in both continents should show whether thispattern is unique to Europe or is more wide-spread.

Jernvall and Fortelius point out that theirresults are consistent with the life-historydifferences we see in modern ungulates.Hypsodont species are somewhat larger, andare more likely to form large herds7 and dis-perse more widely across landscapes, and theincrease in hypsodonty among commontaxa about 11 million years ago might heraldthe evolution of these life-history traits. Butshowing that these traits are associated with atrend of increasing hypsodonty in commonherbivores, and are not found in the rarerones, will require more work on the Miocenefossil record.

Aspects of Jernvall and Fortelius’sapproach will influence future work on evo-lutionary trends. The use of a new metric —average hypsodonty — could prove useful asa palaeoenvironmental indicator, and pro-vides a quantitative measure of the temporaland spatial variation in crown heights ofteeth from different sites. The examinationof common and rare groups has allowed the

finer resolution of a major trend and mightlead to a better understanding of the evolu-tion of grassland ecosystems. It also suggeststhat common groups yield more informa-tion about palaeoecology than has previous-ly been recognized. The use of large data setsto examine evolutionary trends has in-creased markedly in recent years, and furtherinsights into other trends, such as changes inbody size, might result from examining com-mon and rare taxa separately. ■

Jessica M. Theodor is in the Department of Geology,Illinois State Museum, 1011 East Ash Street,Springfield, Illinois 62703, USA.e-mail: jtheodor@museum.state.il.us1. Jernvall, J. & Fortelius, M. Nature 417, 538–540 (2002).

2. Jacobs, B. F., Kingston, J. D. & Jacobs, L. L. Ann. Missouri Bot.

Gdn 86, 590–643 (1999).

3. Janis, C. M. & Fortelius, M. Biol. Rev. Camb. Phil. Soc.

63, 197–230 (1988).

4. Janis, C. M. in Teeth Revisited: Proceedings of the VIIth

International Symposium on Dental Morphology (eds

Russell, D. E., Santoro, J.-P. & Sigogneau-Russell, D.) 367–387

(Musée d’Histoire Naturelle, Paris, 1988).

5. Jernvall, J., Hunter, J. P. & Fortelius, M. Science 274, 1489–1492

(1996).

6. Ruddiman, W. F. (ed.) Tectonic Uplift and Climate Change

(Plenum, New York, 1997).

7. Jarman, P. J. Behavior 48, 215–267 (1974).

in his famous equation, E4mc 2, so informa-tion about the mean-field potential can beobtained from studies of nuclear masses.

When nuclear mass is studied as a functionof the number of protons or neutrons insidethe nucleus, bunching and gaps in the quan-tum spectrum of the mean-field potential arerevealed. This variation, called ‘shell struc-ture’, is a general phenomenon in quantumsystems with a small number of particles —such as electrons in an atom or a quantum dot,or clusters of atoms. The gaps in the quantumspectrum arise as particles fill up closed shellsof quantum-mechanically allowed orbits. Inthe spectrum of nuclear masses, local minimasuggest energetically favourable configura-tions of neutrons (and protons), in which thenumber of neutrons (or protons) is sufficientto complete a shell (Fig. 1).

The shell energy is a quantum-mechani-cal quantity, but in principle it can beexpressed in classical terms, by summing upall possible periodic orbits for a particle as itmoves in the classical mean-field potential2,3.Bohigas and Leboeuf 1 consider the dynam-ics of the atomic nucleus as being made upof layers of regular motion, overlaid withchaotic motion. The shell energy thendepends on contributions from regularorbits and from chaotic orbits. Provided thatthe two contributions are independent, theshell energy can be split into the two parts,each expressed in terms of asymptotic seriesof classical periodic orbits, with fluctuationscalculated analytically1,3. The contributionfrom chaotic orbits can then be comparedwith the difference between experimentallymeasured and calculated shell energy, dE(Fig. 1), and the calculated shell energy canbe compared with the contribution fromregular orbits.

Although the fluctuation of calculatedshell energies for all nuclei is about 3.5 MeV,the fluctuation in dE is considerably smaller,averaging about 0.7 MeV. The discrepancybetween measured and calculated massesdecreases with increasing neutron number(Fig. 1). In their analysis, in terms of regularand chaotic orbits, Bohigas and Leboeufreproduce this behaviour, supporting theinterpretation of the deviation between

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NATURE | VOL 417 | 30 MAY 2002 | www.nature.com 499

The atomic nucleus is central to manyphysical and astrophysical processes,including the nuclear processes that

power the Sun and synthesize the elements.Which atoms exist depends on the stability ofdifferent types of nuclei, and the stability ofan individual nucleus depends on its mass. Intheoretical nuclear physics, it has long been achallenge to calculate the masses of known(and as yet unknown) nuclei — but experi-mentally measured and theoretically calcu-lated masses do not fully agree. Writingin Physical Review Letters, Bohigas and

Leboeuf 1 suggest that chaotic motion in thenucleus could influence nuclear mass, andcould ultimately limit how accurately nuclearmasses can be calculated theoretically.

The atomic nucleus consists of neutronsand protons held together by the strongnuclear force. Mathematically, this compli-cated ‘many-body’ problem can be describedfairly accurately by assuming that the inter-action of a given particle inside the nucleuswith all other particles can be approximatedby an averaged energy term, the ‘mean-fieldpotential’. Einstein equated energy and mass

Nuclear physics

Weighing up nuclear massesSven Åberg

It is difficult to match theoretically calculated masses of atomic nuclei totheir experimentally measured values. It seems that chaotic motion insidethe nucleus may be the reason for this discrepancy.

Figure 1 Nuclear mass varies with the number ofneutrons in the nucleus, showing a ‘shellstructure’ (upper panel). The minima at neutronnumbers 28, 50, 82 and 126 correspond toespecially stable nuclei, in which the number ofneutrons is sufficient to complete a shell (a bandof masses is shown because, for a given numberof neutrons, several nuclei with differentnumbers of protons can exist). Theoreticalmodels cannot quite describe this variation, anda discrepancy is always seen between calculatedand measured mass values (lower panel).Bohigas and Leboeuf 1 suggest that this can beexplained by taking into account chaotic motioninside the nucleus. (Graphic derived from ref. 5.)

0 20 40 60 80 100 120 140 160

0

Nuclear mass(arbitrary scale)

Differencebetween measured

and calculatedmass values, δE

Number of neutrons in nucleus

© 2002 Nature Publishing Group

measured and calculated energies as result-ing from chaotic motion. It seems thatchaotic motion has to be considered iftheoretical descriptions of nuclear massesare to be improved.

A plausible explanation for the appear-ance of chaotic motion is that there are many-body effects that are not included in the mean-field potential. Such effects give rise to verycomplicated spectra when nuclei are excitedto neutron or proton resonances. The effort toexplain these complicated spectra led Wignerin the 1950s to introduce a theory — randommatrix theory4— that is now a key concept ofquantum chaos. According to this theory,properties related to individual energies canbe calculated only in a statistical sense. If this isthe situation for nuclear masses as well, it willcertainly be extremely hard to improve thetheoretical description of nuclear masses.Even in the best theoretical models5–7, it isexceedingly difficult to keep the model uncer-tainties lower than about 0.7 MeV, the level ofthe expected shell-energy contribution fromchaotic motion.

In their calculations, Bohigas andLeboeuf 1 have assumed that there are no cor-relations between chaotic and regular layersof motion. But the existence of non-zero cor-relations between the calculated shell energyand the difference in measured and calculat-ed values (a correlation of about –16% is

obtained from ref. 5) — as well as someuncertainties in estimating fluctuations ofthe chaotic contribution to shell energy —suggests that there is room for improvementbefore the chaotic limit is reached.

Indeed, much is being done in theoreticalnuclear-physics research to try to improvethe description of nuclear masses7, somegoing beyond the mean-field description8–10.Such efforts could allow us to extrapolatemore effectively into nuclear-mass regionsthat have yet to be explored — in particular,regions where the astrophysical creation ofelements takes place. ■

Sven Åberg is in the Department of MathematicalPhysics, Lund Institute of Technology, PO Box 118,S-221 00 Lund, Sweden.e-mail: sven.aberg@matfys.lth.se1. Bohigas, O. & Leboeuf, P. Phys. Rev. Lett. 88, 092502 (2002).

2. Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics

(Springer, New York, 1990).

3. Leboeuf, P. & Monastra, A. G. Ann. Phys. 297, 127–156 (2002).

4. Wigner, E. P. in Statistical Theories of Spectra: Fluctuations

(ed. Porter, C. E.) 88–140, 145–161, 176–180, 188–211,

223–227, 446–461 (Academic, New York, 1965).

5. Möller, P., Nix, J. R., Myers, W. D. & Swiatecki, W. J. Atom. Data

Nucl. Data Tables 59, 185–382 (1995).

6. Goriely, S., Tondeur, F. & Pearson, J. M. Atom. Data Nucl. Data

Tables 77, 311–381 (2001).

7. Samyn, M., Goriely, S., Heenen, P.-H., Pearson, J. M. &

Tondeur, F. Nucl. Phys. A 700, 142–156 (2002).

8. Otsuka, T. Hyperfine Inter. 132, 409–416 (2001).

9. Valor, A., Heenen, P.-H. & Bonche, P. Nucl. Phys. A

671, 145–164 (2000).

10.Rodriguez-Guzman, R., Egido, J. L. & Robledo, L. M.

Phys. Lett. B 474, 15–20 (2000).

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Nervous systems are built for speed,using electrical signals that are propa-gated by the activities of ion channels.

Two responsibilities are incumbent on thesemembrane-spanning proteins. First, theymust provide a selective conduction path-way, the pore — a hospitable environmentthrough which appropriate ions can beescorted across a cell’s outer membrane.Second, most ion channels must undergorapid conformational changes in responseto metabolic cues, opening or closing thepore. These ‘gating’ cues can be changes involtage across the membrane, or molecules(ligands), or both. Just how ion channelsconduct ions, how ligands bind to channelproteins, and how gating cues are transducedinto conformational rearrangements are thebiggest questions facing this field. Someanswers are provided by two papers fromRoderick MacKinnon’s laboratory, on pages515 and 523 of this issue1,2. In the first paperthe authors describe the crystal structure of

a Ca2&-gated K& channel, revealing details ofthe Ca2&-occupied ligand-binding regionand the open pore1. In the second they pro-vide the first comparison of the structures ofopen and closed K&-channel pores, and offera general model for gating2.

MacKinnon’s group has presented aseries of stunning papers over the past fouryears, taking advantage of the ability toobtain large quantities of purified ion-chan-nel proteins from bacteria. Crystal structuresof closed KcsA channels (K& channels fromStreptomyces lividans) containing residentK& ions revealed the essential features of K&

conduction3–5. These channels consist offour protein subunits, and each subunit con-tributes one membrane-spanning segment(a helical structure) to the lining of the pore.At the interface of the channel with the cyto-plasm inside the cell, the helices cometogether to form a crossed bundle rather likean inverted teepee, which makes up the phys-ical gate of the channel. Farther into the

Ion channels

An open and shut caseMaria Schumacher and John P. Adelman

Most ion channels open and close — they are ‘gated’ — in response tocues in their environment. A crystal structure of a Ca2+-gated K&-ionchannel provides insight into how gating works.

NATURE | VOL 417 | 30 MAY 2002 | www.nature.com 501

membrane is an inner cavity that perfectlyaccommodates a hydrated K& ion. This cavi-ty is found just beneath the selectivity filter,which discriminates between K& and otherions. Four rings of backbone carbonyls formthe filter, and during dynamic conductionthey are probably occupied alternately bywater and K& ions. In short, every aspect ofKcsA seems to have evolved for K&-specificconduction.

What about channel gating? Elegantdatabase analysis6 suggested that an intra-cellular structural domain shared by manybacterial K& channels and other transportsystems, and also found in mammalianchannels that are gated by the combinationof Ca2& and voltage, is important for gating.Previous work6 also hinted that this domain,called RCK for ‘regulates conduction of K&

ions’, might be a ligand-binding motif.The latest work from MacKinnon’s group

brings our knowledge of gating and conduc-tion together. In the first paper1 they presentthe structure of a Ca2&-activated K& channel(MthK, from Methanobacterium thermo-autotrophicum), with its pore and attachedRCK domains. The structure shows thatfunctional K& channels consist of four pore-forming subunits, as well as four RCK dimers(that is, eight RCK domains) that form a‘gating ring’ just inside the cytoplasm (Fig. 1,overleaf). Two important interfaces betweenthe RCK domains keep the ring in place. Oneinterface is fixed and probably occurs in alltypes of RCK domain. The other is flexible,made up in part by subdomains that pro-trude from each RCK domain and form aligand-binding pocket between adjacentdomains; in the crystal structure, two Ca2&

ions are clearly positioned at the base of thiscleft between two RCK domains.

In the same paper, the authors comparethe Ca2&-bound RCK domain from MthKwith the unbound RCK domain from anEscherichia coli K& channel, providinginsights into how Ca2& binding might exertmechanical force to open the attached pore-lining helices. The fixed interfaces in the twostructures are similar, indicating that Ca2&

does not alter their relative orientations —they are indeed fixed. But the flexible inter-faces are remarkably different. It seems likelythat Ca2& binding reshapes these clefts, andthat this compels a tilt of the rigid interfaces,expanding the diameter of the gating ringand exerting a force on the attached pore-lining helices that opens the pore (Fig. 1). Inthis way, MthK uses the chemical energy ofCa2& binding to perform the mechanicalwork of gating.

A special consequence of the structure ofMthK is that it captures the pore in an openstate, and this has allowed the authors tocompare it with the previously determinedclosed-pore structure of KcsA. As theydescribe in the second new paper2, the con-trast is remarkable. The outer region of the

© 2002 Nature Publishing Group