conformations — right- and left-handed mirror images of each other, known as enantio-mers. Such molecules crystallize primarily in one of two forms: as a ‘racemate’, meaning that the crystal is an ordered arrangement of equal numbers of both enantiomers; or as two sets of distinct enantiomorphic crystals, each con-taining only left-handed molecules or only right-handed molecules. The most famous example of this second process is Louis Pasteur’s seminal discovery of the spontaneous separation of sodium ammonium tartrate into left- and right-handed crystals3.

In the laboratory, racemic crystals occur more frequently than enantiomorphic crys-tals. This has been attributed to Wallach’s rule4,5, which states that racemic crystals tend to be denser — the left and right hands are arranged around centres of symmetry, and their packing is thus more efficient — than their enantio morphic counterparts, and there-fore more stable. Furthermore, symmetry con-straints limit the number of favourable packing arrangements available to chiral molecules.

Hutin et al.2 used the low solubility of a crys-tal racemate to retrieve the left-handed and right-handed versions of a specific pair of chi-ral molecules from a mixture of many closely related, interconverting conformations of the same molecule, which are likely to have very similar energies. The authors started by attach-ing a chiral building-block (1-amino-2,3-propanediol, or APD) to opposite ends of each of two bridging phenanthroline (phen) ligands. These ligands are themselves bound together by two copper ions (Fig. 1a). The resulting ‘dicopper double dihelicate’ molecule has four APD chiral centres, each of which can be either left handed (written S) or right handed (R).

Things became interesting when, rather like putting a bundle of assorted socks in a drawer, the authors used a racemic mixture of S-APD and R-APD building-blocks. Under this condition, nuclear magnetic resonance studies indicated the presence of a mixture of numerous isomers. Thus (using a hyphen to denote the phen bridge between two chiral centres and a colon to separate the two dif-ferent phen ligands), all the forms R-R:R-R, S-R:R-R, S-S:R-R, S-R:R-S, S-R:S-R and S-S:R-S were present in the solution, together with their partners of opposite chirality, S-S:S-S, R-S:S-S, R-R:S-S, R-S:S-R, R-S:R-S and R-R:S-R (Fig. 1b).

After being left to stand for two weeks, this racemic solution produced crystals. Even when large quantities of crystalline mate-rial were retrieved, the product was always a racemate containing only the enantiomeric R-R:R-R and S-S:S-S forms, in equivalent amounts and arranged in separate columns within the crystal (Fig. 1c). This surprising finding seems to indicate that these two forms are removed continuously and exclusively by crystallization, to be continuously replenished in the solution through the conversion of the other forms listed above. This in turn requires

reversible exchange of S-APD and R-APD build-ing-blocks among the molecules. The overall effect is a self-sorting process, in which a single crystalline racemate — presumably the most stable, least soluble version — forms from many possibilities.

This is a remarkable result, as it illustrates that crystallization can produce a single out-come among many possibilities from a mix-ture under thermodynamic control, even when the energetic differences between the many possible single outcomes is probably small.

A particularly attractive feature of a dynamic combinatorial library is the ability to adjust its composition, and so the stability ranking of its components, through changes in exter-nal factors such as temperature, pressure and light exposure. Libraries containing large numbers of interconverting chiral components, such as that described by Hutin et al.2, repre-sent a unique opportunity to explore the factors that determine whether a racemate or its cor-responding enantiomorphs will be formed6–9, a poorly understood phenomenon. Such libraries

might also prove useful for optimizing and regulating crystal poly morphism and crystal-lization outcomes in general10, a feature that would interest academic and commercial labo-ratories alike — particularly those dealing with pharmaceutical compounds and other special-ist chemicals. ■

Michael D. Ward is at the Molecular Design Institute, Department of Chemistry, New York University, 100 Washington Square East, New York, New York 10003-6688, USA.e-mail: mdw3@nyu.edu

1. Corbett, P. T. et al. Chem. Rev. 106, 3652–3711 (2006).2. Hutin, M. et al. J. Am. Chem. Soc. 129, 8774–8780 (2007).3. Pasteur, L. Ann. Chim. Phys. 24, 442–459 (1848).4. Wallach, O. Justus Liebigs Ann. Chem. 286, 90–143 (1895).5. Brock, C. P., Schweizer, W. B. & Dunitz, J. D. J. Am. Chem.

Soc. 113, 9811–9820 (1991).6. Schipper, P. E. & Harrowell, P. R. J. Am. Chem. Soc. 105,

723–730 (1983).7. Custelcean, R. & Ward, M. D. Cryst. Growth. Des. 5,

2277–2287 (2005).8. Coquerel, G. Top. Curr. Chem. 269, 1–51 (2006).9. Jacques, J., Collet, A. & Willen, S. H. Enantiomers,

Racemates, and Resolutions (Krieger, Malabar, FL, 1994).10. Bernstein, J. Polymorphism in Molecular Crystals (Oxford

Univ. Press, 2002).

ECOLOGY

Scaling laws in the drier Ricard Solé

The vegetation of arid ecosystems displays scale-free, self-organized spatial patterns. Monitoring of such patterns could provide warning signals of the occurrence of sudden shifts towards desert conditions.

Once upon a time the Sahara was green — it was covered by vegetation. The evidence for this comes from many different sources, including the former existence of lakes. Around 5,500 years ago, the wet environmental conditions suddenly came to an end. Despite the absence of abrupt, external climatic change, plant productivity declined and the topsoil was lost. Eventually, the green Sahara became the desert Sahara that we know today1. The changes experienced by the biosphere over the past century, particularly increased desertifica-tion due to rising temperatures and declining rainfall, have raised concerns about the pos-sibility of rapid shifts from green to desert states2,3. Arid and semi-arid ecosystems cover one-third of Earth’s land surface, so there is a pressing need for quantitative ways to help forecast such shifts.

In this issue, Scanlon et al.4 (page 209) and Kéfi et al.5 (page 213) explore the problem of how vegetation in semi-arid ecosystems is organized in space and time. These studies point the way to how forecasting might be achieved. They involve analyses of the size distribution of vegetated patches in the Kala-hari Desert4 and in three different areas of the Mediterranean basin5, and they cover different spatial scales and types of vegetation.

A notable finding by both groups is that the size distribution of vegetation clusters in undisturbed plots falls off as a power law: most patches of vegetation are of small size, but a few of them are very large. Specifically, if S is the size of a given vegetation cluster, then its frequency decays as 1/S γ (with scaling expo-nents within the range 1<γ<2). Such power laws occur in other types of ecosystem6 and are a fingerprint of self-organization: that is, they are the result of internal dynamic pro cesses driven by local interactions. This principle applies to the field data reported by both Scanlon et al.4 and Kéfi et al.5. It indicates that plant interactions play a central role in shaping these ecosystems, which as a whole are characterized by productivity levels that largely depend on precipitation.

The authors also identify the origin of the mechanism underlying self-organization: a process of ‘local facilitation’ among plants, set against the background of overall control by water availability. Water is the limiting resource, but short-range interactions among plants involve positive effects that are a nec-essary condition for power laws to exist. The plants create a local environment that mini-mizes water run-off and facilitates the survival of other plants and seeds (Fig. 1, overleaf).

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Figure 2 | Vegetation states in arid and semi-arid ecosystems. a, Basic interactions between different states of ecosystem patches — vegetated (V), empty (E) and degraded (D). Transitions from one state to another are possible at given rates (blue arrows), some of which are enhanced by the presence of neighbouring vegetation and the resulting improvement in physical conditions through the process of local facilitation (red arrow). b, A consequence of the nonlinear interactions between the three types of

patch is the possibility of bistable behaviour between the well-vegetated (green) state and the desert state. Theoretical models show that a sudden transition from the first state to the second can occur under a continuous change in external stress, such as decreased water or increased grazing5. Like a marble rolling over a folded surface, the ecosystem changes continuously until a critical boundary is reached. At that point, the system suddenly shifts to the new state.

Figure 1 | Mutual benefits. Semi-arid ecosystems, such as the Kalahari (pictured here), are characterized by harsh conditions dominated by water availability. In a process of positive feedback, called ‘local facilitation’4,5, plants that are specialized for such conditions create microenvironments that help other plants to survive. In consequence, neighbouring bare, degraded ground (D in Fig. 2a) can revert to fertile ground (E in Fig. 2a).

The two groups4,5 support this claim by means of modelling with cellular automata, which in both cases successfully reproduces the observed spatial patterns and their scaling-law behaviour. These computer simulations involve the construction of a regular grid of cells, each of which has a particular state. In this case, the basic model considers three possible states: vegetated (V), empty (E) and degraded (D) (Fig. 2a). The first two designate fertile patches that are or are not occupied by plants. The third refers to degraded soil that cannot be colonized by plants. These three types of patch

are related to each other through transitions governed by dynamical rules. Some transitions are affected by the presence of neighbouring vegetated patches. Here the positive effect of local facilitation is clear: it allows bare desert patches to revert to fertile soil that can later be colonized by local seedlings. This type of model leads to scaling laws under a wide range of conditions. Importantly, the work provides the first well-documented example of so-called robust criticality theory7.

An implication of these results is that it might be possible to predict the transition from

a vegetated to a desert state. It has been con-jectured that arid ecosystems might suddenly shift towards a desert condition as external conditions deteriorate2,3. The typical exam-ple of a changing condition cited is decreas-ing rainfall. But increased grazing pressure by animals has a similar impact, and Kéfi et al.5 reveal that more intense grazing leads to a departure from the power-law behaviour, with large patches becoming less and less common. Models consistently predict that such changes in size distribution might be warning signals of the approach of a transition to the desert state (Fig. 2b). The seriousness of such a pos-sibility is highlighted by the fact that some of these transitions are catastrophic and largely irreversible2.

The findings presented in these two papers4,5

are compelling and seem robust. They open up the prospect of testing previous models against reality, and of forecasting future changes in arid ecosystems. For those purposes, however, we will need a rigorous theory that can help in explaining the exact origins of the scaling behaviour and in identifying other spatial measures that are needed to properly test the accuracy of a model. Previous work has been couched in terms of average quantities7, and we require a modelling approach that allows pre-dicted scaling exponents (and their variability) to be estimated more precisely and interpreted in terms of measurable parameters (such as water availability or degree of facilitation).

Moreover, changes in plant productivity are only the first layer in understanding how the whole food web will react to external stresses, and theoretical work will also need to take account of other levels in the food web. In this context, organisms inhabiting arid regions tend

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to have high levels of genetic diversity within species, and to be endemic to their particular ecosystem. They are thus of great relevance in considering biodiversity. A fuller theory of scaling behaviour is required to provide firmer connections between predicted and observed patterns of desertification — and so to provide a better understanding of the nature of transi-tions between green and desert phases. These belong to the family of non-equilibrium phase transitions seen in several different fields of research8,9. There is a great opportunity here for interdisciplinary work that would have potentially far-reaching consequences in conservation biology. ■

Ricard Solé is in the ICREA-Complex Systems Laboratory, Universitat Pompeu Fabra, Dr Aiguader 80, 08003 Barcelona, Spain.e-mail: ricard.sole@upf.edu

1. Foley, J. A. et al. Ecosystems 6, 524–532 (2003).2. Rietkerk, M. et al. Science 305, 1926–1929 (2004).3. Kéfi, S. et al. Theor. Popul. Biol. 71, 367–379 (2007).4. Scanlon, T. M., Caylor, K. K., Levin, S. A. & Rodriguez-

Iturbe, I. Nature 449, 209–212 (2007).5. Kéfi, S. et al. Nature 449, 213–217 (2007).6. Solé, R. V. & Bascompte, J. Self-Organization in Complex

Ecosystems (Princeton Univ. Press, 2006).7. Pascual, M. et al. Phil. Trans. R. Soc. Lond. B 357, 657–666

(2002).8. Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in

Lattice Models (Cambridge Univ. Press, 2007).9. Hinrichsen, H. Physica A 369, 1–28 (2006).

Particles of antimatter might be rare, fleet-ing and seemingly unwelcome guests in our matter-dominated world, but they offer many opportunities to study new science and develop new technologies. Antimatter, and how the laws of physics apply to it, is there-fore of fundamental interest, notwithstanding the challenges of making, manipulating and storing the stuff. On page 195 of this issue1, Cassidy and Mills report the breaking of new ground — the creation of the first-ever molecule of a species of matter–antimatter atom known as positronium.

The laws of physics, as we understand them, are symmetrical: for each type of ordi-nary-matter particle there is a corresponding antiparticle. The proton has the negatively charged antiproton; the electron has the posi-tively charged positron. Such pairs of particles and antiparticles are now regularly created in laboratories around the world. But almost as soon as they are made, they disappear again with a puff and a flash of light, anni-hilating each other to leave only a trace of other particles or photons. The electron and positron, for example, annihilate into two or three photons with a total energy of 1,022 kilo electronvolts (keV) — twice the electron mass, 511 keV. These photons are nothing other than highly energetic X-rays or γ-rays, and are routinely used to characterize materials for high-speed electronics, as well as to study metabolic activity in the brain in the technique known as positron emission tomo graphy (PET).

Just as the electron and proton bind to form atomic hydrogen (H), so the electron and positron bind, albeit fleetingly, to form

a positronium atom (Ps). The existence of positronium was predicted2 in 1946 by the theoretical physicist John Wheeler, and the atom was first isolated experimentally3 by Martin Deutsch in 1951. Wheeler also posited the existence of a dipositronium molecule, Ps2, and even a triatomic variant, Ps3. It is Ps2 that Cassidy and Mills have only now succeeded in producing1 — and with it the first many-posi-tron, many-electron system to be made in the laboratory.

It is tempting to view Ps2 as a diatomic mol-ecule similar to the hydrogen molecule, H2, but there are important differences. Unlike the proton and electron in a hydrogen atom, the positron and electron in positronium have the same small mass of 511 keV. As a result of the quantum uncertainty principle, neither the electrons nor the positrons in Ps2 can be localized in the same way as the much heavier protons in H2. Thus, each of dipositronium’s four particles has one repulsive partner (of the same type) and two attractive partners (of the opposite type). The four do a merry dance around each other in a fuzzy, lumpless soup with matter and antimatter flavours4.

Cassidy and Mills performed their experi-ments1 by accumulating some 20 million positrons in a specially designed trap. They focused these positrons in a burst lasting less than a nanosecond onto a small spot on the surface of a porous silica sample. The positrons diffuse into voids in the silica, where they cap-ture electrons to form positronium atoms. Before these atoms can annihilate, they form about 100,000 Ps2 molecules on the interior surfaces of the voids. The presence of a surface is crucial — because the energy of the bound

ATOMIC PHYSICS

A whiff of antimatter soupClifford M. Surko

A molecule consisting of two electrons and two anti-electrons is similar to, but different from, the familiar hydrogen molecule H2. Its creation heralds a new chapter in the formation of matter–antimatter states.

50 YEARS AGO“British public schools and the future” — [Another] demand which parents make on the public schools is impossible to justify on educational grounds and has social, political and moral implications. Many parents…send their sons to public schools because membership of these schools will be of service to them in their future careers. Although this charge may be exaggerated — the full effects of the establishment of grammar schools under the Education Act of 1902 have not yet been seen — the public school system undoubtedly confers advantages on its products which are denied to those from State schools. A system which enables less able men to come to the top and prevents the abler from doing so cannot be justified on human, economic or moral grounds. But even this does not provide a case for abolishing schools which have so much to commend them educationally; if the wrong boys are getting into public schools, other means of selecting them must be found. From Nature 14 September 1957.

100 YEARS AGOTheoretically at least most observers admit that the adoption of the scientific method in the management of the affairs of State is a preliminary necessity if national efficiency is to be secured… There is growing evidence, also, that politicians in most countries are beginning to realise that statesmanship is no exception to this rule, but, like other skilled labour, is most satisfactory when conducted on scientific principles. But whether British statesmen appreciate this truth to the same extent as those of other great nations is a matter of grave doubt. Their education generally has been of such a character as to leave them with a colossal ignorance of science and scientific methods; and it is only by overcoming the bias received at the public school and university that most of them come to understand the modern outlook.From Nature 12 September 1907.

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