity as "adaptation to different environments." It means, "in some cases, the same gene could be expressed in different ways" (more or less expressed), or "some genes have an impact only in one environment." In other word, she associated plasticity with behavioral ecology rather than with a direct explanation by quantitative genetics.

In line with the model for the evolution of plasticity proposed with Lande in 1985, Via 16 offered, in 1993, a theoretical definition of phenotypic plasticity in accordance with their common model. In the article of 1993, she tried to show that there was a kind of agreement within population geneticists' community interested on plasticity (e.g., 15 It should be noted that there were many earlier studies of genetic divergence vs phenotypic plasticity that should be intimately linked to Via's own work (e.g., [START_REF] Berven | The Genetic Basis of Altitudinal Variation in the Wood Frog Rana sylvatica. I. An Experimental Analysis of Life History Traits[END_REF][START_REF] Stearns | The evolution of life history traits: a critique of the theory and a review of the data[END_REF]. 16 Because of a personality issue between Sara Via and Samuel Scheiner, the dispute concerning the understanding of the evolution of phenotypic plasticity would then mainly be seen as being between those two authors. Schlichting, Stearns, etc.), based on the idea that adaptive plasticity evolves under natural selection. She argued that the disagreement only arose between them concerning the description of the precise means of its evolution. According to Via, "the assertion that phenotypic plasticity is a character that is independent of trait means and the attendant implication that plasticity itself is the target of selection" (Via, 1993, p. 352) was based on an allegedly erroneous analysis of certain phenomena by Bradshaw. She thought that this erroneous analysis was due to the lack of sufficient data in quantitative genetics in 1965, when Bradshaw wrote his article. But the reasons of Via's disagreement with her colleagues should rather be related to a certain confusion expressed by Bradshaw, himself, concerning the definition of "adaptive plasticity," since he was not properly doing quantitative genetics. Furthermore, from her observations of genetic studies with Lande (Via andLande 1985, 1987;[START_REF] Via | Genetic constraints on the evolution of phenotypic plasticity[END_REF], Via concluded that "partial genetic independence between the character states expressed in two environments is a minimal requirement for the evolution of a new reaction norm." From the idea that two genetic mechanisms can produce some degree of genetic independence between character states, she assumed in 1993 that "(1) two character states expressed in different environments will have some genetic independence if allelic sensitivities to the environment differ. […] (2) Partial independence between character states in different environments can also result if some loci influence the character state only in one of the environments, that is, if there is some environmentspecific gene expression […]" (Via, 1993, pp. 355-356). Via was then convinced that the evolution of the adaptive norm of reaction could only occur through the phenotypic traits themselves. Selection did not directly act on plasticity, and therefore plasticity could not be considered as a separate trait with its own genetic etiology. However, one problem she had to deal with was the correlation between the evolution of plasticity and the evolution of the trait expressed in the environment. Therefore, from 1986, Carl Schlichting opened the controversy (he would be followed by others) by defending a very different position. By comparing two species of purslane (Portulaca grandiflora and Portulaca oleraca), Schlichting attempted to show that the plasticity of a trait might evolve independently of the trait. And after Via's paper of 1993, Peter Van Tienderen, in 1994, and Gerdien de Jong, in 1995, would establish quantitative genetic models to highlight the variation of plasticity in accordance with Schlichting's model. In these models, the existence of "plasticity genes 17 " appears attested, since the independence between trait mean evolution and the evolution of plasticity was empirically established.

The controversy between Via and Schlichting illustrates the persistence of theoretical and conceptual disputes, rather than actual empirical obstacles to the identification of a "genetic" basis of plasticity. While some biologists thought that a genetic basis of plasticity exists, the problem remains precisely what they understood by "genetic basis." This issue ties in with a certain degree of uncertainty concerning the proper conceptual approaches of plasticity to adopt, sometimes assumed to refer to another phenotypic trait that would be subject to selection, sometimes assumed to refer to a property of living beings for which the evolution should be analyzed without knowing the causes of the property.

Samuel Scheiner and Richard Lyman's model for the evolution of plasticity: defining "plasticity genes"

17 The expression "plasticity genes" highlights the fact that the direct linear model between a single gene and a specific phenotype is rarely true. Models that are developed around the 1990's are polynomial models, which assess the plurality of genes implied in the expression of a phenotypic trait.

In line with a conception of plasticity as a specific "phenotypic trait," and in the context of researches in quantitative genetics, Scheiner and Lyman established, in 1991, a classification of what they considered as the "genetic basis of plasticity. 18 " According to them, three distinct categories cover the genetic basis of the plastic response. First, "overdominance 19 " expresses the fact that there is an inverse relationship between heterozygosity 20 and plasticity: the more a genotype is homozygous, the more its norm of reaction (its phenotypic response in different environments) is plastic. This model assumes, without formal demonstration, that plasticity is somehow an "accident" that is the result of a loss, or a reduction, of genotypic homeostasis 21 , which leads to an excess of genotypic homozygosity (see [START_REF] Lerner | Genetic homeostasis[END_REF][START_REF] Gillespie | Genotype-Environment Interactions and the Maintenance of Polygenic Variation[END_REF]. Second, "pleiotropy" shows that plasticity is a function of the differential expression of the same gene (the same set of alleles) in different environments (e.g., [START_REF] Falconer | Introduction to quantitative genetics[END_REF][START_REF] Via | Genotype-environment interaction and the evolution of phenotypic plasticity[END_REF], 1987;[START_REF] Via | Genetic constraints on the evolution of phenotypic plasticity[END_REF]. It is analogous to the case, in classical quantitative genetics, where several traits share a common genetic control due to the pleiotropic effects of genes. Finally, "epistasis" indicates that two classes of genes control two fundamental features of a norm of reaction: its plasticity and its overall mean. Therefore, for Scheiner and Lyman, plasticity is the result of the interaction between the genes that determine the magnitude of the response to the environmental effects with the genes that determine the trait mean expression (see [START_REF] Lynch | Environmental Tolerance[END_REF][START_REF] Jinks | The genetic basis of environmental sensitivity[END_REF][START_REF] Scheiner | The genetics of phenotypic plasticity I. Heritability[END_REF]. 18 Scheiner and Lyman conception is less sharp cut than De Jong's (1995) and [START_REF] Van Tienderen | Evolution of generalists and specialist in spatially heterogeneous environments[END_REF] conceptions because it does not focus on "genes of plasticity" but on the basis of plasticity's genetic expression. See [START_REF] Scheiner | The genetics of phenotypic plasticity. II. Response to selection[END_REF]. 19 The concept of overdominance is based on [START_REF] Lerner | Genetic homeostasis[END_REF] and [START_REF] Waddington | Genetic assimilation[END_REF] works. 20 A heterozygous individual is a diploid individual that has different alleles at one or several genetic loci. A homozygous individual is a diploid individual that has equal alleles at one or several genetic loci. 21 Homeostasis, or the "return to a genetic equilibrium", is linked to a homogenous repartition between maternal alleles and paternal alleles leading as a result to heterozygosity. See Pigliucci (2001) p. 69;p. 88-91. This model assumes, in contrast to Via's, that the trait mean and the environmental variance are two independent features. For [START_REF] Scheiner | The genetics of phenotypic plasticity. II. Response to selection[END_REF], these three categories are not mutually exclusive and their uses apply to the effects that occur not in a single environment, but in different environments over time. For this reason, their conception is mainly phenomenological, that is to say, based on the observation of "types" rather than on an investigation of the "real causes" of plasticity. However, the underlying idea is that all the different types could be analyzed with the same tools, those of genetics.

In this context, statistical quantitative genetic studies should be adequate to study any kind of models without knowing what is the actual role of the genes (de Jong, 1995). The fact that the model functions properly, together with its predictive ability, was somehow sufficient to prove its validity and to question the specificity of Via's conception. While Via's conception was facing challenges by the polynomial approach 22 (Van Tienderen, 1991; Scheiner, 1993; Van Tienderen & Koelewijn, 1994), she kept arguing that it was not entirely rejected by the latter approach [START_REF] Via | Adaptive phenotypic plasticity: consensus and controversy[END_REF]. For Via, who deeply disagreed with Scheiner, the "presupposed" independence between the "trait mean" and the plasticity, supported by the polynomial approach, remained to be confirmed. Via showed that the "trait mean" could be both measured for a single environment (independent environmental variability) and from a potential range of expression of the trait, which reflected, this time, the environmental variation in which the trait would be expressed. In the latter case, Via called it the "grand mean." Finally, she suggested that with a distinction between these two measures -the trait mean and the grand mean -it becomes possible to address, in a new way, the issue of the correlation between the evolution of the trait and the evolution of plasticity. For instance, in the case of a trait such as plant growth, the trait mean might be the same for two different species in two different 22 The polynomial approach of plasticity involves considering the measure of plasticity as a polynomial function, which represents its genetic determinism, see [START_REF] Scheiner | Genetics and Evolution of Phenotypic Plasticity[END_REF]. environments (i.e., comparatively, the two species will grow likewise), while, in a single environment, the trait mean might be different for each species (i.e., one of the species will grow more than the other in the environment E 1 and vice versa in the environment E 2 ). The consequence of such a difference, in Via's view, is that, in the same environment, different species of plants might grow differently, whereas, in changing environments, their average growth might be the same. From this observation, Via concluded that phenotypic plasticity should be considered as a "by-product" of the selection exerted on different phenotypic trait means in changing environments and not as a specific trait like Scheiner suggested (Via, 1993, p. 164). In her understanding of the evolution of plasticity, she tried to include what she considered to be essential when trying to characterize plasticity: the specificity of genes-environment interaction (an aspect that Scheiner had voluntarily left out).

Unfortunately, it seems that her attempt was not considered as a success because it appeared useless in a quantitative genetic approach. Indeed, after the paper of 1995 [START_REF] Via | Adaptive phenotypic plasticity: consensus and controversy[END_REF], the debate was considered solved in favor of Scheiner's conception, redefined by [START_REF] Schlichting | Control of Phenotypic Plasticity Via Regulatory Genes[END_REF]. However, some of the issues opened by this debate remained unsolved. The most contentious issue was not about how plasticity evolves, or what were its genetic bases, but it was about how plasticity should be defined, including or not the recognition of its causes. If neither side ever discussed the genotype-phenotype map (which is the underlying process23 ), Via's interest to the genotype-environment interactions in the beginning of her work, certainly drove that issue forward, despite the fact that she distanced herself from the topic when she started paying attention to plasticity, in the way it had been outlined by Bradshaw.

Therefore, a common aspect between the two views (Via and Scheiner's views), and despite their differences, is that, like Bradshaw, they never analyzed precisely the genotype-phenotypic map. This might help to explain why it remains difficult nowadays to associate this question with plasticity.

Conclusion

In order to solve the controversy between Via's approach and the polynomial approach, defended by Scheiner, Schlichting and Pigliucci proposed, in 1993, a new definition for "genes of plasticity" as: "regulatory loci that exert environmentally dependent control over structural gene expression and thus produce a plastic response.

Such loci represent a genetic mechanism for plastic response that is distinct from that assumed in quantitative genetic models of reaction norm evolution" (Schlichting & Pigliucci, 1993, p. 366). The two authors concluded that the debate between Via and Scheiner could be solved, since they saw it as a result of a "semantic ambiguity" rather than as the expression of a fundamental divergence between two distinct positions. Therefore, they saw the distinction between the two opposite conceptions -"existence of plasticity genes" versus "plasticity as a by-product of selection" -only as something apparent, since both options were not mutually exclusive. However, even if this precise debate between Via and Scheiner appeared to be solved for quantitative geneticists, the impression that it might have at least two different possible views concerning plasticity in an evolutionary perspective remained (e.g., depending if one look, or not, at the causes of plasticity). If this problem exists concerning any phenotypic traits, it seems even more problematic concerning plasticity. Recently, Mary Jane West-Eberhard has suggested a third way forward, by taking into account a developmental perspective 24 (West-Eberhard, 2003, p. 33).

http://dx.doi.org/10.1016/j.shpsc.2015.01.003

29

The debate between Via and Scheiner concerning the evolution of phenotypic plasticity illustrates the fact that, even if empirical proofs of a genetic basis of plasticity have become more and more numerous since the mid-1980's, the change linked to a better understanding of the underlying molecular mechanisms (not only to the genetic ones) at the origin of phenotypic traits has led biologists to look at the underlying causes of plasticity. But if understanding how plasticity evolves -that is to say understanding the role of natural selection on plasticity -does not necessarily require referring to its causes, it does require having an idea of what plasticity is. With Bradshaw, plasticity had acquired a clear definition, at least in a certain field of biology -that of quantitative genetics -and a renewed position within scientific concepts. With the emergence of genetics and Bradshaw's works, the term "plasticity" became mainly used as an operating concept of genetics, whereas in the past it was more a metaphorical concept. This new status relies on seminal Bradshaw's article of 1965 but also on a renewed enthusiasm for the area of research of plasticity, first in ecology in the 1970s, then with the development of new tools in quantitative genetics in the 1980s. Despite their disagreement Via and Scheiner were both referring to Bradshaw's definition of plasticity (which did not give any detail concerning the cause of the process, or the genotype-phenotypic map). After the resolution of the debate between Via and Scheiner, Bradshaw's definition remained quite unchanged in biological literature probably because neither of the two camps had questioned it.

That being said, definitions rely on biological tools and since biological tools are continuously evolving it should be the same for their attached notions. Finally, the debate that occurred between Via and Scheiner illustrates the fact that in the 1980s, and despite Bradshaw's precise definition of phenotypic plasticity, it had already become fairly difficult for biologists to overcome some of the difficulties Bradshaw already encountered when he defined the notion of plasticity, as well as to separate it from the field of quantitative genetics. If the debate concerning the evolution of phenotypic plasticity could be solved in the 1990s, it left open to considerations that did not follow from that theoretical framework, since it urged a precise analysis of the phenotype-genotype map.

For this reason, I suggest that biologists who have an interest in "phenotypic plasticity" within the context of an extended synthesis of evolution including development should be aware of this aspect, and should maybe learn in the light of this knowledge how to distance themselves from Bradshaw's legacy.

http://dx.doi.org/10.1016/j.shpsc.2015.01.003

For further details on Bradshaw's works, see E. Peirson, this issue.

The field was founded by some of the proponents of the Modern Synthesis, R.A. Fisher, S. Wright and J.B.S. Haldane, and aimed to predict the response to selection given data on the phenotype and relationships of individuals.

When the book was republished in 1962, a new section entitled "gene activity during development" is added.

For Nilsson-Ehle's explicit reference to plasticity, see[START_REF] Nilsson-Ehle | Vilka erfarenheter hava hittills vunnits rörande möjligheten av växters acklimatisering?[END_REF].

That is not the current standard usage anymore, which mainly limits physiological plasticity to physiological traits (like photosynthetic rate). However there is a current tendency to say that phenotypic plasticity should be seen as developmental plasticity (e.g.,[START_REF] West-Eberhard | Developmental plasticity and evolution[END_REF][START_REF] Gilbert | Ecological Developmental Biology : Integrating Epigenetics, Medicine, and Evolution[END_REF]; for more on this topic, seeNicoglou, 2013a).

It seems that the personality of the two biologists has been a major reason in the crystallization of the debate. This clearly appears in the interviews of Scheiner and Via conducted by Erick Peirson, that he has kindly brought to my attention.

It was the British zoologist Gavin de Beer who first offered, in 1940, such a distinction between the different kinds of plasticity in his book Embryos and Ancestors.

Few quantitative geneticists did, although there were exceptions, e.g.,Atchley's work (e.g., Atchley, 1984).

Acknowledgments:

I thank Erick Peirson and Marci Baranski for offering me the opportunity to be part of the symposium they organized in 2013 at the meeting of the History of Science Society and for their fruitful comments on earlier version of this paper. I also thank David Munns for his comments. I am also grateful to two anonymous reviewers who helped me to improve, I hope, this article. Finally, I thank my PhD advisors Jean Gayon and Philippe Huneman for encouraging me to present this work. I bear full responsibility for all remaining mistakes or deficiencies.


Rémi Coulom

Computing Elo Ratings of Move Patterns in the Game of Go

Move patterns are an essential method to incorporate domain knowledge into Go-playing programs. This paper presents a new Bayesian technique for supervised learning of such patterns from game records, based on a generalization of Elo ratings. Each sample move in the training data is considered as a victory of a team of pattern features. Elo ratings of individual pattern features are computed from these victories, and can be used in previously unseen positions to compute a probability distribution over legal moves. In this approach, several pattern features may be combined, without an exponential cost in the number of features. Despite a very small number of training games (652), this algorithm outperforms most previous pattern-learning algorithms, both in terms of mean log-evidence (-2.69), and prediction rate (34.9%). A 19x19 Monte-Carlo program improved with these patterns reached the level of the strongest classical programs.

Introduction

Many Go-playing programs use domain knowledge encoded into patterns. The kinds of patterns considered in this paper are heuristic move patterns. These are general rules, such as "it is bad to play in the corner of the board", "it is good to prevent connection of two opponent strings", "don't fill-up your own eyes", or "when in atari, extend". Such knowledge may be used to prune a search tree, order moves, or improve random simulations in Monte-Carlo programs [START_REF] Bouzy | Associating domain-dependent knowledge and Monte-Carlo approaches within a Go program[END_REF][START_REF] Sylvain Gelly | Modification of UCT with patterns in Monte-Carlo Go[END_REF].

Move patterns may be built by hand, or generated automatically. A popular approach to automatically generate patterns is supervised learning [START_REF] Araki | Move prediction in Go with the maximum entropy method[END_REF][START_REF] Bouzy | Bayesian generation and integration of Knearest-neighbor patterns for 19x19 Go[END_REF][START_REF] Fredrik | a Go-playing program using neural nets[END_REF][START_REF] Enderton | The Golem Go program[END_REF][START_REF] De | Moyo Go Studio[END_REF][START_REF] Stern | Bayesian pattern ranking for move prediction in the game of Go[END_REF][START_REF] Stoutamire | Machine learning, game play, and Go[END_REF][START_REF] Van Der Werf | Local move prediction in Go[END_REF]: frequent patterns are extracted and evaluated from game records of strong players. In this approach, expert knowledge is used to produce a relevant encoding of patterns and pattern features, and a machine-learning algorithm evaluates them. The advantage of automatic pattern learning over hand-made patterns is that thousands of patterns may be generated and evaluated with little effort, and little domain expertise.

This paper presents a new supervised pattern-learning algorithm, based on the Bradley-Terry model. The Bradley-Terry model is the theoretical basis of the Elo rating system. The principle of Elo ratings, as applied to chess, is that each player gets a numerical strength estimation, computed from the observation of past game results. From the ratings of players, it is possible to estimate a probability distribution over the outcome of future games. The same principle can be applied to move patterns: each sample move in the training database can be considered as a victory of one pattern over the others, and can be used to compute pattern ratings. When faced with a new position, the Elo ratings of patterns can be used to compute a probability distribution over all legal moves.

Related Work

This algorithm based on the Bradley-Terry model is very similar in spirit to some recent related works, but provides significant differences and improvements.

The simplest approach to pattern learning consists in measuring the frequency of play of each pattern [START_REF] Bouzy | Bayesian generation and integration of Knearest-neighbor patterns for 19x19 Go[END_REF][START_REF] De | Moyo Go Studio[END_REF]. The number of times a pattern is played is divided by the number of times it is present. This way, the strongest patterns get a higher rating because they do not stay long without being played. A major weakness of this approach is that, when a move is played, the strengths of competing patterns are not taken into consideration. In the Elo-rating analogy, this would mean estimating the strength of a player with its winning rate, regardless of the strength of opponents. By taking the strength of opponents into account, methods based on the Elo rating system can compute more accurate pattern strengths.

Stern, Herbrich, and Graepel [START_REF] Stern | Bayesian pattern ranking for move prediction in the game of Go[END_REF] address the problem of taking the strength of opponents into account by using a model extremely similar to Elo ratings. With this model, they can compute high-quality probability distributions over legal moves. A weakness of their approach, however, is that they are restricted to using only a few move features, because the number of patterns to evaluate would grow exponentially with the number of features.

In order to solve the problem of combining move features, Araki, Yoshida, Tsuruoka, and Tsujii [START_REF] Araki | Move prediction in Go with the maximum entropy method[END_REF] propose a method based on maximum-entropy classification. A major drawback of their approach is its very high computational cost, which forced them to learn on a restricted subset of moves, while still taking 8.75 days of computation to learn. Also, it is not clear whether their method would be able to provide a good probability distribution over moves, because, like the frequency-based approach, it doesn't take the strength of opponent patterns into account.

A generalized Bradley-Terry model, when combined with the minorizationmaximization algorithm to compute its maximum likelihood, addresses all the shortcomings of previous approaches, by providing the algorithmic simplicity and efficiency of frequency-based pattern evaluation, with the power and theoretical soundness of methods based on Bayesian inference and maximum entropy.

Paper Outline

This paper is organized as follows: Section 2 explains the details of the theory of minorization-maximization and generalized Bradley-Terry models, Section 3 presents experimental results of pattern learning, and Section 4 describes how these patterns were applied to improve a Monte-Carlo program.

Minorization-Maximization and Generalized Bradley-Terry Models

This section briefly explains, independently of the problem of learning patterns in the game of Go, the theory of minorization-maximization and generalized Bradley-Terry models. It is based on Hunter's paper [START_REF] Hunter | MM algorithms for generalized Bradley-Terry models[END_REF], where interested readers will find more generalizations of this model, with all the convergence proofs, references, and mathematical details.

Elo Ratings and the Bradley-Terry Model

The Bradley-Terry model allows to make predictions about the outcome of competitions between individuals. Its principle consists in evaluating the strength of each individual i by a positive numerical value γ i . The stronger i, the higher γ i . Predictions are made according to a formula that estimates the probability that i beats j:

P (i beats j) = γ i γ i + γ j .
The Elo rating of individual i is defined by r i = 400 log 10 (γ i ).

Some Generalizations of the Bradley-Terry Model

The Bradley-Terry model may be generalized to handle competitions involving more than two individuals. For n players:

∀i ∈ {1, . . . , n}, P (i wins)

= γ i γ 1 + γ 2 + . . . + γ n .
Another interesting generalization consists in considering not only individuals, but teams. In this generalization, the γ of a team is estimated as the product of the γ's of its members. For instance:

P (1-2-3 wins against 4-2 and 1-5-6-7) = γ 1 γ 2 γ 3 γ 1 γ 2 γ 3 + γ 4 γ 2 + γ 1 γ 5 γ 6 γ 7 .
Note that the same γ may appear in more than one team. But it may not appear more than once in a team.

Relevance of Bradley-Terry Models

The choice of a Bradley-Terry model makes strong assumptions about what is being modeled, and may not be appropriate in every situation. First, a Bradley-Terry model cannot take into consideration situations where individual 1 beats individual 2 consistently, individual 2 beats individual 3 consistently, and individual 3 beats individual 1 consistently. The strengths are on a one-dimensional scale, which does not allow such cycles. Also, the generalization to teams assumes that the strength of a team is the sum (in terms of Elo ratings) of the strengths of its members. This is also a very strong assumption that may not be correct all the time.

Bayesian Inference

Bradley-Terry models, as described in the previous sections, provide a probability distribution over the outcomes of future competitions, given the strength of individuals that participate. Most of the time the exact value of parameters γ i are unknown, and have to be estimated from the outcome of past competitions. This estimation can be done with Bayesian inference. With γ, the vector of parameters, and R, past results, Bayes formula is:

P (γ|R) = P (R|γ)P (γ) P (R) .
It gives a likelihood distribution over γ, from P (R|γ), that is to say the Bradley-Terry model described in the previous sections, P (γ), a prior distribution over parameters, and P (R), a normalizing constant. Parameters γ may be estimated by finding γ * that maximizes P (γ|R). This optimization can be made more convenient by choosing a prior that has the same form as the Bradley-Terry model itself. That is to say, virtual results R will serve as a prior: P (γ) = P (R |γ). This way, the estimation of parameters of the model will consist in maximizing P (R, R |γ).

A Minorization-Maximization Algorithm

Notations. γ 1 , . . . , γ n are the strength parameters of n individuals. N results R 1 , . . . , R N of independent competitions between these individuals are known. These competitions are of the most general type, as described in Section 2.2. The probability of one competition result may be written as

P (R j ) = A ij γ i + B ij C ij γ i + D ij ,
where A ij , B ij , C ij , and D ij are factors that do not depend on γ i . With this notation, each P (R j ) can be written in n different ways, each time as a function of one particular γ i . E j is defined as

E j = C ij γ i + D ij , and W i = |{j|A ij = 0}|
is the number of wins of individual i. The objective is to maximize:

L = N j=1 P (R j )
Derivation of the Minorization-Maximization Formula. (Readers who do not wish to understand all the details may safely skip to the formula) Minorization-maximization is an iterative algorithm to maximize L. Its principle is illustrated on Figure 1. Starting from an initial guess γ 0 for γ, a function m is built, that minorizes L at γ 0 . That is to say, m(γ 0 ) = L(γ 0 ), and, for all γ, m(γ) ≤ L(γ). The maximum γ 1 of m is then computed. Thanks to the minorization property, γ 1 is an improvement over γ 0 . The trick is to build m so that its maximum can be computed in closed form. This optimization algorithm is often much more efficient than traditional gradient-ascent methods.

L = N j=1 A ij γ i + B ij C ij γ i + D ij
is the function to be maximized. L can be considered as a function of γ i , and its logarithm is:

log L(γ i ) = N j=1 log(A ij γ i + B ij ) - N j=1 log(C ij γ i + D ij ) .
Terms that do not depend on γ i can be removed, and, since either B ij = 0 or A ij = 0, the function to be maximized becomes:

f (x) = W i log x - N j=1 log(C ij x + D ij ) .
The logarithms in the right-hand part may be minorized by their tangent at x = γ i , as shown on Figure 2. After removing the terms that do not depend on x, the minorizing function to be maximized becomes

m(x) = W i log x - N j=1 C ij x E j .
The maximum of m(x) is at

x = W i N j=1 Cij Ej
.

Minorization-Maximization Formula. So, minorization-maximization consists in iteratively updating one parameter γ i according to this formula:

γ i ← W i N j=1 Cij Ej
.

If all the parameters are initialized to 1, and the number of participants in each competition is the same, the first iteration of minorization-maximization computes the winning frequency of each individual. So, in some way, minorizationmaximization provides a Bayesian justification of frequency-based pattern evaluation. But running more than one iteration improves parameters further.

When players have different strengths, C ij indicates the strength of team mates of i during competition j, and E j is the overall strength of participants. With the minorization-maximization formula, a win counts all the more as team mates are weak, and opposition is strong.

Batch Updates. The minorization-maximization formula describes how to update just one γ i . It is possible to iteratively update all the γ i one by one, but it may be inefficient. Another possibility is to perform batch updates. A set of mutually exclusive γ i 's may be updated in one single pass over the data. Mutually exclusive means that they cannot be members of the same team. The batch-update approach still has good convergence properties [START_REF] Hunter | MM algorithms for generalized Bradley-Terry models[END_REF], and offers the opportunity to re-use computations. In particular, 1/E j can be computed only once in a batch.

Pattern-Learning Experiments in the Game of Go

A generalized Bradley-Terry model can be applied to supervised learning of Go patterns, by considering that each sample move is a competition, whose winner is the move in question, and losers are the other legal moves. Each move can be considered as a "team" of features, thus allowing to combine a large number of such features without a very high cost.

Data

Learning was performed on game records played by strong players on KGS. These game records were downloaded from the web site of Kombilo [START_REF] Görtz | Game records in SGF format[END_REF]. The training set was made of the 652 games with no handicap of January, 2006 (131,939 moves). The test set was made of the 551 games with no handicap of February, 2006 (115,832 moves). The level of play in these games may not be as high as the professional records used in previous research on pattern learning, but they have the advantage of being publicly available for free, and their level is more than high enough for the current level of Go-playing programs.

Features

The learning algorithm used 8 tactical features: pass, capture, extension, selfatari, atari, distance to border, distance to the previous move, and distance to the move before the previous move. Some of these features may take more than one value, as explained in Table 1.

The 9th feature was Monte-Carlo owner. It was computed by running 63 random games from the current position. For each point of the board, the number of final positions owned by the player to move was counted.

The 10th feature was shape patterns. Nested circles of radius 3 to 10 according to the distance defined in Table 1 are considered, similarly to [START_REF] Stern | Bayesian pattern ranking for move prediction in the game of Go[END_REF]. 16,780 shapes were harvested from the training set, by keeping those that appear at least 5,000 times.

Each value that these features can take is considered as a separate "individual", and is associated to one strength parameter γ i . Since values within one feature are mutually exclusive, they were all updated together within one iteration of the minorization-maximization algorithm.

Prior

The prior was set by adding, for each γ i , one virtual win, and one virtual loss, against a virtual opponent whose γ is 1. In the Elo-rating scale, this produces a symmetric probability distribution, with mean 0 and standard deviation 302.

Results

Table 1 lists the values of γ for all non-shape features.

Figure 3 plots the mean log-evidence per stage of the game, against the data of Stern, Herbrich, and Graepel [START_REF] Stern | Bayesian pattern ranking for move prediction in the game of Go[END_REF]. This mean log-evidence is the mean logarithm of the probability of selecting the target move according to the Bradley-Terry model, measured over the test set. The overall mean log-evidence is -2.69, which corresponds to an average probability of 1/14.7. Uniform probability gives a mean log-evidence of -5.49, which corresponds to an average probability of 1/243.

Figure 4 is a plot of the cumulative distribution of the probability of finding the target move at a given rank, measured over the test set, and compared with other authors. 1. Model parameters for non-shape features. Each feature describes a property of a candidate move in the current position. A feature my either be absent, or take one of the values indicated in the Level column.

-

Discussion

The prediction rate obtained with minorization-maximization and the Bradley-Terry model is the best among those published in academic papers. De Groot [START_REF] De | Moyo Go Studio[END_REF] claims a 42% prediction rate, so his results are still significantly better. Despite the similarity of the cumulative distributions, the mean log-evidence per stage of the game has a very different shape from that of Stern, Herbrich, and Graepel. Their algorithm provides much better predictions in the beginning of the game, and much worse in the middle. It is worth noting also that their learning experiments used many more games (181,000 instead of 652) and shape patterns (12,000,000 instead of 16,780). So they tend to learn standard opening sequences by rote, whereas our algorithm learns more general rules.

The learning process of our algorithm is not particularly optimized, and took about one hour of CPU time and 600 Mb of RAM to complete. So it is very likely that prediction performance could be improved very easily by using more games, and more shape patterns. Most of the computation time was taken by running the Monte-Carlo simulations. In order to learn over many more games, the slow features could be trained afterward, over a small set of games.

Usage of Patterns in a Monte-Carlo Program

Despite the clever features of this pattern-learning system, selecting the move with the highest probability still produces a terribly weak Go player. It plays some good-looking moves, but also makes huge blunders because it really does not "understand" the position. Nevertheless, the domain knowledge contained in patterns is very precious to improve a Monte-Carlo program, by providing a good probability distribution for random games, and by helping to shape the search tree. This section briefly describes how patterns are used in Crazy Stone [START_REF] Coulom | Efficient selectivity and backup operators in Monte-Carlo tree search[END_REF].

Random Simulations

The pattern system described in this paper produces a probability distribution over legal moves, so it is a perfect candidate for random move selection in Monte-Carlo simulations. Monte-Carlo simulations have to be very fast, so the full set of features that was described before is much too slow. Only light-weight features are kept in the learning system: 3x3 shapes, extension (without ladder knowledge), capture (without ladder knowledge), self-atari, and contiguity to the previous move. Contiguity to the previous move is a very strong feature (γ = 23), and tends to produce sequences of contiguous moves like in Mogo [START_REF] Sylvain Gelly | Modification of UCT with patterns in Monte-Carlo Go[END_REF].

Progressive Widening of the Monte-Carlo Search Tree

Crazy Stone also uses patterns to prune the search tree. This is performed at a much slower rate, so the full power of complex features can be used. When a node in the Monte-Carlo search tree is created, it is searched for a while without any pruning, selecting the move according the policy of random simulations. As soon as a number of simulations is equal to the number of points of the board, this node is promoted to internal node, and pruning is applied. Pruning consists in restricting the search to the n best moves according to patterns, with n growing like the logarithm of the number of random simulations. More precisely, the nth (n ≥ 2) move is added when 40 × 1.4 n-2 simulations have been run. On 19x19, because of the strength of the distance-to-the-previous-move feature, progressive widening tends to produce a local search, again like in Mogo [START_REF] Sylvain Gelly | Modification of UCT with patterns in Monte-Carlo Go[END_REF].

Performance against GNU Go

Conclusion

The research presented in this paper demonstrates that a generalized Bradley-Terry model is a very powerful technique for pattern learning in the game of Go.

It is simple and efficient, can combine several features, and produces a probability distribution over legal moves. It is an ideal tool to incorporate domain knowledge into Monte-Carlo tree search. Experiment results clearly indicate that significant progress can be made by learning shapes over a larger amount of training games, and improving features. In particular, the principle of Monte-Carlo features is very powerful, and could be exploited more, as Bouzy did with history and territory heuristics [START_REF] Bouzy | History and territory heuristics for Monte-Carlo Go[END_REF].

Also, the validity of the model could be tested and improved. First, using all the moves of one game as sample data breaks the hypothesis of independence between samples, since consecutive positions are very similar. Sampling one or two positions per game might be better. Also, the linearity hypothesis of the generalized Bradley-Terry model, according to which the strength of a team is the sum of the strengths of its members, is likely to be wrong. Estimating the strength of some frequent feature pairs separately might improve predictions.

Fig. 1 .

 1 Fig. 1. Minorization-maximization

+ -log x 1 -

 1 x/x0 -log x0

Fig. 2 .

 2 Fig. 2. Minorization of -log x at x0 = 0.5 by its tangent.

Fig. 3 .

 3 Fig. 3. Mean log-evidence per stage of the game (each point is an average over an interval of 30 moves).

Fig. 4 .

 4 Fig. 4. Cumulative distribution: probability of finding the target move within the n best estimated moves.

Table 2 summarizes

 2 Crazy Stone's performance against GNU Go 3.6. Tests were run on an AMD Opteron at 2.2 GHz, on one CPU. Crazy Stone ran, per second, from the empty position, 15,500 simulations on 9x9, and 3,700 on 19x19.

	Pat. P.W.	Size Min./game GNU Level Komi Games Win ratio
	-	-	9x9	1.5	10	6.5	170	38.2%
	x	-	9x9	1.5	10	6.5	170	68.2%
	x	x	9x9	1.5	10	6.5	170	90.6%
	-	-	19x19	32	8	6.5	192	0.0%
	x	-	19x19	32	8	6.5	192	0.0%
	x	x	19x19	32	8	6.5	192	37.5%
	x	x	19x19	128	8	6.5	192	57.1%

Table 2 .

 2 Match results. P.W. = progressive widening. Pat. = patterns in simulations.

Acknowledgments

I thank David Stern, Ralf Herbrich and Thore Graepel for kindly providing files with their performance data, so that I could plot mine against theirs. I am also grateful to the workshop reviewer and the readers of the computer-go mailing list for their comments that helped to improve the first version of this paper.