For nearly 40 years, one of the classicicons of modern nonlinear dynamicshas been the Lorenz attractor. With its

intriguing double-lobed shape and chaoticdynamics, the Lorenz attractor has symbol-ized order within chaos (Fig. 1). The onlyproblem is: does it exist? Mathematicianshave lacked a rigorous proof that exact solu-tions of the Lorenz equations will resemblethe shape generated on a computer bynumerical approximations, and they alsocould not prove that its dynamics are gen-uinely chaotic. Perhaps the calculationsshowed something that merely looked likechaos — a numerical illusion. The smartmoney has always been on chaos in theLorenz system being real, but the rigoroustechniques of dynamical mathematics wereunable to prove it. Until last year, that is,when Warwick Tucker of the University ofUppsala completed a PhD thesis showingthat Lorenz’s equations do indeed define arobust chaotic attractor. The proof has sincebeen published (W. Tucker, C. R. Acad. Sci.328, 1197–1202; 1999), and an excellentsummary has been provided by MarceloViana (Math. Intell. 22, 6–19; 2000).

Tucker’s work is hugely significant, not

just because it provides the Lorenz attractorwith a solid foundation, but because histechniques will be widely applicable. At last,the embarrassing gap between what wethink we know about a nonlinear dynamicalsystem from numerical simulations, andwhat we actually know in full logical rigour,is starting to close. At the moment the meth-ods are limited to dynamics in three dimen-sions, but after Tucker’s breakthrough thatmay well change. Dimensions greater thanthree are of considerable interest because the dimension of a dynamical system is notthe dimension of ordinary space, but thenumber of variables in the equations. Forexample, the motion of a three-body systemconsisting of the Earth, Mars and a spaceprobe requires six variables for each body —three of position and three of velocity — andso is an 18-dimensional dynamical system.

Why bother with rigorous proofs? Surelyany practical implications of the equationsare already embodied in the numericalresults — do we need to be obsessed with logical rigour? Yes, we do. There are severalreasons for taking numerical solutions ofnonlinear differential equations with a pinchof salt. Numerical methods are approxima-

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resilient beast. On all counts, the project hasmet its goals.

Pseudomonas aeruginosa has a largegenome, about one-third larger than that ofEscherichia coli. In fact it has about as manygenes as the budding yeast Saccharomycescerevisiae, and about half as many as thefruitfly. The large size of the P. aeruginosagenome allows for a wide diversity of genes.Although this is no great surprise, the moredetailed annotation of the genome has beenquite revealing. Two features that jump outcan be summarized as regulation, regulationand more regulation, and pumps, pumpsand more pumps.

Stover et al. discovered a large number ofgenes — between 8% and 10% of the predict-ed total of 5,570 genes — that encode proteinswith sequences similar to those of known reg-ulators of gene expression. The authors note acorrelation between genome size and the per-centage of predicted regulators in sequencedbacterial genomes: the larger the genome, thehigher the percentage of regulatory genes.For example, Helicobacter pylori — a highlyspecialized bacterial pathogen that can colo-nize the acidic environment of the humanstomach — has 1,709 predicted genes. Fewerthan 20 of these (about 1%) are expected to beregulatory. Smaller genomes have even lowerpercentages of regulatory genes. The incredi-ble potential for P. aeruginosa to turn genes on and off according to the conditions in which it finds itself is consistent with its nutritional versatility.

The P. aeruginosa genome also encodes

a number of pumps, which may be the keyto the bacterium’s ability to withstand anti-biotics in its human host. It has beensuggested that Gram-negative bacteria, agroup to which P. aeruginosa belongs, canresist the lethal effects of many antibioticsby pumping them out of their cells fasterthan the chemicals can accumulate. Mem-bers of the RND protein family of ‘multi-drug efflux’ pumps can render bacteriaimpervious to antibiotics4. It is not clearhow or why these pumps evolved; perhapsthey were first needed to eliminate nat-urally occurring environmental toxins. Wealready knew of four RND pumps in P.aeruginosa (E. coli has the same number,whereas Mycobacterium tuberculosis, thepersistent pathogen that causes tuberculo-sis, has none). The sequencing project hasnow revealed six more.

When and where are these pumps active?One idea is that they might be expressedwhen P. aeruginosa exists as a biofilm. Theseare stationary communities of bacteria thatare enclosed by a self-produced extracellularmatrix. When bacteria exist as biofilms, theycan resist antimicrobial agents. A hypothesisto explain the persistence of P. aeruginosa inthe lungs of a cystic fibrosis patient is that itmay establish a biofilm in this environment5.

The abundance of regulatory genes andquestions about the regulation of antibioticpumps in P. aeruginosa call for investigatorsto look into gene expression in this bac-terium. To this end, the US Cystic FibrosisFoundation is again taking a uniqueapproach, by building on its initial invest-ment in P. aeruginosa genomics and under-writing the cost of constructing a P. aerugi-nosa gene-expression array. The charityplans to make the array available to inter-ested researchers at a cost consistent with the budgets of academic scientists. It has also created a Cystic Fibrosis BioinformaticsCenter, to support scientists using the arrays.One hopes that the genomic informationreported by Stover et al.1, and the informa-tion to come from the gene-expressionanalysis, will allow us finally to tame thistesty pathogen. !

E. Peter Greenberg is in the Department ofMicrobiology, College of Medicine, University ofIowa, Iowa City, Iowa 52242-1109, USA.e-mail: everett-greenberg@uiowa.edu1. Stover, C. K. et al. Nature 406, 959–964 (2000).

2. Fleischmann, R. D. et al. Science 269, 496–512 (1995).

3. The Pseudomonas Genome Project.

http://www.pseudomonas.com/

4. Nikaido, H. Clin. Infect. Dis. 27, S32–S41 (1998).

5. Costerton, W. C., Stewart, P. & Greenberg, E. P. Science 284,

1318–1322 (1999).

Figure 1 The rod-shaped Pseudomonasaeruginosa on cultured epithelial cells from thehuman respiratory tract. Each P. aeruginosa cellmeasures about 0.5 by 2–3 !m. The bacteriumshows extraordinary nutritional versatility. This might require the large number of genes in its genome (the sequence of which is nowpublished1) that are predicted to be involved inregulating gene expression.

Mathematics

The Lorenz attractor existsIan Stewart

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tions, and chaotic systems are highly sensi-tive to approximations; it is well known thatnumerics can sometimes give seriously mis-leading results. But the deepest reason is thatuntil we can prove what our computers seemto be telling us, then we are ignoring a hugegap in our mathematical technique. Oftensuch a gap is a clue that important conceptualideas are lurking nearby, as is the case here.

The Lorenz attractor dates from 1963,when the meteorologist Edward Lorenz pub-lished an analysis of a simple system of threedifferential equations that he had extractedfrom a model of atmospheric convection. Hepointed out that they possess some surpris-ing features. In particular, the equations are‘sensitive to initial conditions’, meaning thattiny differences at the start become amplifiedexponentially as time passes. This type ofunpredictability is a characteristic feature ofchaos. Conversely, there is also ‘order’ in thesystem: numerical solutions of the equa-tions, plotted in three dimensions, consist ofcurves winding round and round a curioustwo-sheeted surface, later named the Lorenzattractor.

The geometry of the attractor is closelyrelated to the ‘flow’ of the equations — thecurves corresponding to solutions of the differential equations. There is an unstableequilibrium, a saddle point, at the origin.The curves repeatedly pass this point, and arepushed away to the left or right, only to circleround to pass back by the saddle. As theyloop back, adjacent curves are pulled apart— this is how the unpredictability is created— and can end up on either side of the saddle. The result is an apparently randomsequence of loops to the left and right.

The central technical issue in proving thatthe Lorenz system is a chaotic attractor istranslating these statements into suitablyprecise mathematics. Tucker’s proof com-bines two main ideas. One is a conceptualcharacterization of chaotic attractors interms of ‘singular hyperbolicity’, introducedby C. Morales, M. J. Pacifico and E. Pujals (C. R. Acad. Sci. 326, 81–86; 1998). Previouswork on dynamical systems had concentratedon ‘hyperbolic’ systems, where the flow ofthe equations can always be split into a set ofcontracting directions and a complementaryset of expanding ones. But the Lorenz systemis not hyperbolic. Singular hyperbolicityreplaces the idea of expanding directions inthe flow by the condition that part of the flowshould expand in volume. If some sides of abox expand and others contract, the volumemay nonetheless expand if the amount ofexpansion beats the amount of contraction.So singular hyperbolicity is less restrictivethan hyperbolicity.

Tucker’s other important idea is usingcomputer calculations in a rigorous way toestablish certain features of the geometry ofthe solutions of differential equations —normal numerics plus precise error esti-

mates. Tucker obtains a rigorous approxima-tion to the way curves loop back towards theorigin by following the time evolution of lotsof tiny boxes (up to 10,000). But numericalsolutions of differential equations encounterproblems near saddle points, because theflow slows down exponentially fast. To getround this difficulty, Tucker uses the well-

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NATURE | VOL 406 | 31 AUGUST 2000 | www.nature.com 949

Figure 1 The Lorenz attractor was first derivedfrom a simple model of convection in the Earth’satmosphere. Previously, the Lorenz attractorcould be generated only by numericalapproximations on a computer, as shown here.Now we have a rigorous proof that confirms itsexistence.

established technique of a ‘normal form’.Near an equilibrium, any differential equa-tion can be transformed into an equationthat can be solved by an exact formula, to ahigh degree of approximation. This formulacan be used instead of the numerical proce-dure whenever the flow gets too close to theequilibrium.

The task then reduces to finding a set ofboxes such that curves starting within one box eventually return to a box (usually a different one), which in effect says that the collection of boxes forms an attractor. Somefurther technical conditions, related to singu-lar hyperbolicity, are required to establish thatthe flow is chaotic. The search for suitableboxes begins with an informed guess based on raw numerics; any boxes that cause troubleby growing too fast are subdivided until theycease to cause trouble. When the subdivisionprocess of all boxes stops, the proof is com-plete. Finding a suitable collection of boxestook about 30 hours on a fast computer.

Technicalities aside, the main idea here isthat, with care, naive numerics can be usedwith precise error estimates to establish sig-nificant features of the flow of a nonlineardifferential equation. When these featuresare combined with appropriate conceptualinsights, the existence of chaotic attractorsbecomes irrefutable. So, thanks to Tucker,dynamical systems theorists can at last stopworrying about whether their most potenticon might suddenly fall apart. And Lorenz’soriginal insight, that the strange behaviourof his equations was not a numerical artefact,can no longer be disputed. !

Ian Stewart is at the Mathematics Institute,University of Warwick, Coventry CV4 7AL, UK.e-mail: ins@maths.warwick.ac.uk

Malaria

Channelling nutrientsKiaran Kirk

Malaria, an infectious disease responsi-ble for an estimated 300 million to500 million clinical cases and 1.5 mil-

lion to 2.7 million human deaths each year1,is caused by a single-celled parasite thatinvades the red blood cells of its host. In the48 hours after it invades a red blood cell, theparasite grows to many times its original size,and then divides to produce 20–30 new para-sites. To fuel this high rate of growth andmultiplication, the malaria parasite needsnutrients from outside the infected cell.However, a normal red blood cell is unable totake up at least some nutrients fast enough tosatisfy the parasite’s voracious appetite. Onpage 1001 of this issue2, Desai and colleaguesprovide new insights into how this problemis solved. They describe a versatile channel

that is found in the outer membrane ofinfected (but not uninfected) red blood cells,and which helps to supply nutrients to thehungry parasite.

It has long been known that following the invasion of a human red blood cell byPlasmodium falciparum — the most virulentof the four malaria parasites that are infec-tious to humans — the membrane of theinfected cell undergoes a dramatic increase in its permeability to a range of small sol-utes, both charged and uncharged3–6. Thepathways responsible for the increased permeability have previously been shown to have a strong preference for negativelycharged ions (anions) over positively chargedions (cations), and to be blocked by drugsknown to inhibit anion-selective channels6

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