The Mathematics of the Area Law: Kepler's Successful Proof in Epitome Astronomiae Copernicanae (1621)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (Received April 19, 2002) Published online April 2, 2003 Communicated by N. M. Swerdlow     

Ankyrin and spectrin associate with voltage-dependent sodium channels in brain

The segregation of voltage-dependent sodium channels to specialized regions of the neuron is crucial for propagation of an action potential. Studies of their lateral mobility indicate that sodium channels are freely mobile on the neuronal cell body but are immobile at the axon hillock, presynaptic terminal and at focal points along the axon. To elucidate the mechanisms that regulate sodium channel topography and mobility, we searched for specific proteins from the brain that associate with sodium channels. Here we show that sodium channels labelled with 3H-saxitoxin (STX) are precipitated in the presence of exogenous brain ankyrin by anti-ankyrin antibodies and that 125I-labelled ankyrin binds with high affinity to sodium channels reconstituted into lipid vesicles. The cytoplasmic domain of the erythrocyte anion transporter competes for the latter interaction. Neither the neuronal GABA (gamma-aminobutyric acid) receptor channel complex nor the dihydropyridine (DHP) receptor bind brain ankyrin. The results indicate that brain ankyrin links the voltage-dependent sodium channel to the underlying cytoskeleton and may help to maintain axolemmal membrane heterogeneity and control sodium channel mobility.     

Non-function of a Moloney murine leukaemia virus regulatory sequence in F9 embryonal carcinoma cells

Moloney murine leukaemia virus (M-MuLV) infection of embryonal carcinoma (EC) cells results in the integration of proviral DNA into the host cell genome, but not in virus production. One suggested explanation for the lack of viral gene expression in EC cells has been methylation of the integrated viral DNA. However, subsequent reports indicated that integration of the M-MuLV DNA occurs soon after infection, but that viral DNA methylation occurs considerably later. Nevertheless, viral gene expression is not observed even at early times. One possible explanation is that certain M-MuLV regulatory sequences do not function in EC cells. We now present evidence which supports this hypothesis.     

Movement and segregation of kinetochores experimentally detached from mammalian chromosomes

The kinetochore is a specialized structure at the centromere of eukaryotic chromosomes that attaches chromosomes to the mitotic spindle. Recently, several lines of evidence have suggested that kinetochores may have more than a passive role in the movement of chromosomes during mitosis and meiosis. Kinetochores seem to attract and 'capture' microtubules that grow from the spindle poles and microtubules may lengthen or shorten by the addition or subtraction of tubulin subunits at their kinetochore-associated ends. An attractive hypothesis is that kinetochores function as 'self-contained engines running on a microtubule track'. Here, we show that kinetochores can be experimentally detached from chromosomes when caffeine is applied to Chinese hamster ovary cells that are arrested in the G1/S phase of the cell cycle. The detached kinetochore fragments can still interact with spindle microtubules and complete all the mitotic movements in the absence of other chromosomal components. As these cells enter mitosis before DNA synthesis is completed, chromosome replication need not be a prerequisite for the pairing, alignment and segregation of kinetochores.     

Structure, expression and divergence of T-cell receptor beta-chain variable regions

Analysis of three new T-cell receptor beta-chain variable regions together with those in the literature indicates that they have both remarkable similarities and differences with those of immunoglobulin. Less than 10 V regions appear to predominate in the thymus. V beta sequences are much more heterogeneous at the amino acid level than are immunoglobulin V regions and they appear to diverge between species much more quickly, apparently the result of additional hypervariable regions. Three of these putative new hypervariable regions lie outside of the classical immunoglobulin binding site, an indication that important interactions may be occurring in these regions with polymorphic MHC determinants.