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Let X be an n-dimensional, finite, simply connected CWcomplex and setWhen 0 < αX < ∞, we give upper and lower bounds for for k sufficiently large. We also show for any r that αX can be estimated from the integers rk πi (X), i ≤ nr with an error bound depending explicitly on r.

We report for the first time steady-state and time-resolved emission properties of photosystem I (PSI) complexes isolated from the cyanobacterial strain Synechococcus WH 7803. The PSI complexes from this strain display an extremely small fluorescence emission yield at 77 K, which we attribute to the absence of so-called red antenna chlorophylls, chlorophylls with absorption maxima at wavelengths longer than those of the primary electron donor P700. Emission measurements at room temperature with picosecond time resolution resulted in two main decay components with lifetimes of about 7.5 and 18 ps and spectra peaking at about 685 nm. Especially in the red flanks, these spectra show consistent differences, which means that earlier proposed models for the primary charge separation reactions based on ultrafast (∼1 ps) excitation equilibration processes cannot describe the data. We show target analyses of a number of alternative models and conclude that a simple model (Ant2)* ↔ (Ant1/RC)* → RP2 can explain the time-resolved emission data very well. In this model, (Ant2)* represents chlorophylls that spectrally equilibrate in about 7.5 ps and in which RP2 represents the "final" radical pair P700(+)A0(-). Adding an equilibrium (Ant1/RC)* ↔ RP1, in which RP1 represents an "intermediate" radical pair A(+)A0(-), resulted in the same fit quality. We show that the simple model without RP1 can easily be extended to PSI complexes from cyanobacteria with one or more pools of red antenna chlorophylls and also that the model provides a straightforward explanation of steady-state emission properties observed at cryogenic temperatures.

MSC : 55P35; 54N45; 18E30; 55U35; 57T30; International audience; The closed geodesic problem has been solved by many authors under additional hypothesis. In this paper we develop a new way to solve this problem, by defining finitely many obstruction classes. This permits to enlarge significantly the family of manifolds for which this problem is solved. Our method which relies on the concept of A ∞ -section of a homomorphism of differential graded algebras, shows explicitly how the natural structure of shc-algebra on the singular cochains of a space comes into play.

International audience; Most cyanobacteria harvest light with large antenna complexes called phycobilisomes. The diversity of their constituting phycobiliproteins contributes to optimize the photosynthetic capacity of these microorganisms. Phycobiliprotein biosynthesis, which involves several post-translational modifications including covalent attachment of the linear tetrapyrrole chromophores (phycobilins) to apoproteins, begins to be well understood. However, the biosynthetic pathway to the blue-green-absorbing phycourobilin (lambda(max) approximately 495 nm) remained unknown, although it is the major phycobilin of cyanobacteria living in oceanic areas where blue light penetrates deeply into the water column. We describe a unique trichromatic phycocyanin, R-PC V, extracted from phycobilisomes of Synechococcus sp. strain WH8102. It is evolutionarily remarkable as the only chromoprotein known so far that absorbs the whole wavelength range between 450 and 650 nm. R-PC V carries a phycourobilin chromophore on its alpha-subunit, and this can be considered an extreme case of adaptation to blue-green light. We also discovered the enzyme, RpcG, responsible for its biosynthesis. This monomeric enzyme catalyzes binding of the green-absorbing phycoerythrobilin at cysteine 84 with concomitant isomerization to phycourobilin. This reaction is analogous to formation of the orange-absorbing phycoviolobilin from the red-absorbing phycocyanobilin that is catalyzed by the lyase-isomerase PecE/F in some freshwater cyanobacteria. The fusion protein, RpcG, and the heterodimeric PecE/F are mutually interchangeable in a heterologous expression system in Escherichia coli. The novel R-PC V likely optimizes rod-core energy transfer in phycobilisomes and thereby adaptation of a major phytoplankton group to the blue-green light prevailing in oceanic waters.

The architecture of the entire photosynthetic membrane network determines, at the supramolecular level, the physiological roles of the photosynthetic protein complexes involved. So far, a precise picture of the native configuration of red algal thylakoids is still lacking. In this work, we investigated the supramolecular architectures of phycobilisomes (PBsomes) and native thylakoid membranes from the unicellular red alga Porphyridium cruentum using atomic force microscopy (AFM) and transmission electron microscopy. The topography of single PBsomes was characterized by AFM imaging on both isolated and membrane-combined PBsomes complexes. The native organization of thylakoid membranes presented variable arrangements of PBsomes on the membrane surface. It indicates that different light illuminations during growth allow diverse distribution of PBsomes upon the isolated photosynthetic membranes from P. cruentum, random arrangement or rather ordered arrays, to be observed. Furthermore, the distributions of PBsomes on the membrane surfaces are mostly crowded. This is the first investigation using AFM to visualize the native architecture of PBsomes and their crowding distribution on the thylakoid membrane from P. cruentum. Various distribution patterns of PBsomes under different light conditions indicate the photoadaptation of thylakoid membranes, probably promoting the energy-harvesting efficiency. These results provide important clues on the supramolecular architecture of red algal PBsomes and the diverse organizations of thylakoid membranes in vivo.

Let X be a simply connected CW complex with finitely many cells in each degree. The first part of the paper is a report on the different conjectures for the behavior of the sequence rk pi(i) (X). In the second part, we give conditions on the Lusternik-Schnirelmarm category of X and on the depth of the Lie algebra pi(*) (Omega X) circle times Q that imply the exponential growth in k of the sequence Sigma(d)(i=1) rk(pi k+i)(X) for some d. (c) 2006 Elsevier GmbH. All rights reserved.

Cyanobacteria respond to nutrient-limiting conditions by degrading their phycobilisomes (PBS), the light-harvesting complexes for photosynthesis. In Synechococcus sp. PCC 7942, the expression of nblA, an essential gene in this process, is controlled by the response regulator NblR and the sensor NblS. Here we study the effect of inactivation of dspA (an nblS homologue) and an nblR-like gene on phycobilisome degradation in Synechocystis sp. PCC 6803, under nitrogen starvation. In each mutant, the expression of nblA was found to be unaffected and sequential PBS degradation occurred after nitrogen deprivation (although it was slightly delayed). Our results demonstrate that dspA and nblR-like do not exert a major control of PBS degradation in Synechocystis sp. PCC 6803.

Special Issue in honor of Prof. Hans-Bjørn Foxby; International audience; Let L be a graded connected Lie algebra of finite type and finite depth (for instance the rational homotopy Lie algebra of a finite simply connected CW complex). Let L ( p ) = { L p k } k ≥ 1 . Then for any prime p, lim n ⁡ log ⁡ dim ⁡ L ( p ) ≤ n log ⁡ dim ⁡ L ≤ n = 1 . In particular for a space X, the Lie algebra L X = π ⁎ ( Ω X ) ⊗ Q and its even dimensional part L X ( 2 ) have the same log index.

Let C be a differential graded chain coalgebra, U C the reduced cobar construction on C and C-V the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies, HH*(C-V, C-V) and HH*(U C; U C). This result yields to a Hodge decomposition of the loop space homology of a closed oriented manifold, when the field of coefficients is of characteristic zero. C 2004 Elsevier B.V. All rights reserved.

In this paper we use May's algebraic approach to Steenrod operation in conjunction with Munkholm's notion of shc-algebra in order to introduce what we call a π-shc-algebra. (π is a cyclic group of order a fixed prime p.) We prove that the homology and the Hochschild homology of a π-shc-algebra have natural Steenrod operations. Our main topological example is the free loop space. We prove that the Jones’ isomorphism preserves Steenrod operations.

Let M be a closed orientable manifold of dimension d and $\mathcal{C}^*(M)$ be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, $H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M))$ is a graded commutative and associative algebra. The augmentation map $\varepsilon: \mathcal{C}^*(M) \to{\textbf{\textit{k}}}$ induces a morphism of algebras $I : H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M)) \to{H\!H^*(\mathcal{C}^*(M);{\textbf{\textit{k}}})}$ . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H\!H^*(\mathcal{C}^*(M);{\textbf{\textit{k}}})$ , which is in general quite small. The algebra $H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M))$ is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.

Iron plays a key role in the synthesis and functioning of the photosynthetic apparatus. Conditions of partial iron deficiency that lead to a relatively stable phenotype were established and the effects of starvation studied in the unicellular red alga, Rhodella violacea. Synthesis of the photosynthetic pigments were found to decrease, with phycobiliproteins being affected to a lesser extent than chlorophyll a. Biophysical, biochemical and immunological approaches were used to show that the PSI content is highly diminished and the PSII/PSI stoichiometry increased by a factor of 5 compared to standard conditions. Meanwhile light-harvesting complex (LHC) was still assembled in the thylakoid membranes at unchanged levels. The use of translation inhibitors for either nuclear- or plastid-encoded polypeptides revealed that uncoupled LHC may be responsible for the high wavelength-fluorescence contribution observed around 700–710 nm. There is no evidence for the synthesis of new chlorophyll–protein complexes.

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if Ext*(A) (M, A) not equal 0 for some A-module M of at most polynomial growth. THEOREM 1: If f : X --> Y is a continuous map of finite category, and if the orbits of H*(OmegaY) acting in the homology of the homotopy fibre grow at most polynomially, then H*(OmegaY) has finite polydepth. THEOREM 5: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, Sigma(i=k+1)(k+d) dim L-i greater than or equal to V, k greater than or equal to some k(r). (C) 2003 Elsevier SAS.

Let M be a simply-connected closed oriented N-dimensional manifold. We prove that for any field of coefficients lk there exists a natural homomorphism of commutative graded algebras Gamma : H-*(Omega aut(1)M) --> H*(M-S1) where H-*(M-S1) = H*+N(M-S1) is the loop algebra defined by Chas and Sullivan. As usual aut(1)X denotes the monoid of self-equivalences homotopic to the identity, and OmegaX the space of based loops. When lk is of characteristic zero, Gamma yields isomorphisms H-(1)(n+N) (M-S1) congruent to (pi(n)(Omegaaut(1)M) x lk)(boolean OR) where +(infinity)(l=1) H-(l)(n)(M-S1) denotes the Hodge decomposition on H*(M-S1).

The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions on mapping spaces that are induced by an action on the source., Comment: 25 pages

Let $X$ be a simply connected space and $\Bbb K$ be any field. The normalized singular cochains $N^*(X; {\Bbb K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $HH_* N^*X$ of the space $X$. We prove that, endowed with this product, $HH_*N^*X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\Bbb K$. We also show how to construct a simpler Hochschild complex which allows direct computation., Comment: 21 pages, to appear in Topology

To any path connected topological space X X , such that rk ⁡ H i ( X ) > ∞ \operatorname {rk} H_i(X) >\infty for all i ≥ 0 i\geq 0 , are associated the following two sequences of integers: b i = rk ⁡ H i ( Ω X ) b_i= \operatorname {rk} H_i(\Omega X) and r i = rk ⁡ π i + 1 ( X ) r_i= \operatorname {rk} \pi _{i+1}(X) . If X X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that H i ( X ; Q ) = 0 H_i(X;\mathbb Q)=0 , for all i ≫ 0 i\gg 0 , it is a classical result that the sequence of Betti numbers ( b i ) (b_i) grows polynomially or exponentially, depending on whether the sequence ( r i ) (r_i) is eventually zero or not. The purpose of this note is to prove, in both cases, that the r t h r^{\mathrm {th}} Betti number b r b_r is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space X S 1 X^{S^1} .

International audience; The Hochschild cohomology of a DG algebra A with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. The purpose of this paper is to prove that a certain DG Lie algebra of derivations appears as a finite codimensional graded sub Lie algebra of this Lie algebra when A is a strongly homotopy commutative algebra whose homology is concentrated in finitely many degrees. This result has interesting implications for the free the loop space homology which we explore here as well.

Nitrogen (N) limitation in cyanobacteria is well documented: a reduced growth rate is observed, accompanied by a cessation of phycobiliprotein synthesis and an ordered degradation of phycobilisomes (PBS). This leads to a dramatic bleaching phenomenon known as chlorosis. In Synechococcus strain PCC 7942, bleaching due to PBS degradation is also observed under sulfur (S) or phosphorus (P) limitation, and all three are under the control of the nblA gene product, a 59-amino-acid polypeptide which is overexpressed under N, S, and P starvation (J. L. Collier, and A. R. Grossman, EMBO J. 13:1039–1047, 1994). Cyanobase sequence data for Synechocystis strain PCC 6803 indicate the presence of two tandem open reading frames (sll0452 and sll0453) homologous to nblA . We cloned the two genes, identified a unique 5′ mRNA end suggestive of a single transcription start site, and studied nblA expression under conditions of N or S starvation by Northern hybridization: transcripts were detected only under N starvation (no signal is detected in replete medium or with S starvation), whether nblA1 or nblA2 was used as a probe. Mutations in nblA1 and nblA2 were constructed by insertion of a kanamycin cassette; both mutations were nonbleaching under N starvation. Synechocystis strain PCC 6803 does not bleach under S starvation, consistent with the absence of nblA induction in these conditions. These results were confirmed by analysis of the PBS components: sequential degradation of phycocyanin and associated linkers was observed only under conditions of N starvation. This indicates differences between Synechocystis strain PCC 6803 and Synechococcus strain PCC 7942 in their regulatory and signaling pathways leading to N- and S-starved phenotypes.

In a fibration OmegaF (Omegaj)under right arrow OmegaX (Omega pi )under right arrow OmegaB we show that finiteness conditions on F force the homology Serre spectral sequence with F-p-coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, H-*(OmegaX) is "almost" isomorphic to H-*(OmegaB) circle times H-*(OmegaF). One consequence is the conclusion that X is elliptic if and only if B and F are.

The unicellular rhodophyte Rhodella violacea can adapt to a wide range of irradiances. To create a light stress, cells acclimated to low light were transferred to higher irradiance and the kinetics of various changes produced by the light shift were analyzed. The proton gradient generated by excess light led to a non-photochemical quenching of the chlorophyll fluorescence and some photoinhibition of photosystem II centers was also produced by the light stress. After the shift to higher irradiance, the mRNA levels of three chloroplast genes that encode phycoerythrin and phycocyanin apoproteins and heme oxygenase (the first enzyme specific to the bilin synthesis) were negatively regulated. A change in the amount of thylakoids and in the total pigment content of the cells occurred during light acclimation after a light stress. The change in the size of the phycobilisome was limited to dissapearance of the terminal phycoerythrin hexamers in some of the rods. The ability of R. violacea to photoacclimate depends both on large changes in thylakoid number and pigment content and on smaller changes in the antenna size of photosystem II.

We describe here the spectral and biochemical properties of a novel biliprotein belonging to the phycoerythrin family, purified from the phycobilisome of a unicellular red alga, Rhodella reticulata strain R6. This biliprotein is assembled from a unique beta-type subunit, chloroplast-encoded, whose hexameric or dodecameric aggregates are stabilized by unusually large linkers (87 and 60 kDa) encoded by the nuclear genome. Although each beta-type subunit bears two phycoerythrobilins and one phycocyanobilin per chain, the linker polypeptides are non-chromophorylated. The apoprotein of the beta-subunit of the R. reticulata R6 phycoerythrin is specified by a monocistronic rpeB chloroplast gene that is split into three exons. We discuss the relationships between R6 beta-phycoerythrin and the previously published polypeptide sequences, the structural consequences due to the absence of an alpha-subunit, and its evolutionary implications.

Let $X$ be a connected CW complex and $[X]$ be its homotopy type. As usual, $\mbox{SNT}(X)$ denotes the pointed set of homotopy types of CW complexes $Y$ such that their $n^{th}$-Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ are homotopy equivalent for all $n$. In this paper we study a particularly interesting subset of \mbox{SNT}$(X)$, denoted SNT$ _{\pi } (X)$, of strong $n$ type; the $n^{th}$-Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ are homotopy equivalent by homotopy equivalences satisfying an extra condition at the level of homotopy groups. First, we construct a CW complex $X$ such that $\mbox{SNT}_\pi(X) \neq \{ [X] \}$ and we establishe a connection between the pointed set $\mbox{SNT}_\pi (X)$ and sub-groups of homotopy classes of self-equivalences via a certain $\displaystyle\lim_{\leftarrow}{}^1 $ set. Secondly, we prove a conjecture of Arkowitz and Maruyama concerning subgroups of the group of self equivalences of a finite CW complex and we use this result to establish a characterization of simply connected CW complexes with finite dimensional rational cohomology such that $\mbox{SNT}_\pi(X) = \{[X]\}$.

The structure and function of phycobilisomes in the rhodophyte Porphyridium sp. were investigated by comparing the properties of the wild type with a pigment mutant called C12. When grown under low light, cells of C12 were bright orange, while wild-type cells were deep red. The results obtained from a characterization of purified phycobilisomes of the mutant C12 led us to propose the existence in Porphyridium sp. phycobilisomes of two types of rods, some containing only phycoerythrin and others containing phycoerythrin bound to phycocyanin, which is in turn linked to the core by the linker LRC. By studying the partitioning of phycobiliproteins between phycobilisomes and pools of free phycobiliproteins, we found that phycocyanin in the C12 mutant was only present in the pool of free proteins and that its specific linker, LRC, was totally absent. Phycoerythrin was present in the free pool and in the purified phycobilisomes as well. One of the three specific phycoerythrin linkers γ was missing. In light of the fact that in the C12 mutant, the linker LRC is absent and that there is no phycocyanin bound to the phycobilisomes, we propose that the rods in the mutant contain only phycoerythrin. These phycobilisomes are nevertheless functional and exhibit an efficient excitation transfer from phycoerythrin directly to allophycocyanin. Electron microscopy showed the purified phycobilisomes of C12 to be less dense than those of the wild type. This change was attributed to the disappearance of the rods containing the combination phycocyanin/phycoerythrin. Light still regulates phycobiliprotein synthesis in the mutant, as shown by the change in the color of the culture, which turned green-yellow when cells were shifted from low light to high light growth conditions. Light also regulates the structure of the phycobilisomes, which have fewer rods under high light growth conditions.

Special Issue: In Honor of Dennis Sullivan, Part 1 of 2; International audience; We prove that for a wide class of spaces X the homology of the free loop space H∗(XS1;Q) has a very strong exponential growth. We call this convergence, controlled exponential growth, and we prove the good behavior of the controlled exponential growth with respect to fibrations.

In this short paper we try to describe the fundamental contribution of Quillenin the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

The triacylglycerols and fatty acids of Fucus serratus were analysed during the course of a year. The amount of triacylglycerols was maximal in July (19.2% total lipids), but minimal in February (6.6%). Triacylglycerols accumulated more in summer (2.8 mg g−1 dry wt) and autumn (2.6 mg g−1 dry wt) than spring (0.7 mg g−1 dry wt) and winter (0.5 mg g−1 dry wt). In the total lipids, the dominant fatty acids were palmitic (16:0, 24.1%), oleic (18:1, 22.4%) and arachidonic (20:4, 14.4%), but in the triacylglycerols, palmitic 16:0 (22.8%), oleic 18:1 (36.4%) and linoleic acids (18:2, 16.4%) were most abundant. C16 fatty acids were predominant in winter and C18 ones in summer and autumn. The ratios 16:0/18:1 and 20:4/20:5 appeared to be important indicators of the status of the seaweed according to variation of environmental conditions. We conclude that, although the emersion of F. serratus in situ is more effective during winter and spring, correlating with good growth and productivity of thalli, synthesis of triacylglycerols is activated during the summer.

We studied the synthesis and binding of phycoerythrin and its associated linkers to the phycobilisome (PBS) in Rhodella violacea (Kornmann) Wehrmeyer and compared the effects of high light and translation inhibitors on these processes. Rhodella violacea has a simple hemidiscoidal PBS structure with a well-known composition. The number of PBSs per cell decreases when irradiance is increased, and at higher irradiances the rods are shortened with a specific loss of the terminal hexamer of phycoerythrin (PE) and its associated linker. To test whether or not the observed variations were due to a coordination between the expression of the chloroplast-encoded PE and the nuclear-encoded linkers, we inhibited the expression of the chloroplast genes by the translation inhibitor chloramphenicol. In the few PBSs synthesized, the linker associated to the terminal PE hexamer was missing while that associated with the intermediate PE hexamer was still present. The inhibition by cycloheximide of the translation of the nuclear-encoded linkers did not influence the synthesis of the chloroplast-encoded phycobiliproteins. The absence of linkers prevented the formation of PE hexamers and their binding to the PBSs. We therefore propose the existence of two levels of regulation for PE and associated linkers: the intermediate PE hexamer and associated linker are always present even though their amount is reduced when irradiance is increased. In contrast, the terminal hexamer of PE and its associated linker are no longer present under high light. Their absence can be due to a feedback control between the level of PE and the synthesis of the linker: when the level of PE is lowered below a given value by the action of light on the chloroplast, a signal coming from the chloroplast reaches the nucleus and the synthesis of the linker is repressed. There is no sign of nuclear regulation of the synthesis of PE, but the nuclear-encoded linkers have a structural role in the formation of PE hexamers.

Ultrastructural features and immunological properties of some thylakoid proteins were examined in two strains of the prochlorophyte Prochlorococcus and compared to those of other photosynthetic prokaryotes and eukaryotes. Both strains exhibited two or three rows of tightly appressed thylakoidal membranes, located at the cell periphery. However, thylakoids were concentrically arranged in the strain from the Sargasso Sea (SARG) and horseshoe-shaped in the Mediterranean isolate (CCMP 1378). Although lacking phycobilisomes, both cell types shared with cyanobacteria the presence of carboxysome-like structures and glycogen granules as storage compounds. The main thylakoid polypeptides separated by sodium dodecyl sulfate—polyacrylamide gel electrophoresis were characterized by Western blotting using several antibodies. The 30-kDa polypeptide of the light-harvesting complex (LHC) of Prochlorococcus showed a weak positive immunological cross-reaction with an antibody raised against the 32-kDa apoprotein of the LHC of the prochlorophyte Prochlorothrix hollandica. In contrast, it showed no immunological relationships with the chlorophyll a/b (Chl a/b) LHCs of green algae and higher plants. Protein membranes from Prochlorococcus strongly cross-reacted with antibodies raised against reaction center polypeptides of photosystems II and I (PSs II and I) of other photosynthetic organisms, confirming the high degree of conservation of these basic compounds of the photosynthetic machinery during evolution. Immunolocalization of thylakoid proteins showed that the LHC proteins, the major PS II reaction center proteins (CP 43 and D2), and the PS I reaction center proteins were equally distributed within the thylakoid membranes in contrast to the segregation observed in higher plants and green alga thylakoids. We also identified ribulose-1, 5-bisphosphate carboxylase/oxygenase in the carboxysomes. These results suggest that Prochlorococcus is more closely related to cyanobacteria than to green plastids even though it contains Chl b.

Protoplasts from two green pigment mutants of Porphyridium sp. (UTEX 637) containing a low phycoerythrin level were fused by exposure to polyethylene glycol (MW 6000) combined with a short heat shock (45° C, 5 min). Following regeneration on agar plates, red colonies arose in which complementation of the phycoerythrin deficiency had occurred. The complementation frequency was estimated to be 0.2%. Eight progeny showing red pigmentation were isolated and purified by consecutive transfers on agar plates. Characterization of the fusion progeny revealed that their phycobiliprotein and chlorophyll contents per cell were higher than those of their parental mutant strains and, in most strains, similar to that of the wild type. The fusion products proved to be stable over many growth cycles. The DNA content of the wild type and of the parental mutant strains was about 0.05 pg-cell−1. Fusion progeny strains showed a variable DNA content: a few fusants contained the same amount of DNA as the wild type and the parental strains, while others had about 50% more DNA per cell. The DNA content of one of the progeny strains (CF1c) was double that of the wild type (0.1 pg. cell−1). Cells of this fusion progeny contained one nucleus per cell, which suggests that nuclear fusion and the formation of a stable diploid followed cell fusion. Analysis of phycobilisome components of CF1c revealed complementation of linker polypeptides associated with phycoerythrin (γ subunits). CF1c contained, like the wild-type strain, four linker polypeptides; all of these were absent in one parental strain and one was absent in the second. To the best of our knowledge, this is the first report of protoplast fusion, formation of somatic hybrids, and the apparent completion of a parasexual cycle in a red microalga.

Resume Les problemes ouverts qui figurent dans cet article se situent dans le cadre des travaux entrepis depuis une dizaine d'annees en collaboration avec Y. Felix et S. Halperin. Ils concernent le type d'homotopie et l'homologie des espaces de lacets a coefficients dans un corps F o = Q ou F p , p premier. Nos methodes sont plus proches de celles utilisees dans l'etude des H -espaces finis et en algebre locale, que des methodes traditionelles des “homotopistes stables”.

Nous comparons et calculons trois sous-espaces vectoriels remaquables de la cohomologie d'un CW-complexe.We compare three natural subvectorspaces of the cohomology of a simply connected pointed space. These spaces are related to, respectively, the “last cells” of the space, the “dual Gottlieb groups” and the “evaluation map” defined by the Spivak construction.

Only little is known concerning H ∗ ( Ω X ; k ) {H_\ast }(\Omega X;{\mathbf {k}}) , the loop space homology of a finite CW {\text {CW}} complex X X with coefficients in a field k {\mathbf {k}} . A space X X is called an r r -cone if there exists a filtration ∗ = X 0 ⊂ X 1 ⊂ ⋯ ⊂ X r = X \ast = {X_0} \subset {X_1} \subset \cdots \subset {X_r} = X , such that X i {X_i} has the homotopy type of the cofibre of a map from a wedge of sphere into X i − 1 {X_{i - 1}} . Denote by A X {A_X} the sub-Hopf algebra image of H ∗ ( Ω X 1 ) {H_\ast }(\Omega {X_1}) . We prove then that for a graded r r -cone, r ≤ 3 r \leq 3 , there exists an isomorphism A X ⊗ T ( U ) → ≅ H ∗ ( Ω X ) {A_X} \otimes T(U)\xrightarrow { \cong }{H_\ast }(\Omega X) .