Odd circuits in dense binary matroids

We show that, for each real number $\alpha > 0$ and odd integer $k\ge 5$ there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M| \ge \alpha 2^{r(M)}$ and with no $k$-element circuit, then $M$ has critical number at most $c$. The result is an easy application of a regularity lemma for finite abelian groups due to Green

The critical number of dense triangle-free binary matroids

We show that, for each real number $\epsilon > 0$ there is an integer $c$ such that, if $M$ is a simple triangle-free binary matroid with $|M| \ge (\tfrac{1}{4} + \epsilon) 2^{r(M)}$, then $M$ has critical number at most $c$. We also give a construction showing that no such result holds for any real number less than $\tfrac{1}{4}$. This shows that the "critical threshold" for the triangle is $\tfrac 1 4$. We extend the notion of critical threshold to every simple binary matroid $N$ and conjecture that, if $N$ has critical number $c\ge 3$, then $N$ has critical threshold $1-i\cdot 2^{-c}$ for some $i\in \{2,3,4\}$. We give some support for the conjecture by establishing lower bounds

On minor-closed classes of matroids with exponential growth rate

Let \cM be a minor-closed class of matroids that does not contain arbitrarily long lines. The growth rate function, h:\bN\rightarrow \bN of \cM is given by h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$). The Growth Rate Theorem shows that there is an integer $c$ such that either: $h(n)\le c\, n$, or ${n+1 \choose 2} \le h(n)\le c\, n^2$, or there is a prime-power $q$ such that $\frac{q^n-1}{q-1} \le h(n) \le c\, q^n$; this separates classes into those of linear density, quadratic density, and base-$q$ exponential density. For classes of base-$q$ exponential density that contain no $(q^2+1)$-point line, we prove that $h(n) =\frac{q^n-1}{q-1}$ for all sufficiently large $n$. We also prove that, for classes of base-$q$ exponential density that contain no $(q^2+q+1)$-point line, there exists k\in\bN such that $h(n) = \frac{q^{n+k}-1}{q-1} - q\frac{q^{2k}-1}{q^2-1}$ for all sufficiently large $n$

High efficiency protoplast isolation from in vitro cultures and hairy roots of Maesa lanceolata

In vitro cultures of the medicinal plant Maesa lanceolata were established to enable the cultivation of plant material for the production of protoplasts. Callus cultures were initiated using leaves collected from shoot cultures and the root tips from hairy root cultures obtained upon Agrobacterium rhizogenes transformation. For the isolation of protoplast, the different explant material of M. lanceolata was exposed to an enzyme mixture consisting of 1.5% cellulase, 0.5% macerozyme R-10 and 0.5 M mannitol. About 6 x 106 protoplasts g-1 fresh weight were obtained from leaf material and 5 x 105 protoplasts g-1 fresh weight from callus. To obtain high amounts of hairy root protoplasts, the cultures were pretreated with the auxin indole-3-butyric acid (IBA) that stimulated the formation of novel root tips. Using the dissected root tips as starting material, 8 x 105 protoplasts g-1 fresh weight were obtained per preparation. The protoplast isolation method will enable further studies on the transformation and fusion of protoplasts from M. lanceolata

Dynamics of male meiotic recombination frequency during plant development using Fluorescent Tagged Lines in Arabidopsis thaliana

Meiotic homologous recombination plays a central role in creating genetic variability, making it an essential biological process relevant to evolution and crop breeding. In this study, we used pollenspecific fluorescent tagged lines (FTLs) to measure male meiotic recombination frequency during the development of Arabidopsis thaliana. Interestingly, a subset of pollen grains consistently shows loss of fluorescence expression in tested lines. Using nine independent FTL intervals, the spatio-temporal dynamics of male recombination frequency was assessed during plant development, considering both shoot type and plant age as independent parameters. In most genomic intervals assayed, male meiotic recombination frequency is highly consistent during plant development, showing no significant change between different shoot types and during plant aging. However, in some genomic regions, such as I1a and I5a, a small but significant effect of either developmental position or plant age were observed, indicating that the meiotic CO frequency in those intervals varies during plant development. Furthermore, from an overall view of all nine genomic intervals assayed, both primary and tertiary shoots show a similar dynamics of increasing recombination frequency during development, while secondary and lateral shoots remain highly stable. Our results provide new insights in the dynamics of male meiotic recombination frequency during plant development

Cold-induced male meiotic restitution in Arabidopsis thaliana is not mediated by GA-DELLA signaling

Short periods of cold stress induce male meiotic restitution and diploid pollen formation in Arabidopsis thaliana by specifically interfering with male meiotic cytokinesis. Similar alterations in male meiotic cell division and gametophytic ploidy stability occur when gibberellic acid (GA) signaling is perturbed in developing anthers. In this study, we found that exogenous application of GA primarily induces second division restitution (SDR)-type pollen in Arabidopsis, similar to what cold does. Driven by the close similarity in cellular defects, we tested the hypothesis that cold-induced meiotic restitution is mediated by GA-DELLA signaling. Using a combination of chemical, genetic and cytological approaches, however, we found that both exogenously and endogenously altered GA signaling do not affect the cold sensitivity of male meiotic cytokinesis. Moreover, in vivo localization study using a GFP-tagged version of RGA protein revealed that cold does not affect the expression pattern and abundance of DELLA in Arabidopsis anthers at tetrad stage. Expression study found that transcript of RGA appears enhanced in cold-stressed young flower buds. Since our previous work demonstrated that loss of function of DELLA causes irregular male meiotic cytokinesis, we here conclude that cold-induced meiotic restitution is not mediated by DELLA-dependent GA signaling