Maximum size binary matroids with no AG(3,2)-minor are graphic

We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank

Molecular nature of anomalous L-type calcium channels in mouse cerebellar granule cells.

Single-channel analysis revealed the existence of neuronal L-type Ca2+ channels (LTCCs) with fundamentally different gating properties; in addition to LTCCs resembling cardiac channels, LTCCs with anomalous gating were identified in a variety of neurons, including cerebellar granule cells. Anomalous LTCC gating is mainly characterized by long reopenings after repolarization following strong depolarizations or trains of action potentials. To elucidate the unknown molecular nature of anomalous LTCCs, we performed single-channel patch-clamp recordings from cerebellar granule cells of wild-type, Ca(v)1.3(-/-) and Ca(v)1.2DHP(-/-) [containing a mutation in the Ca(v)1.2 alpha(1) subunit that eliminates dihydropyridine (DHP) sensitivity] mice. Quantitative reverse transcription-PCR revealed that Ca(v)1.2 accounts for 89% and Ca(v)1.3 for 11% of the LTCC transcripts in wild- type cerebellar granule cells, whereas Ca(v)1.1 and Ca(v)1.4 are expressed at insignificant levels. Anomalous LTCCs were observed in neurons of Ca(v)1.3(-/-) mice with a frequency not different from wild type. In the presence of the DHP agonist (+)-( S)- 202- 791, the typical prepulse- induced reopenings of anomalous LTCCs after repolarization were shorter in Ca(v)1.2DHP(-/-) neurons than in Ca(v)1.3(-/-) neurons. Reopenings in Ca(v)1.2DHP(-/-) neurons in the presence of theDHPagonist were similar to those in wild- type neurons in the absence of the agonist. These data show that Ca(v)1.2 alpha(1) subunits are the pore- forming subunits of anomalous LTCCs in mouse cerebellar granule cells. Given the evidence that Ca(v)1.2 channels are specifically involved in sustained Ras- MAPK (mitogen-activated protein kinase)- dependent cAMP response element- binding protein phosphorylation and LTCC-dependent hippocampal long- term potentiation (LTP) ( Moosmang et al., 2005), we discuss the hypothesis that anomalous rather than cardiac-type Ca(v)1.2 channels are specifically involved in LTCC-dependent and gene transcription- dependent LTP

Specific kinetic alterations of human Ca(V)2.1 calcium channels produced by mutation S218L causing familial hemiplegic migraine and delayed cerebral edema and coma after minor head trauma

Mutation S218L in the Ca(V)2.1 alpha(1) subunit of P/Q-type Ca2+ channels produces a severe clinical phenotype in which typical attacks of familial hemiplegic migraine (FHM) triggered by minor head trauma are followed, after a lucid interval, by deep (even fatal) coma and long lasting severe cerebral edema. We investigated the functional consequences of this mutation on human Ca(V)2.1 channels expressed in human embryonic kidney 293 cells and in neurons from Ca(V)2.1 alpha(1)(-/-) mice by combining single channel and whole cell patch clamp recordings. Mutation S218L produced a shift to lower voltages of the single channel activation curve and a consequent increase of both single channel and whole cell Ba2+ influx in both neurons and human embryonic kidney 293 cells. Compared with the other FHM-1 mutants, the S218L shows one of the largest gains of function, especially for small depolarizations, which are insufficient to open the wild-type channel. S218L channels open at voltages close to the resting potential of many neurons. Moreover, the S218L mutation has unique effects on the kinetics of inactivation of the channel because it introduces a large component of current that inactivates very slowly, and it increases the rate of recovery from inactivation. During long depolarizations at voltages that are attained during cortical spreading depression, the extent of inactivation of the S218L channel is considerably smaller than that of the wild-type channel. We discuss how the unique combination of a particularly slow inactivation during cortical spreading depression and a particularly low threshold of channel activation might lead to delayed severe cerebral edema and coma after minor head trauma