Nelo Magalhães

Absent, outside, inside: integrating the "environment" into Regulation Theory

Keywords: Capitalocene, epistemology, environment, political ecology

may come

Introduction

This text aims to explain a paradox: 1 the modest contribution of Regulation Theory (RT), one of the most important macroeconomic theories that takes capitalism as its main object, to the debates on the Capitalocene [START_REF] Hornborg | Global Ecology and Unequal Exchange: Fetishism in a Zero-Sum World[END_REF][START_REF] Bonneuil | L'événement anthropocène[END_REF][START_REF] Malm | Fossil capital: The rise of steam power and the roots of global warming[END_REF]. The method of inquiry borrows from epistemology and the sociology of science, and is based on a review of regulationist literature dedicated to environmental issues. We have selected as our corpus all the articles, in English and French, by authors from all disciplines (mainly economists and geographers) referring in their work to RT and to "the environment", taken in a broad sense. We argue that the weakness of regulationist analysis of environmental crises lies in the (changing) place of the environment in its theoretical framework. We distinguish three periods. The first, from the beginning in the late 70s until the 1990s, when environmental damages were already present, is characterized by the (almost complete) absence of the environment. A second period runs from the mid-1990s to the mid-2010s and acknowledges multiple attempts at its integration. These works are very heterogeneous, but what they have in common is to not question the very definition of their object, implicitly adopting the dominant postpolitical framework, that of the major institutions (national and international) responsible for public policies. Finally, we highlight our proposal, shared with others, to break the deadlock. This work ultimately shows that RT lacks a coherent political ecology. 2

Complete absence (1976-1995)

From Michel Aglietta's first book (1976), generally considered to be the first regulationist work, to the first "État des Savoirs" [START_REF] Boyer | Théorie de la régulation : l'état des savoirs[END_REF], the environment, no matter how one understands it, is totally excluded from the scope of RT. Alain [START_REF] Lipietz | Écologie politique régulationniste ou économie de l'environnement[END_REF] explains this fact by two arguments: first, there are only "institutionalized compromises between humans" -therefore, he emphasizes, no "institutionalized compromises with Nature"; second, because RT has focused on Fordism, whose crisis is "independent of the society-nature relationship", which has thus been set aside. His first point is undeniable and will be discussed below. The second deserves to be examined immediately.

Among the remarkable properties of the regulationist approach, the endogenous dynamics of accumulation regimes, the so-called "endometabolism" [START_REF] Lordon | Modéliser les fluctuations, le changement structurel, et les crises[END_REF], is at the top of the list. To put it simply, the general structure of a regime evolves tirelessly, through small transformations (endogenous, therefore), until it reaches a crisis. Thus, a key result of RT is that the crisis of Fordism (whether US or French) originates in the very mechanisms that made it stable. This thesis is based on the observation that productivity gains, which were the driving force behind the Fordist wage compromise, and hence of the whole system, were exhausted before the energy crisis of the 1970s. It is explicitly opposed to the common-sense hypothesis that gave an inordinate role to the "oil shock" (exogenous, therefore) as the main element responsible for the crisis. The consequence is unfortunate, since the sidelining of oil contributed to the exclusion of all energy and material flows from the framework, and prevented RT from entering the academic debates, already numerous in the 1970s,3 on environmental crises. As an illustration, even important analyses on agriculture [START_REF] Allaire | Le modèle de développement agricole des années 1960[END_REF] ignore all kinds of inputs (synthetic fertilizers, energy) which largely contribute to the sector's productivity gains. Throughout this period, RT had literally no material ground.

The Need for Integration: Programmatic and Empirical Works (1995-2015)

Lipietz's observation on the theoretical vacuum vis-à-vis the environment was constantly renewed and deplored (Rousseau, 2002, p. 14;Rousseau & Zuindeau, 2007, p. 5;Cahen-Fourot & Durand, 2016, p. 2). A few attempts emerged to "integrate" the environment later on. They can be summarized in three non-disjoint groups. In a context of the rising of the Sustainable Development paradigm and strong institutionalization of environmental issues at the international level (e.g., Montreal Protocol in 1987, Rio summit in 1992, Kyoto Protocol in 1997), while some regulationist authors were arguing for normative studies4 , the first group was gathering around a reflection on a post-Fordist regime of accumulation [START_REF] Boidin | Socio-économie de l'environnement et du développement durable : État des lieux et perspectives[END_REF][START_REF] Rousseau | Théorie de la régulation et développement durable[END_REF][START_REF] Lardé | Théorie de la régulation et développement durable : essai d'analyse de la diversité nationale en matière d'environnement et de politiques environnementales[END_REF]. For Rousseau & Zuindeau (2007, p. 13-14, our translation), "Sustainable Development is now an essential contribution to thinking about the future of capitalism, or rather capitalisms. We believe that RT will not emerge unscathed from such a contribution and that it will have to renew itself significantly." This theoretical alliance did not occur -this normative proposition is contradictory to the analytical posture of RT.5

The second group of research was empirical. The most frequent approach,6 that of typologies, provided empirical information without any real theoretical reflection. A standard article can even be defined. Whether in [START_REF] Lardé | Théorie de la régulation et développement durable : essai d'analyse de la diversité nationale en matière d'environnement et de politiques environnementales[END_REF], [START_REF] Elie | Approche régulationniste de la diversité des dispositifs institutionnels environnementaux des pays de l'OCDE[END_REF] or Cahen-Fourot (2020), each time OECD databases were used to propose a classification of capitalist economies "from an environmental perspective". The idea, which is often explicit, was to examine whether the typology of the variety of capitalisms produced by Bruno Amable (2005) could be found, or not, when the "environment" variable is added. 7 Rather than deconstructing this category, or establishing its historical genesis, these articles mobilize the OECD's "environmental" variables, supplemented by others available online, to discriminate between countries (moreover with no consideration, , of their construction,8 their quality or their relevance). Symbolic of this type of approach is the article by [START_REF] Elie | Approche régulationniste de la diversité des dispositifs institutionnels environnementaux des pays de l'OCDE[END_REF], which introduces "environmental institutional arrangements" (EIA). These are supposed to summarize the "economy-environment" linkages and are mainly concerned with environmental management and environmental policies. The regulation of environmental conflicts is explicitly presented (p. 8) as being at the origin of EIAs: "An EIA is a set of institutions dedicated to the regulation of environmental problems. These are quotas, standards, taxes, subsidies, and market instruments whose objective is to regulate conflicts related to access (which includes preservation) to ecosystem goods and services." However, the particularities of those conflicts are never presented (they seem to be reduced to "resources"), neither are their causes nor the logic behind their emergence. They remain abstract or implicit, detached from the logic of capital accumulation. From the "ratio of the price of gasoline to the world average" to the "number of ISO 14001-certified companies per billion dollars of GDP" (sic), one can be skeptical about the intelligibility of environmental conflicts. The way in which these typologies were carried out implicitly leads to the fallacious narrative of the emergence of "environmental states" in the Global North (most of the OECD countries), the very ones that construct and implement public policies designated as "environmental", and make power relations disappear (in particular that of relocating to the Global South all the productions and extractions that are the most degrading to their environment9 ). Alongside these two groups (sustainable development, OECD typologies), which focus on explicit public policies, a group of programmatic works stands out (Drummond & Marsden, 1995;Gibbs, 1996;[START_REF] Zuindeau | Régulation School and environment: Theoretical proposals and avenues of research[END_REF][START_REF] Douai | Economie écologique et économie hétérodoxe : pour une socio-économie politique de l'environnement et du développement durable[END_REF][START_REF] Chester | Determining the economic-environment relation: a regulationist approach[END_REF][START_REF] Douai | Institutions and the environment: The case for a political socio-economy of environmental conflicts[END_REF][START_REF] Cahen-Fourot | Contemporary Capitalisms and Their Social Relation to the Environment[END_REF]. A stimulating theoretical proposal consists in the addition of a new institutional form (IF), which would codify (by definition) a new fundamental social relation, autonomous from the other five: the "society-environment relation" or "social relation to the environment" [START_REF] Becker | Theory of regulation and political ecology: an inevitable separation?[END_REF]. These authors present the environment as an "ecological constraint" (which, according to them, is similar to the monetary constraint) on capitalist production that must be taken into account, insofar as it can be the object of potentially destabilizing class struggles for the regime of accumulation in place. For them, this social relation regulates, "the access to the physical environment and the modalities of its use for the activities of production and reproduction. It therefore also regulates the spatial and temporal distribution of the ecological costs and benefits of these (re-)productive activities." They do not justify why this IF is indispensable (nor why it cannot be integrated into the other five). On the other hand, Chester (2010, p. 33) argues against this new IF by asserting that"the changing nature of the economic-environment relation can only be understood by reference to the macro mode of regulation. Although she proposed a stimulating analytical method for empirical investigation, she never provided any empirical work. Most of the reflections -with the notable exception of Cahen-Fourot (2020) who, in turn, never explicitly defines "the relation to the environment" -are hypothetical, whether they are in favor of this new IF or not, and remain tied to the role of the environment in the sphere of production. This is very problematic. What about, for example, large-transport systems or energy infrastructures, land use planning policies, mass tourism? Starting from a new definition of capitalism, for example based on three production factors (money, labor, land), Polanyi's famous "fictitious commodities", linked to three social relations, these authors attempt to highlight contradictions or a purely theoretical characterization of the social relation to the environment. Those analyses are confused. Becker and Raza define the 6th IF as a social relation to nature, thus adopting implicitly the society/nature dualism, while claiming that nature is a social construction. To support their demonstration, they introduce a new factor (N for nature) into the Marxist M-C-M' (M = Money, C = Commodity) cycle, which becomes M-C/N-C/N'-M', before emphasizing the reification of "social relations to nature" in capitalism without making the connection between these two arguments. Some (theoretical) examples of evidence are often recalled (the capitalist system is based on a "commodification of nature" and subjects the "human environment" to the process of valorization) without contributing to the concrete understanding of an accumulation regime.10 While they claim to link "the environment" to the process of capitalist valorization and to structural contradictions (always situated in the process of commodity production11 ), they never lead to empirical studies and remain indifferent to historical works.

In general, the definition of the environment (and environmental conflict) adopted by these three groups is extremely vague, tautological, or left aside -as if it were self-evident.12 However, it is clear that in the vast majority of cases it overlaps with the definition of neoclassical theory: the environment, a sphere separated from society and the economy, is reduced to inputs and outputs, natural resources (raw materials and energy) and waste, and associated conflicts are never distinctly defined or studied. While the authors denounce neoclassical analyses when introducing their research,13 they regularly use its lexicon (externalities, environmental benefits and costs, etc.). Instead of making a radical break with standard theory, as in the case of labor14 or money, RT has taken the same direction as the latter -the work of Bertrand [START_REF] Zuindeau | L'analyse des externalités environnementales : éléments pour un programme de recherche régulationniste[END_REF][START_REF] Zuindeau | Régulation School and environment: Theoretical proposals and avenues of research[END_REF], for example, on the "management of externalities according to the accumulation regime", is the most striking example. The fact that [START_REF] Boidin | Socio-économie de l'environnement et du développement durable : État des lieux et perspectives[END_REF] and [START_REF] Lardé | Théorie de la régulation et développement durable : essai d'analyse de la diversité nationale en matière d'environnement et de politiques environnementales[END_REF] refer to the absence of a "regulationist theory of the environment", competing with "environmental (neoclassical) economics", suggesting that the "environment" is a separate object / commodity, separated from the social world, symbolizes the blind reduction to "natural resources". RT has thus taken for granted, without questioning them, the dominant environmentalist discourses (which are reflected in the very institutionalization of this "object" by the OECD and the creation of "Ministries of Nature and Environment Protection" in the Global North in the 1970s).

Social metabolism and production of space of accumulation regimes

The fundamental fact, or blind spot, ignored by this corpus deserves to be stated: all social relations have a material dimension and all biophysical flows that concern a society are imbricated in social relations (and are therefore not "natural"). To put it another way: every economic fact is a social and, therefore, material fact. For the past five years, rather than confronting the question of "the environment" with yet another scholastic reflection or empirical work that reduces it to environmental policies, we have chosen to integrate biophysical flows into RT. As a starting point, we recognize that accumulation regimes are maintained not only by institutional forms but also by the reproduction of energy and material flows [START_REF] Cahen-Fourot | The evolving accumulation-metabolism nexus of French capitalism since the postwar era[END_REF]. Rather than studying the links between capitalism (or "the economy") and the environment, conceived as separate entities, our approach, inspired by environmental history, seeks to understand capitalism through the material inscription of its main social relations, with an inductive and historical approach (this method partially overlaps with Chester's proposal). Beyond observing correlations between monetary and biophysical statistical series, listing environmentalist laws or simply describing the inputs and outputs of an accumulation regime, our objective is to reflect on (the evolution of) the material basis of all social relations (a fortiori the most fundamental ones). In our perspective, there is no social relation to the environment, but fully material social relations (and always anchored in concrete soils). More than a simple resource, energy, for example, constitutes a dimension of the social class relations, the form of the state and the wage relation. The dominant historical variants it takes (coal, oil, nuclear) are not neutral in their effects on these social relations [START_REF] Mitchell | Carbon Democracy: Political Power in the Age of Oil[END_REF][START_REF] Huber | Fueling Capitalism: Oil, the Regulation Approach, and the Ecology of Capital[END_REF]. The environment is no longer an "outside" of society, an a-social "resource" or an allegedly "wild" space, and is no longer merely worthy of interest as a factor of production, but it is an "inside", present in all institutionalized social compromises. Capital accumulation generates material flows that go far beyond simple inputs/outputs. Beyond a "social metabolism", each accumulation regime is characterized by a production of space, which necessarily affects very diverse (socio-ecological) territories that can provoke territorial conflicts [START_REF] Magalhães | Matières à produire l'espace. Une histoire environnementale des grandes infrastructures depuis 1945[END_REF]). Lipietz's fear is dismissed: it is forbidden to analyze these flows in an ahistorical and asocial way, as neoclassical economists may do, isolating "resources" from the economic system. Just as Marxist scholars, whether within the conception of capitalism as a world-system [START_REF] Moore | Capitalism in the web of life: Ecology and the accumulation of capital[END_REF] or as unequal exchange [START_REF] Hornborg | Global Ecology and Unequal Exchange: Fetishism in a Zero-Sum World[END_REF], have seized upon biophysical flows to account for the Capitalocene while avoiding physical reductionism, RT has much to gain from detecting this dimension in all fundamental social relations [START_REF] Magalhães | The Physical Economy of France (1830-2015). The History of a Parasite?[END_REF][START_REF] Magalhães | Régimes d'accumulation et flux de matières en longue période[END_REF]. It should be noted that we preserve RT's originality since the crisis of Fordism remains endogenous: we have shown how oil imports consolidate the regime, as was the case especially in the 1960s, before destabilizing it [START_REF] Cahen-Fourot | The evolving accumulation-metabolism nexus of French capitalism since the postwar era[END_REF]. It was not oil that caused the crisis, but the sharp rise in its price helped, along with other factors (e.g., internationalization), to destabilize the most structuring mechanisms (in particular the wage relation and the form of the state).

The purpose of this proposal is not to add a new "variable", but to reveal the materiality that is always present in social facts. These works demonstrate that it is possible to conduct a regulationist research by avoiding scholastic reflection on the role of the "environment" in an accumulation regime. There is no need for the notion of a "relationship to the environment" to produce regulationist analyses that integrate biophysical flows and their spatiality: Huber (2013) has shown, for example, that the role of oil in US Fordism can be studied with RT institutionalism. In this respect, a careful reading of Michel Aglietta's famous study of US Fordism shows that the territorial dimension was already present, in suspense -he designates the car and the individual house as the key goods of the (social) norm of mass consumption [START_REF] Aglietta | Régulation et crises du capitalisme[END_REF]. The new field of research consists in rereading the entire regulationist literature and finding the material base and its territoriality (which includes many socioecological spaces that can be defined as environments), that have been made invisible.

Their ecology -and ours?

Since its origin, RT has had a troubled relation with the environment. This comes from a blind adoption of the common-sense definition. Conceived as asocial, the materiality, including all biophysical flows, was first (coherently) excluded from the framework. After 1995, the few relatively rigorous theoretical attempts at "integration" are often based on some hypothetical-deductive reasoning that is indifferent to history and social sciences.

Environmental history and political ecology have long demonstrated the modern invention of the environment (the term only entered administrative language in the 1970s15 ). On the other hand, what is presented today as environmental reflexivity as well as environmental conflicts have existed since at least the origins of capitalism. 16 RT must acknowledge that the environment, starting with its very definition, is a battle field [START_REF] Keucheyan | La nature est un champ de bataille : Essai d'écologie politique[END_REF] over which antagonistic political ecologies struggle (not to be confused with "ecological" / "green" political parties). The dominant social blocs, and the associated academic (and media) relays, support post-political [START_REF] Kenis | Searching for 'the political' in environmental politics[END_REF] and elitist ecologies: conservationism (or technocratic), preservationism (or moral), postmodern (or constructivist).17 The first is largely hegemonic, and in particular is adopted by the neo-classical current (and its derivatives18 ) as well as most of the major (inter)national institutions (e.g., OECD, European Commission). It is also known as eco-modernism or managerialism [START_REF] Christoff | Ecological Modernisation, Ecological Modernities[END_REF]. Compatible with liberal ideology (which reduces all problems to individual responsibilities), it supports the narrative of "green" growth and ecological modernization (illustrated by "green technologies", "green production", etc.). Green refers only to optimization gains (on the input or output side) -concrete environments are made abstract. The second, often described as neo-Malthusian, brings together currents of deep ecology, wilderness, thinkers of an intrinsic value in nature, etc. The preservation of the environment here amounts to emptying spaces of human presence and protecting them from anthropic actions. 19 The third is mainly academic (around thinkers such as Latour, Stengers, Haraway) and presents the "ecological crisis" as a profound mutation of our relationship to the world, linked to the end of the (modern) concept of "nature" and which requires new ontologies or a new conception of politics. 20 Let us note that these approaches are not rigorously incompatible (some actors support both the optimization of resources and wild spaces and places in which assemblies of hybrids would redefine democracy). Above all, they have in common that they remove political conflict from their framework (in this sense they could all be called post-political). As inputs and outputs (including CO2) are detached from social relations (e.g., production or consumption social relations), it is (ultimately) not necessary to question them in order to proceed with optimization: new technologies, new ethics/morals and new compositions of the world will suffice. The attraction of these depoliticized approaches is undoubtedly due to the absence of an environmental or natural "subject" (as Lipietz rightly states). Unlike other relations of domination (class, gender, race, etc.), there is no "subject" oppressed by another in a specific social relation. While the "subject" of social struggles is present from the definition of the (wage) production relation, the "subject" of environmental conflicts is not defined in a dialectical relation. This partly explains why the (technocratic, ethical or postmodern) narratives of the Anthropocene support, at least implicitly, the (post-political) Humanity-Nature duel [START_REF] Bonneuil | L'événement anthropocène[END_REF]. Strictly speaking, because of the epistemological role it entrusts to politics (and social conflict) in its analyses, RT cannot adhere to these ecologies.

In brief: dominant ecologies are perfectly clear; what about ours? 21

In the early 1970s, the Meadows report and the works of Boulding, Georgescu-Roegen and Daly were widely discussed in the academic field[START_REF] Blanchard | Les limites à la croissance dans un monde global. Modélisations, prospectives, réfutations[END_REF].

See Amable & Palombarini (2005).

This proposal is close to the "moral socio-economy of the environment", which tends to imagine the resolution of conflicts through the creation of "ideal institutions" that would facilitate a common vision of the good.

Much more discrete, a few socio-historical case studies could be included in the corpus[START_REF] Buchs | Comprendre le changement institutionnel. Régimes et crises du mode d'usage de l'eau en Espagne (xixe-xxe siècle)[END_REF].

The hypothesis being tested may be particularly poor: "models of capitalism based on different logics provide different institutional responses to the problems of quantitative and qualitative deterioration of natural resources and to conflicts over these resources" (Elie et al., 2012, p. 9, our translation).

Who constructed these data / nomenclatures / indicators, and how? Are they comparable in countries as diverse as Germany, Mexico or South Africa?

Following this line of reasoning, free trade constitutes a formidable policy, to be maintained and deepened in rich countries, to transform the state into an "environmental state".

That capitalists have a structural interest in transforming their private costs into social costs has already been demonstrated by[START_REF] Kapp | The social costs of business enterprise[END_REF] -a rarely cited author.

The two main "proofs" follow the dialectic of use value and exchange value and the addition of a new factor in the M-C-M' valorization process (James O'Connor's second contradiction thesis).

 12 In[START_REF] Lardé | Théorie de la régulation et développement durable : essai d'analyse de la diversité nationale en matière d'environnement et de politiques environnementales[END_REF], the environment is simply defined as "the elements of nature (air, water, energy, landscape amenities, ecosystem functions of nature) serving as economic resources and consumer goods, but also has to do with issues of waste and pollution, as well as the space in which individuals and activities are located"; in[START_REF] Rousseau | Economie et Environnement : Une analyse régulationniste de la rente environnementale[END_REF], the environment is the "result of the natural elements in themselves and of the society-nature relationship (...) composed of natural objects and objects transformed (by man)

".13 Gendron (2008) reminds us that RT has the theoretical weapons to deconstruct neoclassical conceptions in environmental economics (market neutrality, the principle of optimization, the hypothesis of reversibility, technological determinism, rationality of atomized economic

agents).14 RT has as much interest, in order to maintain its coherence as a political economy, in avoiding the (technocratic) terminology of "natural resources" as that of "human resources".

While it has been mostly careful not to mention "nature" -which is an even more problematic term -it has neglected discussions of alternatives[START_REF] Feuerhahn | Du milieu à l'Umwelt : Enjeux d'un changement terminologique[END_REF].

Far from quantitative databases, one must look to health, aesthetic, habitat, consumption and other issues to grasp what some economists reduce to externalities[START_REF] Fressoz | Introduction à l'histoire environnementale[END_REF].

Other classifications are possible.[START_REF] Bourg | Écologies politiques : Essai de typologie[END_REF] distinguish the following streams of political ecology: (Neo-)Malthusian (mainly preservationists), Arcadian (Illich, Gorz, etc.), authoritarian, institutionalist (in their terminology this is close to the moral approach), catastrophist (which includes collapsologist movements), grassroots (close to preservationists), constructivist (postmodern in our terminology), and anarchist.

The alliance is based on the representation of a problem in terms of "means" and "ends" (Robbins).

By taking into account the number of Greenpeace members and member organizations of the International Union for Conservation of Nature in each country, Cahen-Fourot (2020) mixes both environmentalisms to capture the "political demand for the environment".

Bruno Latour affirms the need for the politicization of ecology, but with a very particular definition of the word "political" which integrates within its perimeter all the non-human actants (endowed with agency according to him), while evacuating the usual power social relations.


Henri Cohen
email: henri.cohen2@free.fr

Elementary Continued Fractions for Linear Combinations of Zeta and L Values

We show how to obtain infinitely many continued fractions for certain Z-linear combinations of zeta and L-values. The methods are completely elementary.

Introduction and Main Goal

In [4], the authors give five continued fractions for certain Z-linear combinations of zeta values, only obtained and checked numerically, as well as other linear combinations involving powers of π, Catalan's constant, etc... The purpose of the present paper is to show that these continued fractions are completely elementary. In fact, we explain three general methods for constructing them. In particular we give seven infinite families of such continued fractions. We also discuss analogous results involving L-values for Dirichlet characters of conductor 3 and 4.

We use the following notation for a continued fraction S, which may differ from notation used in other papers in the literature: S = a(0) + b(0)/(a(1) + b(1)/(a(2) + b(2)/(a(3)

+ • • • ))) ,
and we denote as usual by p(n)/q(n) the nth convergent, so that p(0)/q(0) = a(0)/1, p(1)/q(1) = (a(0)a(1) + b(0))/a(1), etc. We recall the following trivial result due to Euler: Lemma 1.1 Let f (n) be a nonzero arithmetic function, and set by convention f (0) = 0. When the left-hand side converges, we have

n≥1 z n f (n) = [[0, f (n) + zf (n -1)], [z, -zf (n) 2 ]] ,
and in addition the N th partial sum of the series is equal to the N th convergent p(N )/q(N ) of the continued fraction.

As a trivial application, for k ≥ 2, we have the trivial continued fraction

ζ(k) = [[0, n k + (n -1) k ], [1, -n 2k ]] .
The second trivial lemma that we will use is the following:

Lemma 1.2 Let (a(n), b(n))
define a continued fraction with convergents (p(n), q(n)), and let r(n) be an arbitrary nonzero arithmetic function with r(0

) = 1. Then if we set a ′ (n) = r(n)a(n) and b ′ (n) = r(n)r(n + 1)b(n), the corresponding con- vergents (p ′ (n), q ′ (n)) are given by (p ′ (n), q ′ (n)) = r!(n)(p(n), q(n)) with evident notation, and in particular p ′ (n)/q ′ (n) = p(n)/q(n).
Thanks to the first lemma above, we can thus transform any series into a continued fraction, not really interesting. For instance, assume that I want a CF for

ζ(2) + ζ(3): we have ζ(2) + ζ(3) = n≥1 (n + 1)/n 3 ,
so applying the first lemma to f (n) = n 3 /(n + 1) and z = 1, we get

ζ(2) + ζ(3) = [[0, n 3 /(n + 1) + (n -1) 3 /n], [1, -n 6 /(n + 1) 2 ]] ,
and applying the second lemma to r(0) = 1, r(n) = n 2 + n for n ≥ 1, we get

ζ(2) + ζ(3) = [[0, 2n 4 -2n 3 + 2n -1], [2, -(n 8 + 2n 7 )]] .
As mentioned, not very interesting, in particular because continued fractions involving ζ(3) should have a(n) a polynomial of degree at most 3, and b(n) at most 6.

We can now more precisely state our goal, much wider than the simple proofs of the continued fractions given in [4]. First, we set the following definition: Definition 1.3 Let d ≥ 0 be an integer. A rational period of degree k is the sum of a convergent series of the form n≥1 χ(n)f (n), where χ(n) is a periodic arithmetic function taking rational values, and f ∈ Q(x) is a rational function with rational coefficients, whose denominator is of degree k.

Two remarks concerning this definition: first, it is not compatible with the definition of periods as given in [START_REF] Kon-Zag | Periods, in Mathematics unlimited 2001 and Beyond[END_REF]. Second, one could ask the coefficients to be algebraic instead of rational, but this leads to a theory which is too general.

Examples: log(2), π, and L(χ, 1) for a nontrivial Dirichlet character all have degree 1, more generally π k and L(χ, k) are rational periods of degree k. Note also that a Q-linear combination of rational periods of degree ≤ k is again a rational period of degree ≤ k.

Thus, our goal will be as follows: find continued fractions (a(n), b(n)) for rational periods degree k where for n sufficiently large a(n) is a polynomial of degree at most k and b(n) of degree at most 2k, which we abbreviate by saying that it has bidegree at most (k, 2k).

The prototypical "trivial" example is S = n≥1 1/P (n) with P a polynomial of degree k ≥ 2, and S = [[0, P (n) + P (n -1)], [1, -P (n) 2 ]] is a continued fraction of required bidegree (k, 2k), by Euler's lemma above, and similarly for S = n≥1 (-1) n-1 /P (n). On the contrary, as mentioned above, such a continued fraction does not seem to exist for n≥1 (n + 1)/n 3 .

2 First Method: use of Polynomial Multipliers Proposition 2.1 Fix an integer k ≥ 2, and let P ∈ Q[x] be a nonzero polynomial with rational coefficients such that P (x) divides x k P (x+ 1)+ (x-1) k P (x-1), and set R(x) = (x k P (x + 1) + (x -1) k P (x -1))/P (x).

(1) We have the continued fraction expansion

S = n≥1 1 n k P (n)P (n + 1) = [[0, R(x)], [1/P (0) 2 , -n 2k ]] ,
which is a continued fraction of bidegree (k, 2k).

(2) If P (x) and P (x + 1) are coprime polynomials and d = deg(P ) ≤ k, S is a rational period of degree at most k.

Proof. (1). By Euler's lemma above we have

S = [[0, n k P (n)P (n + 1) + (n -1) k P (n -1)P (n)], [1, -n 2k P (n) 2 P (n + 1) 2 ]] .
I claim that P (1) = 0: indeed, if P (1) = 0 we deduce that 1 k P (2) = R(1)P (1) so P (2) = 0, and the recursion (n -1) k P (n) = R(n -1)P (n -1) -(n -2) k P (n -2) for n ≥ 3 implies that P (n) = 0 for all n, so P has infinitely many roots so is identically zero, contradiction. We can thus apply the second lemma to r(n) = 1/P (n) 2 for n ≥ 1 and r(0) = 1 and we obtain immediately

S = [[0, R(n)], [1/P (1) 2 , -n 2k ]], proving (1).
For (2), we first note that using the same proof as in (1) but using the recursion backwards, we have P (0) = 0. We can thus write a partial fraction decomposition in the form

1 x k P (x)P (x + 1) = 1≤j≤k c j x j + N (x) P (x)P (x + 1)
,

where the c j are constants and deg(N (x)) ≤ 2d -1, where d = deg(P ). Since by assumption P (x) and P (x + 1) are coprime, by the extended Euclidean algorithm, there exist polynomials U and V such that U (x)P (x) + V (x)P (x + 1) = N (x), and U and V can be chose of degree less than or equal to d -1.

Thus N (x)/(P (x)P (x + 1)) = U (x)/P (x + 1) + V (x)/P (x), so S is a rational period of degree at most d ≤ k. ⊓ ⊔

Remarks

(1) It is possible that the condition that P (x) and P (x + 1) are coprime can be lifted.

(2) One can prove that P (x) satisfies the identity P (x + 1) = (-1) d P (-x), or equivalently P (1 -x) = (-1) d P (x). I am grateful to "Ilya Bogdanov" from the MathOverflow forum for the proof of this fact.

Exactly the same proposition with an identical proof can be applied to alternating sums: Proposition 2.2 Fix an integer k ≥ 2, and let P ∈ Q[x] be a nonzero polynomial with rational coefficients such that P (x) divides x k P (x+ 1)-(x-1) k P (x-1), and set R(x) = (x k P (x + 1) -(x -1) k P (x -1))/P (x).

(1) We have the continued fraction expansion

S = n≥1 (-1) n-1 n k P (n)P (n + 1) = [[0, R(x)], [1/P (0) 2 , n 2k ]] ,
which is a continued fraction of bidegree (k -1, 2k).

(2) If P (x) and P (x + 1) are coprime polynomials and d = deg(P ) ≤ k, S is a rational period of degree at most k.

The next section consists in searching for suitable polynomials P (x) and writing the corresponding rational period and continued fraction.

Examples

Proposition 3.1 The condition of the proposition P ∈ Q[x] dividing x k P (x + 1) + (x -1) k P (x -1) is satisfied in the following cases:

(1) k ≡ 0 (mod 2) and P (x) = 2x -1.

(2) k ≡ -1 (mod 3) and P (x) = 3x 2 -3x + 1.

(3) k ≡ -1 (mod 4) and P (x) = 2x 2 -2x + 1.

(4) k ≡ -1 (mod 6) and P (x) = x 2 -x + 1.

(5) k = 5 and P (x) = 5x 4 -10x 3 + 19x 2 -14x + 4.

Proof. In the first four cases, it is sufficient to check that any root of P (x) = 0 is also a root of x k P (x + 1) + (x -1) k P (x -1). For instance in the first case P (x) = 2x -1, for a = 1/2 we have P (a + 1) = 2 and P (a -1) = -2, and indeed 2 -k • 2 + (-2) -k • (-2) = 0 when k is even. The last case is done by a direct divisibility test.

⊓ ⊔

The same proof shows the following:

Proposition 3.2 The condition of the proposition P ∈ Q[x] dividing x k P (x + 1) -(x -1) k P (x -1
) is satisfied in the following cases:

(1) k ≡ 1 (mod 2) and P (x) = 2x -1.

(2) k ≡ 1 (mod 4) and P (x) = 2x 2 -2x + 1.

(3) k ≡ 2 (mod 6) and P (x) = x 2 -x + 1.

Note that I have not found any other examples than the ones given in the above two propositions, but I may have missed some.

Thanks to these propositions, it is now just a matter of working out explicitly all the above examples, in other words of computing the partial fraction expansions of the expressions 1/(x k P (x)P (x + 1)), which is routine so not given explicitly. In particular, we will see that in the non-alternating cases, the sum S is a Q-linear combination of 1 and ζ(k) for k ≥ 2 of fixed parity, and in the alternating cases, with in addition log(2).

We now give the corresponding formulas, and give examples after. The following corollary immediately follows from the above propositions:

Corollary 3.3 By convention, set ζ(0) = ζ * (0) = 0, ζ * (1) = log(2), and ζ * (k) = (2 k-1 -1)ζ(k) for k ≥ 2.
We have the following general continued fractions:

k-1 j=0 2 2j ζ(2(k -j)) = [[2 2k-1 , R 1 (n)], [-1, -n 4k ]] , with R 1 (x) = (x 2k (2x + 1) + (x -1) 2k (2x -3))/(2x -1) , k j=0 (-3) 3j (ζ(6(k -j) + 2) + 3ζ(6(k -j))) = [[-(-3) 3k+1 /2, R 2 (n)], [1, -n 12k+4 ]] , with R 2 (x) = (x 6k+2 (3x 2 + 3x + 1) + (x -1) 6k+2 (3x 2 -9x + 7))/(3x 2 -3x + 1) , k j=0 (-3) 3j (ζ(6(k -j) + 5) + 3ζ(6(k -j) + 3)) = [[(-3) 3k+2 /2, R 3 (n)], [1, -n 12k+10 ]] , with R 3 (x) = (x 6k+5 (3x 2 + 3x + 1) + (x -1) 6k+5 (3x 2 -9x + 7))/(3x 2 -3x + 1) , k j=0 (-4) j ζ(4(k -j) + 3) = [[4 k , R 4 (n)], [1, -n 8k+6 ]] , with R 4 (x) = (x 4k+3 (2x 2 + 2x + 1) + (x -1) 4k+3 (2x 2 -6x + 5))/(2x 2 -2x + 1) , k j=0 (ζ(6(k -j) + 5) -ζ(6(k -j) + 3)) = [[-1/2, R 5 (n)], [1, -n 12k+10 ]] , with R 5 (x) = (x 6k+5 (x 2 + x + 1) + (x -1) 6k+5 (x 3 -3x + 3))/(x 2 -x + 1) , k j=0 2 4j ζ * (2(k -j) + 1) = [[2 4k , R 6 (n)], [-2 2k , n 4k+2 ]] , with R 6 (x) = (x 2k+1 (2x + 1) -(x -1) 2k+1 (2x -3))/(2x -1) , k j=0 (-64) j ζ * (4(k -j) + 1) = [[(-1) k 2 6k-1 , R 7 (n)], [2 4k , n 8k+2 ]] , with R 7 (x) = (x 4k+1 (2x 2 + 2x + 1) -(x -1) 4k+1 (2x 2 -6x + 5))/(2x 2 -2x + 1) , k j=0 2 6j (ζ * (6(k -j) + 2) -4ζ * (6(k -j))) = [[2 6k , R 8 (n)], [2 6k+1 , n 12k+4 ]] , with R 8 (x) = (x 6k+2 (x 2 + x + 1) -(x -1) 6k+2 (x 2 -3x + 3))/(x 2 -x + 1) .
We now give corresponding examples:

ζ(4) + 4ζ(2) = [[8, 2n 4 -4n 3 + 10n 2 -8n + 3], [-1, -n 8 ]] , 27ζ(2) -3ζ(6) -ζ(8) = [[81/2, R(n)], [-1, -n 16 ]] , with R(n) = 2n 8 -8n 7 + 46n 6 -110n 5 + 178n 4 -182n 3 + 118n 2 -44n + 7 , ζ(5) + 3ζ(3) = [[9/2, 2n 5 -5n 4 + 22n 3 -28n 2 + 23n -7], [1, -n 10 ]] , 4ζ(3) -ζ(7) = [[4, 2n 7 -7n 6 + 37n 5 -75n 4 + 99n 3 -77n 2 + 31n -5], [-1, -n 14 ]] , ζ(3) -ζ(5) = [[1/2, 2n 5 -5n 4 + 22n 3 -28n 2 + 15n -3], [-1, -n 10 ]] , 4ζ(5) + 11ζ(3) = [[273/16, 2n 5 -5n 4 + 42n 3 -58n 2 + 45n -13], [4, -n 10 ]] , 3ζ(3) + 16 log(2) = [[16, 5n 2 -5n + 3], [-4, n 6 ]] , 15ζ(5) + 48ζ(3) + 256 log(2) = [[256, 7n 4 -14n 3 + 18n 2 -11n + 3], [-16, n 10 ]] , 64 log(2) -15ζ(5) = [[32, 9n 4 -18n 3 + 30n 2 -21n + 5], [-16, n 10 ]] , 127ζ(8) -124ζ(6) + 64ζ(2) = [[64, R(n)], [128, n 16 ]] , with R(n) = 12n 7 -42n 6 + 110n 5 -170n 4 + 154n 3 -82n 2 + 24n -3 .

Second Method: use of the ψ Function and Derivatives

Recall that ψ(z) is the logarithmic derivative of the gamma function (most authors call ψ the digamma function, and ψ ′ , ψ ′′ , etc... the trigamma, tetragamma functions, but this is terrible terminology). By orthogonality of characters, it is immediate to show that for k ≥ 1

ψ (k) (r/m) = (-1) k-1 k!m k+1 φ(m) χ mod m χ(r)L(χ, k + 1) ,
and for k = 0 the same formula is valid if we interpret L(χ 0 , 1) as

L(χ 0 , 1) = d|m µ(d) d log(d) - φ(m) log(m) m .
In particular, for m = 1, 2, 3, 4, and 6, which are the values of m for which φ(m) ≤ 2, we obtain the following table, where as usual we set G = L(χ -4 , 2), Catalan's constant, and

G 3 = L(χ -3 , 2): r/m ψ(r/m) + γ ψ ′ (r/m) ψ ′′ (r/m) 1 0 ζ(2) -2ζ(3) 1/2 -2 log(2) 3ζ(2) -14ζ(3) 1/3 -3 log(3)/2 -π/(2 √ 3) 4ζ(2) + 9G 3 /2 -26ζ(3) -4π 3 /(3 √ 3) 2/3 -3 log(3)/2 + π/(2 √ 3) 4ζ(2) -9G 3 /2 -26ζ(3) + 4π 3 /(3 √ 3) 1/4 -3 log(2) -π/2 6ζ(2) + 8G -56ζ(3) -2π 3 3/4 -3 log(2) + π/2 6ζ(2) -8G -56ζ(3) + 2π 3 1/6 -log(432)/2 -3π/(2 √ 3) 12ζ(2) + 45G 3 /2 -182ζ(3) -12π 3 / √ 3 5/6 -log(432)/2 + 3π/(2 √ 3) 12ζ(2) -45G 3 /2 -182ζ(3) + 12π 3 / √ 3
On the other hand, there exist many continued fractions for ψ(z) and its derivatives. The ones for ψ(z) itself are rather complicated, and those for ψ (k) (z) for k ≥ 3 are trivial transformations of the defining series, so the only remaining interesting ones are those for ψ ′ (z) and ψ ′′ (z). This of course implies that we restrict to rational periods of degree two and three.

We choose the nicest continued fractions, taken from [2]:

For ψ ′ (z) we have

ψ ′ (z) = [[0, (2z -1)(2n -1)], [2, n 4 ]] ,
valid for z > 1/2. However, from the trivial identity ψ ′ (z) = ψ ′ (z + 1) + 1/z 2 , we can deduce infinitely many other continued fractions:

ψ ′ (z) = [[ 0≤j<k 1/(z + j) 2 , (2z + 2k -1)(2n -1)], [2, n 4 ]] ,
now valid for z > 1/2 -k. Referring to the above table and choosing z = 1/3, 2/3, 1/4, 3/4, 1/6, and 5/6, and k = 0, 1, etc..., we obtain as many continued fractions as we like for 8ζ(2) ± 9G 3 , 3ζ(2) ± 4G, and 8ζ(2) ± 15G 3 . For instance, after simplifications:

8ζ(2) -9G 3 = [[0, 2n -1], [12, 9n 4 ]] , 8ζ(2) + 9G 3 = [[18, 10n -5], [12, 9n 4 ]] , 3ζ(2) -4G = [[0, 2n -1], [2, 4n 4 ]] , 3ζ(2) + 4G = [[8, 6n -3], [2, 4n 4 ]] , 8ζ(2) -15G 3 = [[0, 4n -2], [4, 9n 4 ]] , 8ζ(2) + 15G 3 = [[24, 8n -4], [4, 9n 4 ]] .
For ψ ′′ (z) we have

ψ ′′ (z) = [[0, (2n -1)(n 2 -n + 1 + 2z(z -1))], [-2, -n 6 ]] ,
valid for z > 1/2. However, from the trivial identity ψ ′′ (z) = ψ ′′ (z + 1) -2/z 3 , we can deduce infinitely many other continued fractions:

ψ ′′ (z) = [[-2 0≤j<k 1/(z+j) 3 , (2n-1)(n 2 -n+1+2(z+k)(z+k-1))], [-2, -n 6 ]] ,
now valid for z > 1/2 -k. Referring to the above table and choosing z = 1/3, 2/3, 1/4, 3/4, 1/6, and 5/6, and k = 0, 1, etc..., we obtain as many continued fractions as we like for 39ζ(3) ± 2π 3 / √ 3, 28ζ(3) ± π 3 , and 91ζ(3) ± 6π 3 / √ 3. For instance, after simplifications:

39ζ(3) -2π 3 / √ 3 = [[0, (2n -1)(9n 2 -9n + 5)], [27, -81n 6 ]] , 39ζ (3)
+ 2π 3 / √ 3 = [[81, (2n -1)(9n 2 -9n + 17)], [27, -81n 6 ]] , 28ζ (3)
-π 3 = [[0, (2n -1)(8n 2 -8n + 5)], [8, -64n 6 ]] , 28ζ (3)
+ π 3 = [[64, (2n -1)(8n 2 -8n + 13)], [8, -64n 6 ]] , 91ζ(3) -6π 3 / √ 3 = [[0, (2n -1)(9n 2 -9n + 13/2)], [9, -81n 6 ]] , 91ζ (3)
+ 6π 3 / √ 3 = [[216, (2n -1)(9n 2 -9n + 25/2)], [9, -81n 6 ]] .

Third Method: Bauer-Muir Acceleration

This very classical method is just as elementary as the previous ones, but the formulas are slightly more complicated. Let (a(n), b(n)) n≥0 be a continued fraction with convergents (p(n), q(n)), and let r(n) n≥1 be any sequence (for now). For n ≥ 1 we define

R(n) = a(n)+r(n) and d(n) = r(n)R(n+1)-b(n) = r(n)(a(n+1)+b(n+1))-b(n) .
We make the following two essential assumptions: R(1) = a(1) + r(1) = 0, and d(n) = 0 for all n ≥ 1.

We define:

A(0) = a(0) + b(0) R(1) , B(0) = b(0)d(1) R(1) 2 , A(1) = a(1)R(2) + b(1) R(1) , A(n) = R(n + 1) -r(n -1) d(n) d(n -1)
for n ≥ 2, and

B(n) = b(n) d(n + 1) d(n) for n ≥ 1.
The following result is easy to prove by induction:

Proposition 5.1 Let (P (n), Q(n)) be the convergents of the continued fraction defined by (A(n), B(n)). For n ≥ 2 we have

(P (n), Q(n)) = (p(n + 1), q(n + 1)) + r(n + 1)(p(n), q(n)) .
In particular, if p(n)/q(n) and P (n)/Q(n) both tend to a limit as n → ∞, these limits are equal.

This process is called Bauer-Muir acceleration, because if r(n) is chosen appropriately, it accelerates the convergence of the continued fraction. An important fact is that if the accelerated formulas are simple enough, for instance when d(n) is constant, the acceleration process can be iterated. This fact, combined with a suitable diagonal process, is the basis of Apéry's initial proofs of the irrationality of ζ(2) and ζ(3). However, we will not consider this here.

Let us consider some simple examples.

Example 1: log(2) The trivial continued fraction for log(2), directly coming from the series log

(2) = n≥1 (-1) n-1 /n, is log(2) = [[0, 1], [1, n 2 ]].
Applying Bauer-Muir acceleration iteratively, we immediately obtain

log(2) = [[0, 1], [1, n 2 ]] = [[1, 3], [-1, n 2 ]] = [[1/2, 5], [1, n 2 ]] = [[5/6, 7], [-1, n 2 ]] ,
and so on, the general formula being

log(2) = [[ 1≤j≤k (-1) j-1 /j, 2k + 1], [(-1) k , n 2 ]] .
Note that this is not the same as the trivial continued fraction obtained from n>k (-1) n-1 /n, since this converges like (-1) n /n as the initial one, while the accelerated formula given above converges like (-1) n /n 2k+1 .

Example 2: π 2 /6 The trivial continued fraction for ζ(2) = π 2 /6, directly coming from the series and so on, the general formula being

π 2 /6 = n≥1 1/n 2 , is π 2 /6 = [[0, 2n 2 -2n + 1], [1, -n 4 ]].
π 2 /6 = [[2 1≤j≤k (-1) j-1 /j 2 , 2n 2 -2n + k 2 + k + 1], [(-1) k , -n 4 ]] .
Once again, this is not the tail of the series defining π 2 /6, since it converges like 1/n 2k+1 .

Example 3: π 2 /6 (again) Another trivial continued fraction for π 2 /6, directly coming from the series

π 2 /6 = 2 n≥1 (-1) n-1 /n 2 , is π 2 /6 = [[0, 2n-1], [2, n 4 ]].
Applying Bauer-Muir acceleration iteratively, we immediately obtain

π 2 /6 = [[0, 2n -1], [2, n 4 ]] = [[1, 6n -3], [2, n 4 ]] = [[5/4, 10n -5], [2, n 4 ]] = [[49/36, 14n -7], [2, n 4 ]] = [[205/144, 18n -9], [2, n 4 ]] = [[5269/3600, 22n -11], [2, n 4 ]] ,
and so on, the general formula being

π 2 /6 = [[ 1≤j≤k 1/j 2 , (2k + 1)(2n -1)], [2, n 4 ]] .
Note that the constant term of the continued fraction for j≥1 1/j 2 is the partial sum of the series j≥1 (-1) j-1 /j 2 and the constant term of the continued fraction for j≥1 (-1) j-1 /j 2 is the partial sum of the series j≥1 1/j 2 .

Example 4: G = L(χ -4 , 2), Catalan's constant Here, we could take the trivial continued fraction for G, directly coming from the series G = n≥1 (-1) n-1 /(2n-1) 2 , and apply iteratively Bauer-Muir, giving and so on, but this is not pretty, first because b(0) changes, and second because one needs to specify both a(0) and a(1) since n -1 vanishes for n = 1.

G = [[0, 1, 8(n -1)], [1, (2n -1) 4 ]] = [[1/6,
A nicer continued fraction taken from [2], which in fact we are going to prove, is

G = [[1, 8n 2 -8n + 7], [-1/2, -16n 4 ]]. We thus obtain G = [[0, 8n 2 -8n + 3], [1/2, -16n 4 ]] = [[1, 8n 2 -8n + 7], [-1/2, -16n 4 ]] = [[8/9, 8n 2 -8n + 19], [1/2, -16n 4 ]] = [[209/225, 8n 2 -8n + 39], [-1/2, -16n 4 ]] = [[10016/11025, 8n 2 -8n + 67], [1/2, -16n 4 ]] ,
and so on, the general formula being

G = [[ 1≤j≤k (-1) j-1 /(2j -1) 2 , 8n 2 -8n + 4k 2 + 3], [(-1) k /2, -16n 4 ]] ,
and the continued fraction for k = 0 being obtained by reverse Bauer-Muir, with extremely slow convergence in 1/ log(n).

But in turn these formulas prove that the initial continued fraction converges to G, since if we set S k = 1≤j≤k (-1) j-1 /(2j -1) 2 , which is the kth partial sum of the series defining G, the kth continued fraction is S k + (-1) k /2/(4k 2 + 3 -16/(4k 2 + 19 -• • • )), and this clearly tends to lim k→∞ S k = G as k → ∞. The reader can check that unfortunately, the method does not work (i.e., the formulas become extremely complicated) for the alternating sum giving 3ζ(3)/4, nor for ζ(k) for k ≥ 4, explaining in large part why Apéry's method has not been extended to ζ(k) for k ≥ 5. Note that, on the contrary, Apéry's method does work for a large number of other series, and will be the object of a future paper.

Conclusion

We have given three rather different methods to obtain infinitely many continued fractions for certain linear combinations of zeta and L values. Note, however, that they are all polynomially convergent (i.e., in C/n k for some k ≥ 1), while really interesting continued fractions are exponentially or at least subexponentially convergent. As already mentioned, this will be the subject of a future paper [1].

  .. When a(n) and b(n) are polynomials A(n) and B(n) for n ≥ 1, we will write the continued fraction as S = [[a(0), A(n)], [b(0), B(n)]]. For instance, the continued fraction ζ(2) = 2/(1 + 1/(3 + 16/(5 + 81/(7 + 256/(9 + • • • ))))) will simply be written as ζ(2) = [[0, 2n -1], [2, n 4 ]].

  ApplyingBauer-Muir acceleration iteratively, we immediately obtainπ 2 /6 = [[0, 2n 2 -2n + 1], [1, -n 4 ]] = [[2, 2n 2 -2n + 3], [-1, -n 4 ]] = [[3/2, 2n 2 -2n + 7], [1, -n 4 ]] = [[31/18, 2n 2 -2n + 13], [-1, -n 4 ]] = [[115/72, 2n 2 -2n + 21], [1, -n 4 ]] = [[3019/1800, 2n 2 -2n + 31], [-1, -n 4 ]] = [[973/600, 2n 2 -2n + 43], [1, -n 4 ]] ,

Example 5: ζ( 3 )

 3 The trivial continued fraction for ζ(3), directly coming from the seriesζ(3) = n≥1 1/n 3 , is ζ(3) = [[0, (2n -1)(n 2 -n + 1)], [1, -n 6 ]].Applying Bauer-Muir acceleration iteratively, we immediately obtainζ(3) = [[0, (2n -1)(n 2 -n + 1)], [1, -n 6 ]] = [[1, (2n -1)(n 2 -n + 5)], [1, -n 6 ]] = [[9/8, (2n -1)(n 2 -n + 13)], [1, -n 6 ]] = [[251/216, (2n -1)(n 2 -n + 25)], [1, -n 6 ]] ,and so on, the general formula beingζ(3) = [[ 1≤j≤k 1/j 3 , (2n -1)(n 2 -n + 2k 2 + 2k + 1)], [1, -n 6 ]] .


Scalla

Christian Mougin

Ferdinand Cabanne

René Scalla

Additional observations on the chlorotoluron hydroxylase and N-demethylase activities in wheat microsomes

Keywords: Herbicide metabolism, phenylureas, cell suspension cultures, 2, 4-D, chlorotoluron ring-methyl hydroxylase ; 2, 4-D

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Results are consistent with a cytochrome P-450-dependent oxidation of that herbicide in microsomes of wheat cells, which also metabolized diuron and isoproturon.

INTRODUCTION

Although studies of herbicide metabolism have shown that numerous herbicides undergo oxidation in plants, our knowledge of the enzymatic systems involved in these reactions is rather poor.

Frear et al. (1969) published the first results

showing the occurrence of a cytochrome P-450 system in the N-demethylation of monuron in cotton (Gossypium hirsutum). Several years later, [START_REF] Makeev | Microsomal hydroxylation of 2,4-D in plants[END_REF] showed that a similar enzymatic system was responsible for the hydroxylation of 2,4-dichlorophenoxyacetic acid

(2,4-D) in cucumber (Cucumis sp.) leaves.

More recently, the implication of cytochrome P-450 systems in the oxidation of herbicides was established in Jerusalem artichoke (Helianthus tuberosus) (chlorotoluron : Fonné, 1985), corn (Zea mays)(chlorotoluron : Fonné-Pfister and Kreuz, 1990 ;primisulfuron : Fonné-Pfister et al., 1990 ;bentazon : McFadden et al., 1990), and in sorghum (Sorghum bicolor)(metolachlor : Moreland et al ., 1990).

Microsomal fractions from wheat (Triticum aestivum) also catalyzed the oxidation of diclofop [START_REF] Mcfadden | Aryl hydroxylation of diclofop by a cytochrome P-450 dependent monooxygenase from wheat[END_REF]Zimmerlin and Durst, 1990 ;Frear et al., 1991) and other herbicides such as triasulfuron, chlorsulfuron and linuron (Frear et al., 1991).

We previously showed that the oxidative metabolism of chlorotoluron in wheat was catalyzed by cytochrome P-450 enzymes (Mougin et al., 1990(Mougin et al., , 1991)). Microsomes CPUH and CPUDM activities (Mougin et al., 1990).

Conversely, inhibitors of P-450 decreased the CPUH and CPUDM activities when added to microsomal incubation media (Mougin et al., 1991).

In this paper, the postulated involvement of Plant material. Seeds of wheat (Triticum aestivum L., var "Etoile de Choisy") were allowed to germinate on wet paper for 5 days at 25°C in the dark. Etiolated shoots were then excised from the caryopses and stored for 1 h on ice prior to the isolation of microsomes.

Cell cultures and treatments. Achlorophyllous wheat cells (Triticum aestivum L., var. "Koga II") were routinely cultivated as previously described [START_REF] Canivenc | Induced changes of chlorotoluron metabolism in wheat cell suspension cultures[END_REF]. Treatment of cultured cells with 300 µM 2,4-D, in order to stimulate the rates of monooxygenase activity and to enhance the levels of cytochrome P-450 have also been described (Mougin et al., 1991).

Microsomal fractions were isolated 48 h following the beginning of 2,4-D treatment. Cell suspensions were kept for 1 h at 4°C, and then filtered through a stainless steel filter prior to the isolation of microsomes.

Preparation of microsomes. Microsomes were prepared from wheat shoots (30-40 g fresh weight) or cells (55-65 g fresh weight) according to Mougin et al. (1991). Incubation media consisted of 0.1 M phosphate buffer (pH 7.5), 15% (w/v) glycerol, 0.7 mM mercaptoethanol and 1 mM glutathione. Assays were stopped by addition of 1 ml n-hexane and sonicated. Substrates and metabolites of enzymatic reactions were successively extracted with 2 x 1 ml n-hexane, 2 x 1 ml diethylether/n-hexane (1/1 , v/v) and 1 ml pure diethyl-ether.

Enzymatic

Fractions were reduced under vacuum, pooled and spotted on thin layer chromatography silica gel 60F 254 plates (Merck). Chlorotoluron [START_REF] Canivenc | Induced changes of chlorotoluron metabolism in wheat cell suspension cultures[END_REF], diuron (Swanson and Swanson, 1968) and isoproturon [START_REF] Cabanne | Effect of the cytochrome P-450 inactivator 1-aminobenzotriazole on the metabolism of chlorotoluron and isoproturon in wheat[END_REF] extracts were analyzed as previously described.

Peroxidative activities. Assay media for peroxidative reactions contained 15% (w/v) glycerol and 0.3 mg microsomal proteins in a final 1 ml volume of 0.1 M phosphate buffer (pH 7.5). Peroxygenase activity (Ishimaru and Yamasaki, 1977 Peroxidase activity was determined using guaiacol as a substrate by the method of [START_REF] Maehly | The assay of catalases and peroxidases[END_REF].

Changes in absorbance were followed at 436 nm in the presence of 2 mM H 2 O 2 and 2 mM guaiacol. An absorbance coefficient of 25500 M -1 cm -1 was used. Hydrogen peroxide was omitted in controls.

NADPH-cytochrome c reductase activity. NADPH-cytochrome c reductase activity was measured after addition of 2 mM KCN to the reaction mixture to minimize cytochrome c reoxidation by mitochondria, according to the method of [START_REF] Benveniste | Purification and characterization of the NADPH-cytochrome P-450 (cytochrome c) reductase from higher-plant microsomal fraction[END_REF]. An extinction coefficient of 18500 M -1 cm -1 at 550 nm was used for activity determination.

Other methods. Effects of antibodies were measured by incubating microsomes with the IgG fraction of sera of rabbits immunized against several enzymes purified from Jerusalem artichoke tubers : NADPH-cytochrome P-450 reductase [START_REF] Benveniste | Purification and characterization of the NADPH-cytochrome P-450 (cytochrome c) reductase from higher-plant microsomal fraction[END_REF]), CA4H (Gabriac et al., 1991), and 7-ethoxycoumarin O-deethylase (ECOD) (Werck-Reichhart et al., 1990). Quantitative determination of hemoproteins was carried out using the method of Omura and Sato (1964). Type I and II spectra were obtained with fractions that had been enriched in endoplasmic reticulum by the following method :

microsomes were layed onto a sucrose cushion (30% , w/v) and centrifuged for 90 min at 80000xg (Beckman SW 27 rotor). The fraction lying on the cushion was used for interaction spectra. For all recordings of difference spectra, the protein concentration was 1 mg ml -1 . An ethanolic solution of chlorotoluron was added to a solution of bovin serum albumin (1 mg ml -1 ), the solvent was evaporated under vaccum, and the resulting solution of bovin serum albumin plus chlorotoluron was used for spectral measurements. Final concentrations of herbicide were 100 and 160 µM. Procloraz, dissolved in dimethyl sulfoxide (DMSO)15%, was administered to microsomes in the range from 100 to 400 µM. Microsomal proteins were estimated according to [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF] with bovine serum albumin as a standard. Effectors were added to the incubation media dissolved in phosphate buffer or in DMSO. In the latter case, final solvent concentration was 1% (v/v) and did not affect enzymatic activities. Controls were supplemented with the same volumes of solvent. Results are expressed as means of independent experiments (n = 2 to 4) ± SD. Enzymatic activities and cytochrome P-450 levels are expressed in pmol min -1 mg -1 and pmol mg -1 of microsomal proteins, respectively. The fungicide procloraz, an inhibitor of sterol biosynthesis, combines with P-450 from bird and rat livers to give a type II spectrum (Rivière, 1983). It was found to inhibit chlorotoluron oxidation by wheat microsomes when applied at 100 µM (Mougin et al., 1991). In the same concentration range (from 100 to 400 µM), procloraz induced the formation of type II spectra with a minimum at 390-400 nm and a peak at 429 nm (fig.

RESULTS

Spectral

C).

Ability of microsomes isolated from wheat shoots to metabolize chlorotoluron

Microsomes isolated from wheat cell cultures exibited CPUH and CPUDM activities which were enhanced following subculturing of cells in the presence of 400 µM cyometrinil or 300 µM 2,4-D (Mougin et al., 1990(Mougin et al., , 1991)). The activities were not restricted to whole cells, microsomes isolated from wheat shoots were also to stimulate monooxygenase activities (McFadden et al., 1990 ;Mougin et al., 1991 ;Frear et al., 1991).

However, none of these treatments stimulated the microsomal metabolism of chlorotoluron. Consequently, all the following experiments were done with microsomes isolated from 2,4-D treated cell cultures.

Effects of reduced pyridine nucleotides and cofactor analogues on chlorotoluron ring-methyl hydroxylase and

N-demethylase activities

CPUH and CPUDM activities were measured in the presence of cofactors applied under various conditions. [START_REF] Duffel | Changes in dimethylaniline N-oxidase activity in P-450 : the cinnamic acid 4-hydroxylase[END_REF].

It strongly lowered the hydroxylase and N-demethylase activities. Methimazole, an inhibitor of flavin-containing monooxygenases (Sabourin and Hodgson, 1984) had no effect on the tested activities, as already reported concerning the chlorotoluron N-demethylase of microsomes isolated from Jerusalem artichoke tubers (Fonné, 1985).

Some inhibitors of lipid peroxidation and NAD(P)H oxidases were also assayed in order to determine whether detrimental reactions reduced the P-450 activities.

Methylindole, an inhibitor of lipid peroxidation [START_REF] Adams | -3-Methylindole inhibits lipid peroxidation[END_REF] very slightly stimulated the enzymatic activities at 100 µM (tab. 3).

Superoxide dismutase (SOD) stimulated both activities, when added alone or with catalase. In the latter case, the effects corresponded to additional effects of the two compounds applied alone. Antibodies raised against ECOD from Jerusalem artichoke did not inhibit activity of wheat enzymes.

Effects of antibodies on cinnamic

Effects of compounds modifying hydroperoxide-dependent activities

The chlorotoluron N-demethylase was weakly sensitive to carbon monoxide and aminobenzotriazole, but showed a cumene hydroperoxide-dependent activity (Mougin et al., 1990). That could suggest a possible participation of peroxidases or peroxygenases in the N-demethylation of the herbicide. Activities of two hydroperoxide-dependent enzymes, namely aniline oxidase (a peroxygenase) and guaiacol oxidase (a peroxidase)

were assayed using microsomes isolated from 2,4-D treated cells.

In addition, the sensitivity to mercaptoethanol of the cumene hydroperoxide-dependent CPUDM activity was evaluated because this compound is known to inhibit the peroxidative enzymes [START_REF] Blée | Hydroperoxide-dependent sulfoxidation catalyzed by soybean microsomes[END_REF][START_REF] Gullner | Hydrogen peroxide dependent N-demethylase activity in the leaves of normal and heat-shocked bean plants[END_REF].

No activities of the peroxygenase and peroxidase-type enzymes were detected in the absence of hydroperoxyde(s), except a basal CPUDM activity (tab. Also, the identity of the physiological substrates of these enzymes is still an enigma. A study of competition between chlorotoluron and an exogenous substrate or a known physiological substrate of P-450

5). However

was carried out to shed some light on these problems.

Effects of some physiological or exogenous substrates of cytochrome P-450 systems are shown in tab. 6.

Concentrations equivalent to 10 times the apparent Km were chosen for physiological substrates of P-450 cytochromes of plants. Exogenous substrates were applied at dose equivalent, in most cases, to their apparent Km values found in recent papers.

Results indicated that the addition of 300 µM cinnamic acid in the incubation medium containing 500 µM chlorotoluron slightly stimulated CPUH and CPUDM activities. By contrast, lauric acid, ferulic acid, and the exogenous substrates had some inhibiting effects toward the CPUDM activity, except the benzo[a]pyrene.

Ability of microsomes isolated from wheat cell cultures to oxidize diuron and isoproturon

Diuron and isoproturon are structural analogues of chlorotoluron. Diuron is N-dealkylated in wheat plants (Ryan et al., 1981), and isoproturon undergoes a ring-isopropyl hydroxylation and a N-dealkylation in those plants [START_REF] Cabanne | Effect of the cytochrome P-450 inactivator 1-aminobenzotriazole on the metabolism of chlorotoluron and isoproturon in wheat[END_REF]. When added to microsomes in the presence of chlorotoluron and at the same concentration as the herbicide, diuron strongly lowered both CPUH and CPUDM activities by about 70% inhibition, whereas isoproturon had no effect (Mougin et al., 1991).

We tested the in vitro oxidation of these two herbicides by microsomes isolated from 2,4-D treated cells. Microsomes were able to metabolize diuron (fig.

3 A) and isoproturon (fig. 3 B) probably to N-dealkylated metabolites, in the presence of NADPH.

The rates of reactions were in the same range for the two herbicides, corresponding to 3-5 pmol min -1 mg -1 of each metabolite. These rates were lower than those obtained for the ring-methyl hydroxylation and N-demethylation of chlorotoluron. No transformation could be detected in the absence of NADPH cofactor.

DISCUSSION

Wheat plants actively metabolize the herbicide chlorotoluron in vivo [START_REF] Gross | The behavior of chlorotoluron in wheat and soil[END_REF][START_REF] Cabanne | Effect of the cytochrome P-450 inactivator 1-aminobenzotriazole on the metabolism of chlorotoluron and isoproturon in wheat[END_REF]. The plant enzymatic systems metabolizing chlorotoluron are constitutive, but their activities need to be stimulated for in vitro studies. Using wheat cell cultures, [START_REF] Canivenc | Induced changes of chlorotoluron metabolism in wheat cell suspension cultures[END_REF] et al., 1990, 1991). --------------------------------------------------------------Relative activities (%) Incubation conditions -----------------------------CPUH CPUDM - ------------------------------------------------------------ - + NADP + 0.5 mM 90.9 ± 7.9 102.5 ± 6.0 + NADP + 2.0 mM 68.0 ± 5.5 69.0 ± 3.6 --------------------------------------------------------------Activities are expressed as % of control values : a 42.5 and b 55.8 pmol min -1 mg -1 .

------------------------------------------------------------- Relative activities (%) Incubation conditions ----------------------------- CPUH CPUDM -------------------------------------------------------------- NADPH 0.

Table 3. Effects of various compounds on NADPH-dependent chlorotoluron ring-methyl hydroxylase (CPUH) and N-demethylase

(CPUDM) activities.

- ------------------------------------------------------------ - --------------------------------------CA4H CPUH CPUDM ------------------------------------------------------------- --------------------------------------------------------------Enzymatic activities Incubation conditions ---------------------------------------CPUDM aniline ox. guaiacol ox. (pmol min -1 mg -1 ) (nmol min -1 mg -1 ) -------------------------------------------------------------without cofactor 3.8 ± 0.4 0.0 0.0 with cofactor 46.1 ± 1.2 8.0 ± 0.8 1164.2 ± 31.0 +MSH 1.4 mM 45.4 ± 1.5 0.0 18.5 ± 2.2 --------------------------------------------------------------Table 6. Effects of various physiological and exogenous substrates of cytochrome P-450 systems on chlorotoluron ringmethyl hydroxylase (CPUH) and N-demethylase (CPUDM) activities.

------------------------------------------------------------- Relative activities (%) Incubation conditions ---------------------------- CPUH CPUDM -
.7 ± 1.4 -------------------------------------------------------------- Activities
------------------------------------------------------------- Relative activities (%) Incubation conditions -
--------------------------------------------------------------Relative activities (%) Incubation conditions -----------------------------CPUH CPUDM -------------------------------------------------------------- Incubations with diuron (d) and isoproturon (i) produced polar metabolites. Herbicide concentration in assay media were 500 µM. Incubations were run for 90 min with 0.5 mg microsomal protein, under aerobic conditions and in the presence of NADPH.

Summary.

  The ring-methyl hydroxylation and N-demethylation of chlorotoluron were studied in microsomes of wheat (Triticum aestivum) in both materials, but exhibited higher rates in cells than in plants. Treatement of cells with 300 µM 2,4-dichlorophénoxyacetic acid for 48 h gave mean rates of 40 and 55 pmol min -1 mg -1 for the ring-methyl hydroxylation and the N-demethylation, respectively. The reactions required NADPH as cofactor and aerobic conditions. Chlorotoluron ring-methyl hydroxylase and N-demethylase as well as cinnamic acid 4-hydroxylase were drastically inhibited by polyclonal antibodies raised against NADPH cytochrome c reductase.Polyclonal antibodies raised against cinnamic acid 4-hydroxylase and 7-ethoxycoumarin O-deethylase did , respectively. Among tested effectors, those known as inhibitors of cytochrome P-450 enzymes were the most efficient on ring-methyl hydroxylation and N-demethylation of chlorotoluron.

  D or cyometrinil, applied to cells 2 or 3 days before isolation of microsomes, were able to increase in vitro (a) rates of cinnamic acid 4-hydroxylase (CA4H) and lauric acid hydroxylase, two cytochrome P-450 dependent monooxygenases of higher plants, (b) levels of cytochrome P-450 contents, and (c) rates of

  Several difference spectra of wheat microsomes were obtained (fig.1). Reduced carbon monoxide spectra showed that microsomes isolated from wheat cells, grown for 48 h in the presence of 300 µM 2,4-D, contained about 250 pmol cytochrome P-450 per mg microsomal proteins. Low levels of P-420 were present (fig.1 A).The substrate binding spectrum of type I is one cytochrome P-450 in an enzymatic reaction. A spectrum of this type was obtained by incubating wheat cell microsomes with chlorotoluron according to the way indicated in Materials and Methods(fig.1 B). The spectra showed a peak at 389 nm and a minimum near 410 nm for 100 and 160 µM herbicide, cytochrome P-450 systems.

  able to hydroxylate and to N-demethylate chlorotoluron in the presence of NADPH (fig. 2 A). However, CPUH and CPUDM activities of wheat shoot microsomes were about 6-fold lower than those of microsomes of wheat cells treated with 2,4-D (fig. 2 B), amounting to 17.0 and 35.0 pmol min -1 mg -1 in the same conditions of incubation. No peaks corresponding to metabolites could be detected on radio-chromatograms following incubations in the absence of NADPH or microsomal proteins. In order to stimulate the CPUH and CPUDM activities of the plant microsomal preparations, we treated plants with agrochemicals such as herbicide safeners cyometrinil and naphthalic anhydride, reported

  and CPUDM activities of wheat cell microsomes were evaluated in the presence of polyclonal antibodies raised against components of P-450s of higher plants (tab. 4). The three activities were drastically reduced after 10 min incubation with antibodies raised against purified NADPH-cytochrome c (P-450) reductase from Jerusalem artichoke tubers. activity was inhibited by 75% (not shown). Wheat CA4H was drastically inactivated by antibodies raised against Jerusalem artichoke CA4H, whereas CPUH and CPUDM activities were unaffected.

  preferred electron donor for chlorotoluron oxidation, although NADH was able to support appreciable reaction rates. The involvement of NADPH cytochrome P-450 reductase was also evidenced by the effects of structural analogues of NADPH that compete with the coenzyme at the level of the flavoprotein(Salaun et al., 1980). Interestingly, antibodies raised against NADPH cytochrome c reductase from Jerusalem artichoke drastically inhibited both CPUH and CPUDM activities of wheat cells. Because antibodies raised against CA4H did not interact with enzymes responsible for chlorotoluron metabolism, the cytochrome P-450 forms responsible for the CA4H, CPUH and CPUDM activities are probably different.Type I spectra of P-450 were obtained with wheat cell microsomes in the presence of chlorotoluron, demonstrating the binding of the herbicide in a catalytic site of P-450. Carbon monoxide, which is the most classical inhibitor of P-450, inhibits both CPUH and CPUDM, although this latter enzyme seems to be less sensitive. Procloraz induced a type II binding spectra when added to microsomes, suggesting the binding of the fungicide to the heme iron of P-450, in agreement with the inhibition of chlorotoluron oxygenation already described(Mougin et al., 1991). The cytochrome Peffects of o-phenanthroline and noctylamine on both CPUH and CPUDM activities.No strong interactions were observed between CPUH and CPUDM activities and physiological or exogenous substrates of P-450. Again, chlorotoluron metabolism is probably not mediated in wheat by CA4H, laurate in-chain hydroxylase, or ferulic acid 5-hydroxylase. enzymes showed that the properties of the chlorotoluron N-demethylase are distinct from those of these enzymes.Taken together, these new results agree with those previously obtained in vivo and in vitro, and are consistent with the hypothesis that cytochrome P-450 enzymes catalyze the ring-methyl hydroxylation and the N-demethylation of chlorotoluron in microsomal preparations isolated from wheat cell cultures.The phenylureas diuron and isoproturon were also found to be oxidized by microsomes isolated from wheat cells. This result is worth comparing with previous data showing that diuron acted as a good competitor of chlorotoluron metabolism in vitro, whereas isoproturon had no effect(Mougin et al., 1991). Identities of the enzymes involved in these oxidations need to be confirmed by additional experiments.ACKNOWLEDGMENTS. The authors thankDr. P. Gaillardon (I.N.R.A., Dijon, France) for the generous gift of labelled diuron and isoproturon. We are also indebted to Drs I. Benveniste and D. Werk-Reichhart (C.N.R.S., Strasbourg, France) for providing polyclonal antibodies raised against NADPH-cytochrome c reductase, CA4H and ECOD activities from Jerusalem artichoke. mouse liver and kidney induced by steroid sex hormone. Mol. Pharmacol., 19, 134-139. Fonné R., 1985. -Intervention du cytochrome P-450 des végétaux supérieurs dans l'oxydation des composés exogènes : l'aminopyrine et le chlortoluron. R. and Kreuz K., 1990. -Ring-methyl hydroxylation of chlorotoluron by an inducible cytochrome P450-dependent enzyme from maize. Phytochemistry, 29, 2793-2796.Fonné-Pfister R., Gaudin J., Kreuz K., Ramsteiner K. : isolation and characterization of a microsomal mixed function oxidase from cotton.

  -----------------------------------------------------------Activities are expressed as % of maximal activities : a 34.0 and b 42.5 pmol min -1 mg -1 . Incubations were conducted for 25 min in the presence of NAD(P)H regenerating system.

  are expressed as % of control activities : a 40.2, b 51.2, c 37.2 and d 42.6 pmol min -1 mg -1 .

  -----------------------------------------------------------Activities are expressed as % of control activities : a 5397, b 46.3, c 92.6, d 4719, e 49.2 and f 64.8 pmol min -1 mg -1 .

Figure 1 .

 1 Figure 1. Difference spectra of wheat cell microsomes. A, reduced carbon monoxide spectrum ; B, chlorotoluronenzyme type I binding spectrum. Herbicide concentrations were 0, 100 and 160 µM ; C, procloraz-enzyme type II binding spectrum. Final concentrations of fungicide ranged from 100 to 400 µM.

Figure 2 .

 2 Figure 2. Metabolism of chlorotoluron by microsomes isolated from wheat shoots (A) and cultured cells (B). Radioactive peaks on thin layer chromatography elution profiles co-chromatographied with hydroxymethyl-chlorotoluron (a), N-monodemethylated-chlorotoluron (b) and chlorotoluron (c). Chlorotoluron concentration in assay media were 500 µM. Incubations were run for 90 min with 0.5 mg microsomal protein, under aerobic conditions and in the presence of NADPH.

Figure 3 .

 3 Figure 3. Thin layer chromatography elution profiles of 90 min incubations with microsomes isolated from wheat cells in the presence of diuron (A) and isoproturon (B).

Table 2 .

 2

	Effects	of	pyridine	nucleotide	analogues	on
	chlorotoluron ring-methyl hydroxylase (CPUH) and N-demethylase
	(CPUDM) activities. Activities were assayed during 15 min runs
	in the absence of a cofactor regenerating system.

Table 4 .

 4 Effects of polyclonal antibodies on cinnamic acid 4-

	hydroxylase (CA4H), chlorotoluron ring-methyl hydroxylase
	(CPUH) and N-demethylase (CPUDM) activities. Microsomes and
	antibodies were pre-incubated for 10 min at 25°C. Enzymatic
	assays were started by adding the other components to the
	reaction media, and incubated for 25 min. Controls were
	supplemented with pre-immune sera.

Table 5 .

 5 Effects of various compounds on (hydro)peroxide-

	dependent activities. Cofactors were cumene hydroperoxide for
	chlorotoluron	N-demethylase	(CPUDM)	and	aniline	oxidase
	activities, and hydrogen peroxide for guaiacol oxidase. MSH :
	mercaptoethanol.