Andrzej Stasiak and John H. Maddocks- Best packing in proteins and DNA

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from different fish species were found toadsorb at prism, secondary prism and pyra-midal orientations, but not on the basalplane. This explains why the ice grows asneedles parallel to the c-axis of the unit cell.)

Applications of antifreeze proteins, some

based on their being potent inhibitors ofrecrystallization 6, have been sought for main-taining texture in frozen foods, improvingstorage of blood, tissues and organs, cryo-surgery, and protecting crops from freezing.But just what is it that gives these moleculestheir activity? One side of them is relativelyhydrophobic and the other relatively hydro-philic, and there has been disagreement overwhich side binds to ice7. Much of the evidencehas come from freezing hysteresis measure-ments on synthesized versions of the naturalproteins with amino-acid substitutions intro-duced into them.

The observations of Sidebottom et al.1

raise the issue of the relationship between thetwo main effects of antifreeze proteins: freez-ing hysteresis (protection from freezing) andrecrystallization inhibition (in organisms,presumably protection from freezing dam-age once freezing does occur). The generalunderstanding has been that, like freezinghysteresis, recrystallization inhibition is adirect consequence of immobilization ofsolidliquid interfaces in partially frozensamples. Sidebottom and colleagues resultssuggest, however, that the two effects areuncoupled; if they are, then that understand-

ing is probably flawed.The molecular structures of antifreezepeptides described by Graether et al.2 andLiou et al.3 are the first from insects to becompletely characterized. Both show beauti-ful structural matches between ice and thehydrophilic groups on one side of the mol-ecule. Most of the fish antifreeze proteinsshown to fit onto ice were linear molecules.Here, however, the arrays are two-dimen-sional in both cases, so there can now be littledoubt that it is the hydrophilic sides that stickto the ice, leaving the relatively hydrophobicsides in contact with liquid water.

The structures2,3 are similar in being -helices, showing a new way in which peptide-chain folding can bring about the structuralmatch that seems to be necessary for anti-freeze activity. No doubt more ways will befound in the future. The very active spruce-budworm antifreeze of Graether et al.2 is novelin yet another respect, in that the ice grows asbasal plates from solution rather than as nee-dles; this may be the first case of an antifreezeprotein binding to the basal face of ice.

What of more general issues? The usualapproach to antifreeze adsorption to ice hasbeen to analyse the bonding between the anti-freeze molecules and ice. In a different view-point, which I favour, the rather large anti-freeze molecules can be viewed instead as verysmall particles. Their equilibrium positions atthe icewater interface can then be considered

in terms of interfacial energies, and insightsfrom theoretical studies of the interactionsof particles with growing crystals can be

applied8,9

. The important parameter is thedifference in energy between the antifreezeice and antifreezewater interfaces, and theprinciple is minimization of the total inter-facial energy. Accordingly, even if an anti-freeze molecule has no preference betweencontacting ice or water (its interfacial energieswith them are the same), it may adsorbstrongly because the area of the icewaterinterface is reduced when it resides there.

Another point is that antifreeze proteinsmust match the ice structure well enough toprevent water molecules diffusing into theinterface and pushing the protein ahead as

the crystal grows. This pushing ahead is thecrystallization pressure familiar from manyfrost-heaving studies10, in which the particlecan maintain its equilibrium position at thesurface as the crystal grows. Water moleculesdiffuse behind the particles, pushing themahead and forming ice lenses in freezingsoil. Considerable pressures are generatedin this way, even over a supercooling temper-ature range of 1 C or less.

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NATURE | VOL 406 | 20 JULY 2000 | www.nature.com 251

With an array of antifreeze moleculesanchored permanently to the ice and unableto be pushed ahead, ice growth only occurs ata supercooling sufficient to engulf the mol-ecules, which then depends on their spacingon the ice surface. The depressed freezing

point presumably comes from the Kelvineffect, the local decrease in freezing point atthe curved ice interface that bulges forwardbetween the adsorbed antifreeze molecules(Fig. 1). The three-dimensional geometry ofthis is complicated, however, and has notbeen worked out.

The question that seems hardest to answeris that, if the adsorption is effectively perma-nent (as is necessary to stop the ice growth),why does the observed freezing point dependon the solution concentration? Especially atlow solution concentrations, one wouldthink it ought to depend more on the time

available for antifreeze molecules to adsorb,but experimentally it does not seem to.All in all, the three papers13discussed here

do indeed provide thought-provoking infor-mation on the structure and behaviour ofantifreeze proteins. But clearly there remainsa lack of consensus on some of the fundamen-tal issues to do with antifreeze action. sCharles A. Knight is at the National Center for

Atmospheric Research, PO Box 3000, Boulder,

Colorado 80307-3000, USA.

e-mail: [email protected]

1. Sidebottom, C. et al. Nature406, 256 (2000).

2. Graether, S. P. et al. Nature406, 325328 (2000).

3. Liou, Y.-C., Tocilj, A., Davies, P. L. & Jia, Z. Nature406,

322324 (2000).4. DeVries, A. L. Science172, 11531155 (1971).

5. Knight, C. A., Cheng, C. C. & DeVries, A. L.Biophys. J. 59,

409418 (1991).

6. Knight, C. A., DeVries, A. L. & Oolman, L. D. Nature308,

295296 (1984).

7. Haymet, A. D. J., Ward, L. G. & Harding, M. M.J. Am. Chem.

Soc. 121, 941948 (1999).

8. Uhlmann, D. R., Chalmers, B. & Jackson, K. E.J. Appl. Phys. 35,

29862993 (1964).

9. Asthana, R. & Tewari, S. N.J. Mater. Sci. 28, 54145425 (1993).

10.Jackson, K. A. & Chalmers, B.J. Appl. Phys. 29, 11781181 (1958).

Mathematics

Best packing in proteins and DNAAndrzej Stasiak and John H. Maddocks

Apragmatic way to store a rope is to coil itloosely, and drop it in an appropriatelyshaped box. But if you are extremely

parsimonious with space, as nature often is,this solution is suboptimal there is a lotof unused space in the centre of the coil.So what is the longest piece of rope you canpack into a particular box? Such questionsof optimal packing are addressed by Maritanet al.1 on page 287 of this issue. Some ofthe optimal shapes they find are the fam-iliar, naturally occurring, helical structuresof proteins and DNA.

Optimal packing is a classic area of dis-

crete geometry. Perhaps the best knownexample is Keplers problem of the densestpacking of identical spheres2, which hasimportant applications in such fundamentalphysical phenomena as crystallization andmelting of condensed matter. But mostphysical objects cannot be modelled as sim-ple spheres. In particular, many polymers,including DNA molecules and portions ofproteins, might more reasonably be mod-elled as deformable tubes. For example, ifwe consider how DNA molecules could bepacked within a small virus3, we arrive at thequestion of optimal packing of tubes. Unlike

Figure 1 Two-dimensional model of ice-growth

inhibition by antifreeze molecules (red). One

side of each molecule sticks to the ice, which

grows between the molecules. But the curvature

lowers the local freezing point through the

Kelvin effect.

Supercooledwater

Ice

2000 Macmillan Magazines Ltd


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spheres whose geometry is fixed, the optimalshape of the centre line of a deformable tubemust be found. Consequently it is not a

simple matter to find the optimal packingof even a single tube.This modelling problem makes sense

only if the tube, like any real rope or molec-ule, has a definite non-zero thickness. Figure1a shows a way of packing (presumably opti-mally) a tube of length 4 and uniformthickness r(or radius r, but be aware thatsometimes thickness means diameter) into acubic box of side length 4r. The centre line ofthe tube is a saddle-like trajectory that is builtfrom four semi-circles. The point of thisexample is that, whenever the parameters ofthe tube and the box are less simply related,

it is hard to even guess the optimal packing.This is the general topic addressed by Mari-tanet al.1 in their numerical simulations.

The basic issue to be overcome in numeri-cal simulations is that the idea of thickness isnot quite as simple as it might first seem4,5. Inthe tube model a non-zero thickness has twopossible effects: first, the centre line cannotbe bent too sharply; and second, any twopoints that are far apart along the curve can-not be too close to each other in space. So, fora given centre line, the maximum possiblethickness is governed by either local bendingor by non-local points of closest approach,or, apparently exceptionally, by both con-ditions simultaneously. For the centre lineof the tube shown in Fig. 1a, both of theseconditions are simultaneously realized atevery point along the red centre line.

Maritan et al. characterize the thicknessof strings or tubes in terms of a quantitycalled global radius of curvature6. For anycurve made up of discrete straight lines join-ing node points, the thickness is taken to bethe minimal radius of all possible circlespassing through any three nodes of the curve.If the smallest radius is achieved by threeadjacent nodes, the thickness is controlledby local bending, whereas if the nodes onthe minimal circle are not all adjacent, thethickness is governed by the non-localcondition of closest approach.

Armed with this tool, Maritan et al. use a

Monte Carlo algorithm to move the nodes ofa centre line with a given length into an opti-mal shape, which maximizes thickness when

subject to one of several compactness con-straints. Perhaps the simplest compactnesscondition is that the centre line is completelycontained in a given box. In spirit, their pro-cedure is similar to that of earlier studies (seeexamples in ref. 7) but, with the exceptionof ref. 8, all previous work has looked at theoptimal shapes of closed, knotted curves,much beloved of mathematicians. (A curveis closed if its two ends are joined or gluedtogether to form a loop. A loop is knotted if itcannot be smoothly deformed to a simplecircle without cutting.) Indeed, from a solelymathematical point of view, Maritan et al.s

contribution is the elegant, and in retrospectdelightfully obvious, idea that the con-straints of closure and knotting can usefullybe replaced by one of several compactnessconditions on the centre line. (In the absenceof any compactness constraint at all, theoptimal centre line is merely a straight lineof infinite thickness.)

What about the physical implications ofthe simulations presented by Maritan et al.?Perhaps their most intriguing results arisewhen they impose local compactness condi-tions that are independent of external con-straints such as a box. They state that, for abroad class of local constraints, the optimalcentre line is a particular helix in which theratio of the pitch (or period)p to radius rissuch that the bending and closest-approachconstraints are realized everywhere simul-taneously (p/r2.512). Maritan et al. thenconsider crystal structures of various -helical polypeptides (one of the basic struc-tural motifs of proteins), and show that thehelices formed by the -carbons in thepolypeptide backbone have almost the sameoptimal shape as found in their simulations(Fig. 1b).

Are optimal packings of tubes related toother basic structural motifs in biology? Forexample, does the DNA double helix alsoinvolve optimally packed tubes? A relatedproblem studied by Pieranski8 involvesfinding the densest coiling of two identical

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252 NATURE | VOL 406 | 20 JULY 2000 | www.nature.com

Figure 1 Tightly packed tubes. a, Packing of a tube into a box whose sides are four t imes the radius

of the tube. b, A single helical tube whose centre line has the critical pitch/radius ratio of 2.512.c, Double-helical tubes in which the ratio between pitch and tube radius has the critical value 2.

Helices with parameters as in b closely resemble those observed in -helical segments of proteins and

are also obtained in numerical simulations of optimal packing of individual tubes by Maritan et al.1,

whereas the parameters of the double-helical tubes in c closely match those for DNA.

P.PIERANSKI/A.DOBAY

100 YEARS AGO

It is a remarkable sign of the times when the

head of a firm principally distinguished for

the introduction into this country of American

methods of dealing with drugs, i.e. by putting

them up in new and convenient shapes and

doses, goes out of his way to fit up extensive

research laboratories. This is what Mr.

Wellcome has done... A well-built modern

house has been secured at No. 6 King Street,

Snow Hill, and has been converted into a

series of three commodious and well-fitted

laboratories, a library and office, and a

store-room and workshop. Each laboratoryis self-contained and each is connected with

the other and with the directors office by

means of telephones Mr. Wellcome intends

to carry on his laboratories in no narrow

spirit; this means, I presume, that he has

other views then the conversion of his

business into a chemical manufacturing

concern. Though much work is done towards

the perfection of the firms preparations, time

has been found for several researches which

have been published, and other work of this

kind is in hand All interested in the

advance of chemistry, whether pure or

applied, will wish Mr. Wellcome success,and also that he may find imitators among

the numbers of firms who are meditating an

advance in the direction of a more scientific

method of conducting their manufactures.

From Nature19 July 1900.

50 YEARS AGO

Crystalline inclusion bodies in tobacco plants

infected by tobacco mosaic virus have been

known since 1903, and circumstantial

evidence has made it appear likely that these

crystals are composed largely of the virus

protein. The present work makes it appear

even more likely than before that the crystals

are pure virus protein, and shows the crystals

to be of considerable interest from several

quite different but related points of view. On

account of the exceptionally large dimension

of the protein particle, it has been possible

for the first time to make, in part at least, a

structure analysis of the crystal using visible

light in a manner analogous to that of X-ray

diffraction. As a result, it has been possible

to settle the controversial question of the

length of the rod-shaped virus particle in the

living plant. Also, the interpretation of the

appearance of the crystals, as seen with

the microscope, leads to a theory of the

formation of images of three-dimensional

objects. M. H. F. Wilkinset al.

From Nature22 July 1950.



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interwound tubes forming a double-helicalstructure (Fig. 1c). As noted by Pieranski, forthis problem there is a critical pitch-to-radiusratio (p/r26.28) above which the lineof contact between the tubes is straight, andbelow which it is helical. The crystallographic

diameter of the classic DNA double helix is23.7 ngstroms (), which would correspondto a radius for each of the two idealized heli-cal tubes of 23.7/45.92 (here the sugar-phosphate backbones lie on the outside ofthe helical tubes). The generally acceptedvalue for the pitch is 10.5 base pairs or 35.7 ,sop/r35.7/5.926.03, which is within 4%of 2. Is this an insight or just a coincidence?We suspect it is the former. sAndrzej Stasiak is at the Laboratoire dAnalyse

Ultrastructurale, Universit de Lausanne, CH-1015

Lausanne, Switzerland.

e-mail: [email protected]

John H. Maddocks is in the Department of

Mathematics, Swiss Federal Institute of Technology

Lausanne, CH-1015 Lausanne, Switzerland.

e-mail: [email protected]. Maritan, A., Micheletti, C., Trovato, A. & Banavar, R. B. Nature

406, 287290 (2000).

2. Sloane, N. J. Nature395, 435436 (1998).

3. Odijk, T. Biophys. J. 75, 12231227 (1998).

4. Katritch, V. et al. Nature384, 142145 (1996).

5. Litherland, R., Simon, J., Durumeric, O. & Rawdon. E. Topol.

Appl. 91, 233244 (1999).

6. Gonzalez, O. & Maddocks, J. H. Proc. Natl Acad. Sci. USA 96,

47694773 (1999).

7. Ideal Knots(eds Stasiak, A., Katritch, V. & Kauffman, L. H.)

(World Scientific, Singapore, 1998).

8. Pieranski, P. in Ideal Knots (eds Stasiak, A., Katritch, V. &

Kauffman, L. H.) 2041 (World Scientific, Singapore, 1998).

individually and in combination. They thenanalysed the resulting embryos.

Surprisingly, both groups find that -spectrin is not essential for many of theprocesses suggested from previous investi-gations. Their worms do not lose general

membrane integrity, and synaptic vesiclesin nerve endings are clustered normally.The cellular secretory pathways appearunimpaired: the worms deposit cuticle andsecrete collagen and components of thebasement membrane as usual. The cells thatshould be polarized are polarized, suggest-ing that -spectrin has no primary func-tion in this process. Ankyrin is a connectingprotein that is known to link spectrin to avariety of transmembrane proteins, includ-ing cell-adhesion molecules of the L1 family.Moorthyet al.2 find that, in their worms,ankyrin apparently binds to L1 adhesion

molecules normally.However, the worms are paralysed andhave a dumpy appearance1. The maindefects lie in the organization of muscle andnerve cells. The number of neurons is nor-mal, but the patterns of axon outgrowthare altered1,2, with few axons finding theirtargets. The muscle cells have disruptedsarcomeres (contractile units)1,2, and thesarcoplasmic reticulum an intracellularcalcium store is generally missing. Itseems that the dumpy appearance is causedby a failure of the muscles to spring back aftercontracting. The muscle defects may arise

during the course of contraction, as wormswith both the unc-70mutation and the unc-54 mutation, which results in a failure inmuscle contraction, show less severe charac-teristics than the unc-70mutants1.

What, then, has become of the antici-pated functions of spectrin? Dubreuil et al.3

have looked at fruitflies that lack -spectrin,and provide some clues to the role of thisprotein in cell polarization. These flies live just long enough for copper cells in theintestine to be analysed. In the mutant flies,the copper cells are polarized and ankyrinassociates with the membrane of these cellsas normal. So spectrin is not the main driverof cell polarization. It has been suggested8

that what drives cell polarization is the con-tact of transmembrane cell-adhesion mol-ecules with either their extracellular matrixligands or their counterparts on other cells.So, in this case, a transmembrane L1-typeadhesion molecule binds to its ligand andrecruits ankyrin. But a transmembrane ionpump, the Na+/K+ ATPase, does not accu-mulate as normal in the plasma membraneof the mutant copper cells. Why is this?

Spectrin has been described as a protein-sorting machine9. The new data13 showthat this model can be taken a step further:spectrin not only sorts, but also collects,proteins at the plasma membrane. Geneticevidence indicates that spectrin functionsas a tetramer (Fig. 1). Each tetramer has two

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NATURE | VOL 406 | 20 JULY 2000 | www.nature.com 253

We thought we knew what spectrindoes. Is it not the elastic, membrane-bound protein that prevents red

blood cells from rupturing as they circulatein the bloodstream? And does it not have thesame supporting function in other cells? Thesecond assumption has seldom been ques-

tioned over the past two decades, but has justbeen overturned by the power of experimen-tal genetics, as described in three reports13 intheJournal of Cell Biology. The results maybear on human diseases such as musculardystrophy.

Red blood cells with deleterious muta-tions or deficiencies in spectrin have weak-ened outer membranes, are misshapen andlack resilience4, so it is not surprising thatspectrin has long been assumed to be neces-sary for membrane integrity. Over the years,other functions have been ascribed to spec-trin as well. For example, it is thought to be

involved in generating the polarized mor-phology of epithelial cells, and to have rolesin the functioning of the Golgi complex an organelle involved in protein secretionfrom cells and the organization of synap-tic vesicles (see, for example, refs 57).

To probe spectrins function, Hammar-

lund et al.1

and Moorthyet al.2

have takenadvantage of the simplicity of the genome ofthe nematode worm Caenorhabditis elegans.This genome, like that of the fruitflyDrosophila melanogaster, has only threespectrin genes, encoding , and H formsof the protein (Fig. 1). Hammarlund et al.1

looked at worms with a mutation calledunc-70, which they discovered to lie in the-spectrin gene. Moorthyet al.2 used a nowcommon technique for blocking proteinexpression: they injected double-strandedRNA into the worms gonad to block theexpression of one or more spectrin subunits

Cell biology

A protein accumulatorJennifer C. Pinder and Anthony J. Baines

Figure 1 The structure of spectrin12,13. Spectrin is a giant molecule comprising and subunits, ofwhich there are different types. For example, Drosophilaand C. eleganshave , andH forms of

spectrin. (H-Spectrin is a heavy form of-spectrin.) The and subunits associate to form an

elongated ()2 tetramer. Lying near to the interior surface of the plasma membrane, spectrin forms

a hexagonal lattice, the nodes of which are crosslinked by the cytoskeletal protein actin. This network

is attached to the membrane in several ways, for example through the connecting protein ankyrin.

-Spectrins have binding sites for-spectrin, actin and ankyrin (Ank). The pleckstrin-homology (PH)

domain binds to certain membrane lipids. The Src-homology-3 (SH3) domain of-spectrin probably

accumulates signalling molecules close to the membrane. EF-hands are calcium-binding sites. ABD,

actin-binding domain; C, carboxy terminus; N, amino terminus.

PHAnk

C

C

C

C

ABD

EF

NActin

N N

SH3

N

Spectrin

Spectrin Spectrin

Spectrin


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Andrzej Stasiak and John H. Maddocks- Best packing in proteins and DNA