Are All Things Either Animal Vegetable Or Mineral
Interface Focus. 2015 Aug half dozen; 5(iv): 20150027.
Crystals: animal, vegetable or mineral?
Abstract
The morphologies of biological materials, from body shapes to membranes within cells, are typically curvaceous and flexible, in contrast to the angular, facetted shapes of inorganic thing. An alternative dichotomy has information technology that biomolecules typically assemble into aperiodic structures in vivo, in contrast to inorganic crystals. This paper explores the development of our understanding of structures across the spectrum of materials, from living to inanimate, driven by those naive beliefs, with particular focus on the development of crystallography in materials science and biology. The thought that in that location is a clear distinction between these two classes of matter has waxed and waned in popularity through past centuries. Our current understanding, driven largely past detailed exploration of biomolecular structures at the sub-cellular level initiated by Bernal and Astbury in the 1930s, and more recent explorations of sterile soft thing, makes it clear that this is a faux dichotomy. For example, liquid crystals and other soft materials are common to both living and inanimate materials. The older picture of disjoint universes of forms is better understood as a continuum of forms, with pregnant overlap and common features unifying biological and inorganic matter. In addition to the philosophical relevance of this perspective, at that place are important ramifications for science. For example, the debates surrounding actress-terrestrial life, the oldest terrestrial fossils and consequent dating of the emergence of life on the Earth rests to some degree on prejudices inferred from the supposed dichotomy between life-forms and the balance.
Keywords: crystallography, liquid crystals, course
1. Introduction
The title of the coming together 'Bioinspiration of New Technologies' which led to this paper bows to the prevailing raison d'être of modern science: in service of modern technology. Surely, the lessons of billions of years of evolution are worth applying to the pattern and manufacture of new materials and machines. Given my own interest in fundamental inquiry (which after all underpins all game-irresolute breakthroughs in engineering science) the following bug came to mind on reflecting on this theme:
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(1) What is biological science?
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(2) How do the concrete sciences inform biological science?
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(3) How does biology inform the physical sciences?
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(4) Are the biological and physical universes distinct?
These cardinal questions, which guide the newspaper, lead to conclusions that—I hope—help to shed some light on how we are to go forward as physical and biological scientists.
ii. Biological grade
'Animal, vegetable or mineral' was a game we played equally children. Someone thought of an object: mayhap a cloud, or a motorcar bike, or a kangaroo. The object was to identify the object with as few questions as possible, answered by 'yeah' or 'no' only. A simple starting question, that narrowed things down pretty apace, was to enquire 'Is it animal, vegetable or mineral?' The game has illustrious antecedents. In the eighteenth century, the pioneering taxonomist, Linnaeus (now known chiefly for his piece of work in constitute classification), catalogued the consummate spectrum of the material world into three Kingdoms: Regnum Animale, Regnum Vegetabile and Regnum Lapideum [one]. Apparently, fauna and vegetables are living, minerals not. And the clues to cataloguing objects into one of those iii classes seem clear. For example, living objects, whether beast or vegetable, are typically sinuous, curved and soft; while sterile minerals are typically angular and hard (figure 1).
The intuitive view of abiotic or non-living and living materials. Whereas biological forms are curved, inorganic materials are typically facetted and divisional by flat faces. (Image courtesy of JuanManuel Garcia Ruiz).
And so, for example, 'biomorphic' art, design and architecture, adult past the artists Jean Arp, Yves Tanguy and Joan Miró in the concluding century characterizes living forms as endowed with curvatures, and seemingly more liquid than solid. A cute contempo example is a sculpture by the Swedish artist Eva Hild [2]. These forms echo the words of the ancient Chinese text Dao De Jing: 'What is supple and yielding goes with life; what is strong and difficult goes with death' [3]. Examples are shown in figure 2.
Biomorphic or inorganic sculptural forms? (a) Jean Arp, Fruit du Pagode (1949); photo copyright Tate, London (2015); (b) Stack sculptures by Donald Judd (1970s), from [iv]; (c) Eva Hild, Funnel Loop 1084 (2007) [2]. (Image courtesy of Eva Hild).
A naive answer to the question 'What is biological science?' emerges from these artistic reflections on biomorphology. Biological science is characterized by curvature. By contrast, the earliest morphological studies of mineral crystals in the seventeenth century past the Danish founder of crystallography, Steno, recognized that crystals are characterized by fixed angles between flat faces. The characteristic flat cleavage faces of crystals were understood by Kepler, Hooke, Huygens and others as arising from regular, ordered arrangements of tiny atoms, much like oranges stacked in a fruit shop.
Those theories were dramatically confirmed with the development of atomic scale crystallography past von Laue and the Braggs, virtually exactly 100 years agone. That development arose from the discovery of X-rays, and characteristic diffraction patterns formed by shining X-rays through crystals, starting time observed by Walter Friedrich and Paul Knipping in 1912. In the hands of the Braggs, and their students, notably William Astbury, Desmond Bernal and Kathleen Lonsdale i crystallography quickly uncovered the atomic arrangements in many inorganic minerals and (later) organic molecular crystals. Crystals are indeed extremely ordered and geometrically rigid stackings of atoms, thereby confirming the earlier ideas of Steno and his successors. While most of Braggs' co-workers continued their explorations of the worlds of mineral and organic crystal, two of their brightest, Astbury and Bernal, decided to explore the biological world via X-ray diffraction.
Diffraction relies on highly ordered, indeed crystalline (or quasi-crystalline) arrangements of scattering constituents, namely atoms or, via small-angle Ten-ray diffraction, molecules. (Here I use the term 'diffraction' in the conventional sense of moving ridge interference producing detached diffraction spots, in contrast to 'scattering', which gives diffuse intensity distributions in reciprocal space. Given contempo developments with very high powered light sources such every bit Ten-ray free-electron lasers, this stardom is fading, with the appearance of 'nano diffraction' techniques [6]). So the question of whether crystallography is helpful in agreement biological structures is worth request. Every bit the Russian theoretician of crystalline symmetries, Fedorov, said 'Crystallisation is decease' [three]! In spite of the perceived gulf between animals and minerals, Bernal and Astbury pushed on, and decided to probe proteins, common to all biological species. They split the potentially unending task into two: Astbury headed off to written report fibrous proteins in his 10-ray apparatus, while Bernal decided to explore globular proteins. Fedorov'south dictum seemed to apply: poly peptide diffraction was a messy affair, dominated by diffuse handful rather than distinct sharp 'Bragg' spots. In dissimilarity to the clean, characteristically 'spotty' diffraction patterns from highly crystalline minerals, biological affair was revealed to be less ordered, with a virtual continuum from discrete diffractions spots in the former, to diffuse structure in the latter, illustrated past the examples of figure 3.
X-ray diffraction patterns from a point source. (a) Zincblende (ZnS) crystal diffraction, recorded by P. Ewald; image from The Crystalline Land past Due west.H. and W.L. Bragg; (b) cellulose fibres, oriented vertically [7]; (c) a lock of Mozart's hair, and (d) α-keratin diffraction blueprint, from Mozart'south hair (oriented obliquely) past William Astbury'southward colleague, Elwynn Beighton, in 1958.
Astbury was before long investigating porcupine quills, hedgehog spines, merino wool and human hair. Later in his career, his banana Elwynn Beighton fifty-fifty imaged a lock of Mozart'southward hair, obtaining an image that moved Astbury—a talented apprentice musician—to shed tears during the delivery of at to the lowest degree one lecture. two
The intrinsic fuzziness of fibre protein diffraction patterns, such equally α-keratin, reflects the lack of long-range crystalline order in their construction. 3 At the atomic scale, crystallographers seemed to have uncovered a natural division between the ordered crystals of the inorganic world and the messier animate cosmos. This finding echoed Fedorov'due south dictum cited higher up.
This apparent separation of biological from inanimate matter is complicated by a striking characteristic of all known life: it is wet. No water; no life. While living 'extremophiles' have been institute in the anaerobic deep sea, at loftier temperatures and extremes of pH [9], no life is known for thousands of foursquare kilometres in the Atacama Desert in Chile, the driest region on the Earth. Owing to their inevitable h2o content, most biomaterials are 'soft materials', defined by the pioneer of soft matter physics, Pierre Gilles de Gennes, as complex (containing, for example, many components) and flexible. With the notable exception of the skeletal (hard) parts of an organism, living tissue is soft matter, typically composed of variously aggregated biomolecules swimming in a pregnant volume fraction of water, all held together by weak (and poorly understood) interactions, including the ubiquitous hydrophobic force and electrodynamic interactions, such equally dispersion forces [10]. A remarkable feature of many wet biomolecules—observed by Bernal in dispersions of Tobacco Mosaic Virus in h2o—is their germination of gel-similar states, intermediate to liquids and crystals, dubbed 'liquid crystals'.
As their name implies, liquid crystals combine structural features of both liquids (disorder) and crystals (society); they are intermediate in nature or mesomorphic (to use Friedel'southward name derived from the Greek word for middle [11]). By convention, we distinguish between thermotropic and lyotropic liquid crystals. Thermotropic examples are formed on heating up a pure chemical compound as an intermediate, and still somewhat ordered stage, of melting. Typical examples are cholesteric liquid crystals, characterized by a relative twist along ane axis between the orientations of neighbouring molecules. Therefore, the structure is described past the helical (single) twist orientation vector field: if a molecule is located at a some point in the sample, its orientation is likely to be that of the field at that point (or its changed). So positional guild—characteristic of a crystal—is absent, though orientational order exists. The cholesteric phase has a characteristic length scale, namely the pitch of the helical twist orientational field, which is uncorrelated with the molecular dimensions [12]. Past dissimilarity, the 'fractional melting' of lyotropic mesophases is effected by the add-on of solvent (rather than heating). Lyotropic mesophases typically form in mixtures of water with detergent (or lipid); these take positional order at mesoscopic length scales, typically tens to hundreds of times the molecular lengths, but are molten at atomic and molecular length-scales. So-called 'bicontinuous cubic mesophases' are particularly interesting examples of the lyotropic country (that I discuss below).
Both lyotropic and thermotropic liquid crystals are far from rare in biological materials. Examples of cholesteric and bicontinuous cubic phases abound. For example, Yves Bouligand, a biologist with strong interests in liquid crystal physics, established the presence of cholesteric society in the skeletal parts of a variety of organisms [thirteen]. Joseph Needham, a fellow member of the Club for Theoretical Biology (a group—including Bernal—who established the intellectual basis for much of modernistic biology), said in 1935 '[Liquid crystals are] not merely a model for what goes on in the living prison cell … but a state of organisation actually found in the living cell' [14]. Indeed, nosotros now know that liquid crystals are so common in biological science that Fedorov's pronouncement should maybe be rephrased to read: 'Liquid crystallisation is life'!.
Bernal and Astbury's visionary project, that kickstarted no less an enterprise than molecular biology, remains every bit vital today as when it was formulated almost ninety years ago. Indeed, a big fraction of pharmaceutical and medical research—and beam hours of the world's synchrotrons—focuses on the atomic and molecular architectures of biological materials, largely probed by X-ray crystallography. This is an extraordinary testament to the original vision of Bragg's junior colleagues, and to the importance of fundamental physics in driving biological studies. Today, the central repository for protein structure, the Protein Data Depository financial institution, hosts nearly a hundred thousand protein and nucleic acid structures, 90% of which are reported from Ten-ray data (and 10% from NMR analyses) [xv].
Although impressive in its scope, the project is not the final discussion in biology. To date, about X-ray structural analyses have required the growth of true poly peptide crystals, necessary in order to obtain precipitous, mineral-similar diffraction patterns. Since biomolecules are essentially 'dead' in the crystalline country, these structures exercise not necessarily reveal their geometric subtleties found in their native, biologically active country in vivo. Furthermore, since many (non-structural) proteins are explicitly constructed to avert aggregation into larger units, they tin can only be coaxed to crystallize with added molecules, such every bit detergents, which surely perturb the usual hydrophobic–hydrophilic residual that is so disquisitional to biological activity. These caveats aside, the standing efforts to deduce the geometry of biomaterials at the diminutive and molecular scales, is driven by a simple principle, common across the life and natural sciences: 'Structure is role'. As Astbury wrote in 1950:
Molecular biological science is predominantly iii-dimensional and structural - which does non mean, however, that it is merely a refinement of morphology. It must of necessity inquire at the same time into genesis and function. [16, pp. 6–7]
A useful lesson can be gleaned from this brief overview of biological structure. At first glance biological thing is very unlike in its macroscopic form from hard, angular crystals. However, the very tool developed to probe crystalline structures—crystallography—is now a powerful primal to probing structural biology at the molecular scale. Thanks to the discovery of X-rays and the subsequent rapid evolution of diffraction physics past the Braggs and their team in England, molecular biology was built-in.
However, equally in all victory tales, this triumphal narrative skirts around a less well-understood issue, namely the nature of crystallinity versus liquid crystallinity and the distinction between structural order and disorder.
3. Generalized crystallography
The vexed and often abused concept of structural order was analysed explicitly by Bernal. Later on in life, inspired by the discovery of the irrational α-helix that Pauling deduced from Astbury'south fibrous protein data, too as virus structures then being uncovered by Caspar and Klug, he chosen for a radical rethink of but what we mean by a 'crystal'. The circuitous structures in biological thing had 'broken formal crystallography, shattered it completely'. He called for a new 'generalised crystallography':
Nosotros clung to the rules of crystallography which gave u.s. the 230 infinite groups every bit long as we could … it needed Pauling to break them down with his irrational α-helix. And then there are no rules, or the sometime rules are enormously changed. What nosotros have called crystallography is a item, small branch of crystallography, three-dimensional lattice crystallography. We are seeing now a generalised crystallography … whatsoever kind of a repeat organisation is a crystal in this general sense. Protein bondage are examples of information technology, so is DNA, and RNA. They accept their ain inner logic, the same kind of logic but a unlike chapter of the logic that applies to the three-dimensional regular lattice crystals [17, preface].
This telephone call remains fresh and challenging to this day. Just as concrete crystallography spawned molecular biology, so generalized crystallography—that emerged from the circuitous assemblies constitute in proteins and viruses—is of increasing urgency and relevance to abiotic matter. The near celebrated example to date is that of quasi-crystals, beginning reported in metal alloys, quintessential examples of non-living thing. More recently, however, quasi-crystals have been found in a diversity of soft materials, from polymer melts, to colloidal crystals whose chemic compositions resemble the constituents of biological matter [18]. To my knowledge, definitive recognition of quasi-crystals in a living system is unknown. (Even so, structural studies of institute chloroplasts past Gunning and colleagues uncovered a plethora of forms, some belonging to Bernal's 'pocket-sized co-operative' of iii-dimensional lattice crystallography and others possibly quasi-crystalline or aperiodic. These are discussed farther below).
Surely, in fourth dimension, many new examples of quasi-crystals will be uncovered, possibly likewise in biology. It is crucial to remind ourselves of Bernal's dictum that emerged from biological studies: that classical crystals are just one realization of patterned structures. Quasi-crystals are just one more species of general crystal. The point is made very articulate by Bernal's inferior colleague, Alan Mackay, who, following Bernal'southward thinking, suggested that quasi-crystals may be establish in nature, prior to their discovery in alloys past Shechtman [19]; a discovery crowned by the 2011 Nobel Prize for Chemical science. According to Mackay, the written report of quasi-crystals is a 'a kind of legalistic discovery. It's a discovery of a material which breaks the laws that were artificially constructed. They were not laws of nature; they were laws of the human classificatory system.' [20].
Bernal called for the evolution of 'statistical geometry' to quantify generalized crystallography. It is sobering to realize that simply today, l years later, is that phone call being treated seriously, driven in part by the written report of granular materials by physicists, also as mathematical developments in discrete geometry. The exploration of jammed states of these soft materials [21], similar to Bernal'due south disordered 'heaps' [22], as well as glass-crystal transitions are at present a respectable—indeed stylish—area of condensed matter science, driven in large office by the massive growth in numerical simulations. Despite that focus, the nature of disordered 'glassy' states, and their place within the spectrum of generalized crystallography, remains unclear. For example, Sharon Glotzer has suggested that glasses of rigid polyhedral forms results from a competition between a number of attainable crystalline forms [23]. And so are glasses merely wannabe crystals, or, equally claimed by Bernal, something quite singled-out? We practice not yet know.
Bernal'southward definition of a generalized crystal as a 'repeat organization' deserves some reflection. Its superficial informality belies a radical agenda. In particular, he chooses to ignore any specification of dimension, or space. Certainly, given that materials are embedded in iii-dimensional Euclidean space, in that location are some constraints. Notwithstanding, every bit he makes clear, the repeat arrangement could refer to whatsoever symmetry or quasi-symmetry, non just those discrete isometries of 3-dimensional Euclidean space found in the International Tables for Crystallography [24]. Clearly, Bernal's agenda was to augment structural and crystalline concepts to include the very materials his colleagues were exploring and so successfully, from proteins to viruses.
Structural studies of viruses revealed regular assemblies of proteins on curved substrates, rather than conventional three-dimensional crystals packed in infinite. This structural 'system' was therefore very different from that of conventional 3-periodic crystals, quasi-crystalline and statistically random arrays in iii-dimensional Euclidean space (
). We now know that many viruses are assemblies of capsid proteins on the two-dimensional surface of a (somewhat deformed) sphere (
). The most symmetric examples result in arrangements with chiral icosahedral signal group symmetry (I in the nomenclature of the crystallographic customs), and form a symmetric ornament of the topological sphere. Co-ordinate to simulations, assembly of two distinct capsomer forms (one viral capsid hexamer and 1 pentamer), is sufficient to mimic many of the forms observed in viruses [25], though a more than basic understanding of but how hexameters and pentamer themselves form is lacking. Even so, the point is articulate: icosahedral order emerges from very generic interactions between binary constituent building blocks.
Icosahedral viruses are therefore prime examples of generalized crystals in Bernal's sense. The underlying structural principle is that of a symmetric reticulation (a 'repeat principle') of curved two-dimensional space (namely 
) rather than conventional (flat) iii-dimensional space. It turns out that symmetric reticulations of another two-dimensional space are every bit relevant to materials, whether living or not.
3.ane. 2-dimensional (non-Euclidean) crystallography
Non-Euclidean geometry admits just three homogeneous two-dimensional spaces, with abiding (Gaussian) curvature at all points. Two are well known: the apartment plane 
and the sphere, 
. The third space, 'two-dimensional hyperbolic space', ℍ2, is more difficult to picture, every bit it is impossible to embed simply in two- or three-dimensional Euclidean space (
). To portray ℍii, some distortions are required. One way, discovered by Poincaré, is to shrink it dramatically radially virtually a single bespeak, so that the entire space sits within a unit disc of 
. This 'Poincaré disc model' of ℍ2, has the merit of beingness conformal, so that all angles in ℍii are preserved by the map into the apartment disc. A design in ℍ2, drawn in this model, is illustrated in effigy 4 a.
(a) Poincaré disc model of the hyperbolic aeroplane (ℍ2), tiled with hyperbolic 246 triangles; (b) The P surface, tiled with 246 triangles; (c) The D surface, tiled with 246 triangles; (d) The Gyroid surface, tiled with 246 triangles. If the blue and white triangles are assumed to be equivalent, all of these patterns display *246 symmetry. (Images courtesy of Myfanwy Evans).
In the past few years, we (and others) have explored repeat organization within ℍii in some detail. Just equally point groups tin can be derived from isometries of 
, nosotros tin enumerate an infinite listing of detached groups that are isometries of ℍtwo [26]. While mathematically interesting, it is of little use unless we tin map ℍ2 into our own space (
). That tin exist done, with the help of a discrete lattice of disclinations that deform ℍtwo simply enough to allow it embedding in 
[27]. Within the linguistic communication of grouping theory, those embeddings are precisely the subset of the discrete groups of ℍii that can exist mapped into 
by projection onto hyperbolic surfaces embedded in our space. If we insist that the hyperbolic surface is strictly embedded, i.e. free of self-intersections, the well-nigh symmetric examples that we know of are those whose hyperbolic symmetry is described by the Conway orbifold symbol * 246 [26]. Remarkably, information technology turns out that hyperbolic surfaces with *246 symmetry embed into 
to course multi-handled sponges with three contained lattice vectors, i.e. the surfaces themselves are conventional (cubic) crystals. There are three known embeddings, and all iii are examples of 'triply periodic minimal surfaces', namely the P, D and Gyroid surfaces, illustrated in figure 4.
Nosotros can motion between patterns in 
and related patterns in ℍii with ease, by editing the order of symmetry elements [28]. Consider, for case, a flat two-dimensional crystal, such equally the pattern of Angels and Devils by Escher, fatigued in 
. This planar crystalline design can exist negatively curved, by the addition of disclinations, to form a generalized hyperbolic (two-dimensional) crystal in ℍ2, as illustrated in figure v a. Note that in this image the hyperbolic crystal is portrayed within the Poincaé disc model, so that the various angels and devils appear to shrink as they arroyo the disc edge. In fact, they exercise not in ℍ2, and this effect is an artefact of the map. In the true crystal (i.e. in ℍ2), all angels and all devils are identical. The image in figure 5 b therefore portrays a regular tessellation—and instance of Bernal'due south 'repeat arrangement'—of ℍtwo with only two tiles, one angelic, the other diabolic. A map of the Angels and Devils tiling onto the P surface is illustrated in figure five c. Structures that course these surfaces tin can therefore exist viewed through ii singled-out perspectives. They can exist comprehended as patterns embedded within 
, and their triplet of lattice vectors makes them members of Bernal's 'small branch of crystallography, three-dimensional lattice crystallography'. That view corresponds to the image in effigy 5 d. Alternatively, they tin exist perceived within the confines of two-dimensional ℍ2 lonely, rather similar the view of a very thin, two-dimensional ant living in the hyperbolic surface. From that perspective, they are 2-dimensional, hyperbolically curved crystals, arranged according to the *246 isometries of ℍtwo. That view corresponds to the tiling of ℍ2 itself, shown in effigy 5 b.
(a) Thou.C. Escher'due south drawing of a tessellation of the Euclidean aeroplane (
), Angels and Devils (with orbifold four*2); (b) Escher's hyperbolic Angels and Devils tessellation, drawn in the Poincaré disc model of ℍ2 (with orbifold 4*3); (c) A fragment of the hyperbolic tiling in (b), excised from ℍ2and mapped into a unmarried (rhombohedral) unit cell of the P-surface (a three-periodic minimal surface) embedded in 
; (d) A larger fragment of the P-surface, made of many unit cells, tiled with Angels and Devils. (Images (b–d) courtesy of Stuart Ramsden).
The examples in figures 4 and v demonstrate a unproblematic principle: regular organizations in ℍii with suitable symmetries (east.g. *246) can be mapped into conventional three-dimensional crystals in 
. In fact, many three-dimensional Euclidean crystals tin can be built via this projection from ℍtwo to 
. We accept enumerated elsewhere the simpler cases of allowed orbifolds [26,29]. Note that well-nigh of these crystallographic hyperbolic orbifolds project into 
via the P, D and Gyroid (cubic) surfaces to form crystals that are themselves not cubic, simply of lower symmetry. That procedure has been used to generate crystalline 3-dimensional nets, via tilings of ℍ2 [xxx–33].
This project procedure tin—in principle—be further generalized to build regular patterns in ℍii that do not project to three-dimensional Euclidean crystals. The state of affairs is somewhat analogous to the formation of irrational helices on a cylinder, formed by rolling upwards a page busy with judiciously oriented parallel lines in 
. The hyperbolic case is more circuitous yet, since 
is replaced past ℍtwo, and the rolling process occurs along a number of axes simultaneously. Although the mathematics has all the same to be worked out, and may prove challenging, there is no fundamental obstacle to a fuller enumeration, generating many additional groups whose translational elements are a subset of those of the P, D and Gyroid surfaces. In the linguistic communication of crystallography, we could cull translational subgroups of arbitrarily large lattice vectors. Indeed, nosotros could extend one, or two, or all iii lattice vectors indefinitely, resulting in a pattern projected onto these surfaces with no visible translational symmetries in 
! Such a design could remain symmetric in ℍii, with well defined ' repeat operations'. In principle then, we tin can construct generalized crystals (in exactly the sense supposed by Bernal), which comprise no translational symmetries. In this style, hyperbolic space offers immense scope to construct generalized crystals, more exotic than quasi-crystals, or the irrational α-helices in proteins.
3.2. Minimal surfaces as frustrated mappings from hyperbolic to Euclidean space
In the previous section, the P, D and Gyroid minimal surfaces have been used as a mathematical scaffold to map from ℍ2 to 
. However, it turns out that, in addition to their mathematical utility, these surfaces are routinely realized in condensed materials. These hyperbolic forms are therefore more than geometric abstractions.
Recall that curvature is a characteristic of biological course (effigy ii). Indeed, the P and D and Gyroid surfaces and their pairs of intertwined volumes describe well the convoluted labyrinths and hyperbolic lipid-poly peptide bilayers found in many membranes inside cell organelles in vivo. These are the then-called 'cubic membranes' [34,35]. The aforementioned geometries (albeit shrunken) are too found in abiotic materials that class lyotropic liquid crystals. These are the bicontinuous cubic mesophases. Although these bicontinuous cubic phases are non living materials, their chemical make-up is similar to that of amphiphilic membranes in vivo.
The formation of these curved shapes is possibly unsurprising, given that biomaterials in vivo and in vitro are typically moisture and soft. Notwithstanding, these curved hyperbolic forms were first explored in detail by scientists, due to their relevance to inorganic materials. In the early 1980s, Sten Andersson in Lund and Alan Mackay in London suggested independently that the covalent silicate networks of porous zeolites were tilings of the P, D and Gyroid surfaces [36,37]. And so structural curvature, generally thought of as the hallmark of biological form, is prevalent too in inorganic structures. To reiterate, precisely the same structures are formed in living cell membranes, organic lyotropic liquid crystals and inorganic silicates (although at very unlike lengths scales) [38,39].
Given the very different physics underlying the formation of inorganic silicate structures and soft amphiphilic membranes, the presence of these specific minimal surfaces beyond such a broad spectrum suggests a fundamental structural feature of these patterns. It is noteworthy that these (and simply these) minimal surfaces have *246 symmetry, the smallest orbifold domain of all known hyperbolic orbifolds that tin be mapped into 
. In other words, equally far as we know, these surfaces are the most symmetric realizations of homogeneous hyperbolic ii-infinite in 
. Therefore, the P, D and Gyroid are probable to represent optimal embeddings of a uniformly curved hyperbolic flick [40,41], a hypothesis supported past calculations but as nevertheless unproven [42]. In other words, the fluctuations of (Gaussian) curvature in the P, D and Gyroid minimal surfaces are smaller than in other more than complex minimal surfaces, therefore their bending energy is also lower [40,41,43,44]. If this hypothesis holds true, information technology implies that the emergence of three-dimensional crystallinity at a global scale in these systems is due solely to a local demand for compatible curvature. Crystallinity so emerges as a frustration-minimizing solution to the incompatibility between ℍii and 
.
Despite its importance as a guide to our understanding of the physics of even the simplest hyperbolic structures, this 'homogeneity' hypothesis remains unproven. Nosotros are hamstrung by mathematical ignorance of culling candidate structures to these crystalline hyperbolic surfaces. Indeed, we do not even know if quasi-crystalline or aperiodic minimal surfaces of space genus exist in 
, let alone how their curvature homogeneity compares with the simpler cubic P, D and Gyroid surfaces.
This gap is meaning, from the perspectives of both fundamental physics and biology. For, in improver to the P, D and Gyroid geometries, other, less recognizable hyperbolic structures have also been observed in vivo in particularly interesting examples of biological membranes, namely young and/or light-starved found chloroplasts. Brian Gunning has reported a number of geometries, all based on hyperbolic surfaces with tetrahedral nodes, forming labyrinths of various forms [45,46]. The structures include conventional crystalline patterns, as well as radially symmetric simply aperiodic examples. None of these correspond to known minimal surfaces. The being of these tetrahedral patterns, rather than the ameliorate understood P, D and Gyroid structures, handsomely exposes our lack of primal agreement of two-dimensional hyperbolic patterns in materials, be they biological or non-living. Are other 'generalized crystals' derived from regular patterns in ℍ2, beyond the conventionally crystalline P, D and Gyroid structures likely to appear in materials? Until we have more fundamental mathematical agreement of hyperbolic surfaces, we do not know. All the same, it is now certain that regular curved forms are as relevant to our agreement of condensed materials every bit the familiar facetted forms of hard crystals and curvature is innate in both breathing and inanimate materials. At a more philosophical level, one time a broader view of simply what constitutes a crystal is adopted [47], distinctions betwixt biological and not-living forms at the atomic, molecular and larger length scales are far less clear.
four. The organic versus the inorganic
Information technology is helpful at this point to briefly survey other modes of distinguishing the animate and inanimate realms. What, for example, are the chemical rather than structural differences between biological and abiotic matter?
Given that most abiotic chemical species can exist coaxed to crystallize with relative ease compared with biomolecules, where does the boundary between bio- and abiotic molecules lie? Or, to rephrase the question, tin we distinguish Regnum Lapideum, Regnum Vegetabile and Regnum Animale at the atomic or molecular scale? Since the nineteenth century, chemistry has been conventionally divided between 'organic' and 'inorganic'. Every bit the names suggest, the division echoes the animal–mineral dichotomy.
Prior to the twentieth century, organic chemistry was believed to exist the study of the fundamental components of living things, those containing a life-force, absent in inorganics. This 'Vitalism' philosophy provided a user-friendly classification principle: in society to belong to Regnum Animale or Regnum Vegetabile, this mystical life-force was required. The natural corollary was that members of the inanimate kingdom, Regnum Lapideum were gratis of this attribute. To rephrase that dichotomy in the words of perhaps the nigh historic classical physicist and contemporary of Steno, Isaac Newton, 'Nature'southward deportment are either vegetable … or purely mechanical' [48, p. 306]. It is very curious then to learn that Newton also wrote of the process of 'vegetation', that governs 'Natures obvious laws and processes' and is the 'sole outcome of a latent spirit and that this spirit is the same in all things' [48]. Clearly, Newton is discussing the mystical life-force, since its presence is certainly needed for vegetation. But, he says, information technology is nowadays in all things—brute, vegetable or mineral—writing 'that metals vegetate subsequently the same laws'. Thus, vitalism permeates all things in Newton's worldview!
That mystical view of Newton'southward was quickly diluted to a more than rational i, in accord with the spirit of the Enlightenment. Until the mid-nineteenth century, the distinction between 'life' and the remainder was clear: living things were infused with that mystical life-force. However, the rise of organic chemistry in the previous century was driven in part past experiments that proved this elementary stardom to be false. Organic molecules could be made in a exam tube, from 'expressionless' precursors: the classic example was Wöhler's synthesis of urea from inorganic ammonium cyanate [49] and, in the twentieth century, the Fischer–Tropsch reactions [50]. More than recently, geochemists have discovered that complex organic molecules, including polycyclic aromatic hydrocarbons ('PAHs', until recently, widely assumed to be molecular signatures of life) can exist formed under relatively benign hydrothermal conditions, mixing inorganic minerals such as siderite (fe carbonate) and water at elevated temperatures (e.k. [51,52]).
Today, the distinction is reduced to atomic book-keeping merely: organic chemistry explores molecules containing carbon atoms, excluding the uncomplicated ionic salts (carbonates, oxides and carbides). Inorganic chemistry explores the residue. Just in many quarters the prejudice lingers that—somehow—inorganic and organic chemicals are qualitatively dissimilar. Given that a typical inorganic crystal (siderite) can be converted to a typical organic molecule (a polyaromatic hydrocarbon) by hydrothermal treatment solitary, the old prejudice is surely overdue for disposal.
v. Bridging Regnum Animale, Vegetabile and Lapideum
With the benefit of the accumulated knowledge of fabric structure and function of all species since the seventeenth century, any attempt to unweave the plaited skein that enfolds materials into the three realms proposed by Linnaeus is complex, at best. The give-and-take in previous sections highlights the pivotal role that liquid crystals play in frustrating a simple classification schema à la Linneaus.
Liquid crystals were commencement observed in a form of cholesterol-based organic molecules extracted from plants by the German chemist Reinitzer in the 1880s. With the assistance of the physicist Lehmann, he had discovered cholesteric liquid crystals. And so dramatic and life-like were the writhing figures visible in the optical microscope during the melting process, that the eminent scientist Ernst Haeckel wrote a book entitled 'Crystal Souls—Studies of Inorganic Life' (frontispiece reproduced in effigy 6) [53]. Haeckel, like Newton before him, guided by his ain mystical (and by that time largely outdated) views, was convinced that these liquid crystals contained the essence of life itself, the vital force.
Image from Haeckel'south book Crystal souls [53].
Haeckel's views are generally taken every bit the final gasp of an outmoded view; the dogma of Vitalism died with him. Ironically, despite the later association of the liquid crystalline state with biological affair by the 2d one-half of the twentieth century, the discovery of liquid crystallinity in the nineteenth century triggered much fence, followed by retreat from the dogma of Vitalism. But that retreat was far from orderly, or unanimous. For example, an early twentieth century text described the contributions of Lehmann (the co-discoverer of liquid crystals, thus):
Professor O. Lehmann … endeavoured to bear witness that no difficult and fast line tin be drawn betwixt the living and the dead. He contended that crystals of numerous substances showed all the characteristics of life as revealed in certain of the lowest organisms; that substances which crystallise do and then in a specific form and resemble many plants; that crystallisation requires a germ to kickoff with; that some crystals are capable of growth, while other poison themselves past arresting substances contained in the medium investing them. He challenged the statement that living things are always fluid or partially so, and that crystals are invariably solid. In support of this last proposition he maintained that liquid crystals tin can now be produced … Those of soft lather beget a skillful example. Professor Lehmann directed attending to some remarkable crystalline forms occurring in mucilaginous fluids which, nether the microscope, are seen to be in a country of abiding movement. The views of Professor Lehmann have, of course, to be subjected to the almost severe criticism on the office of physicists and biologists earlier they can be accepted … [54, p. 28].
This demise of Vitalism was accompanied past quarantining of physicists from biologists, with occasional—temporary—reunions. Nevertheless, the search for life'due south specific markers remains. Erwin Schrödinger's essay What is life? remains influential today, though most probable more so among biologists rather than physicists. Schrödinger detected two principal features of life: heritability and spontaneous self-assembly. A tertiary characteristic is often invoked today: emergence [55], which describes those characteristics of a organisation that cannot exist ascribed to individual constituents, but by their collective activity [56]. The last quality is a challenge to the traditional reductionist mode of scientific explication, and therefore appealing to some (e.k. systems biology [57]) and every bit repellant to others [58]. Necessary though these conditions may exist for life, all three features are found in abiotic systems as much as living systems. Heritability is well known to crystal growers: seed crystals beget nuclei for crystal reproduction in a test tube. Self-associates—at equilibrium, or in dissipative systems—is a characteristic of almost all soft materials, active or passive regardless of their biological content. The liquid crystals described before are self-assemblies, and can exist formed in vivo or in non-living material (though typically the former are far more swollen that the latter). Lastly, emergence is a characteristic of complex systems, such as soft matter, rather than biology sui generis [59]. Are in that location further characteristics of biology that are peculiar to living systems? Some biologists keep to insist at that place are. For example, the following quote from a physiologist is a scarcely bearded update of Vitalism: … life has a third hush-hush not mentioned past Schrödinger. The blueprint of living organisms is not determined by physico-chemic laws [55]. Cleary, the struggle to place biology as a free-standing science remains a precious 1 to biologists.
An alternative view is one that views the biological and non-living realms every bit a continuum, without clear stardom betwixt one and the other. At one extreme, we accept the refractory, facetted matter of classical crystals; at the other the curvilinear, articulated blob that is a developing fetus. Between the two extremes lies a multitude of spatial and temporal organizations common to both the living and the sterile. Information technology remains unknown simply how broad (gauged by structural and functional measures) that common zone is. The challenge for physical and biological scientists today is to explore that eye zone without prejudice.
While the systems biologists will likely protest, studies of material self-assembly and organization at the mesoscopic length scale (somewhere between atoms, molecules and the cellular scale) offer a useful view of biological and related materials. And that view reveals the importance of Bernal's concept of generalized crystals, specially liquid crystals, evoked by Haeckel equally the seat of the life-force. Certainly, the exploration of liquid crystals in biology—still on-going—has inspired a number of new ideas and open up questions in our understanding of the multitude of shapes and assemblies that narrate biological architecture. And much of the intellectual impetus for those newer ideas arose from condensed affair physics and chemistry. The pioneering studies of biological liquid crystals started past Bernal and colleagues in England, were advanced substantially by Vittorio Luzzati [threescore] and Yves Bouligand [61] in France and Kaare Larsson in Sweden [62]. Their piece of work demonstrated unequivocally the importance of exploring non-living soft materials in vitro to understanding biological assemblies in vivo and vice versa.
This lesson—that biology proper (whatever that is) and condensed matter or materials science should exist studied hand-in-paw—is one that remains also ofttimes ignored. The 'tyranny of bailiwick' remains a major obstacle, 4 specially in this era of extreme specialization. I last case written report demonstrates the importance—and relevance—of this arroyo to current science. This example concerns the identification of life's remnants from fossils. Given the mineral composition, induced by templating of earlier biological remnants, the exact location of fossils in Linnaeus' schema is surely problematic. Indeed, structural measures alone are bound to be uncertain. Here is the perfect domain for a challenging game of Beast, Vegetable or Mineral! (east.g. figure 7)
Fauna, vegetable or mineral? Three materials, imaged in an optical microscope. (a–c) An ancient putative micofossil; a bacterium Gallionella ferruginea and a silica-carbonate precipitate, respectively.
In 1996, NASA triumphantly suggested that they had found the 'smoking gun' that demonstrated extra-terrestrial life, namely the presence of bacterial remains inside a meteorite dislodged from Mars, and collected in Antarctica [63]. The annunciation was met with massive interest, so profound that President Clinton appeared on Telly to discuss the finding [64]. The excitement was triggered by a scanning electron micrograph that revealed a curvilinear, segmented crush (figure 8 a), similar to the shapes of modernistic filamentous bacteria, and to the forms found in Archaean rocks from northwestern Commonwealth of australia, and believed to be fossilized bacteria.
(a) NASA'south scanning electron micrograph of a fragment of the Martian meteorite, {"type":"entrez-protein","attrs":{"text":"ALH84001","term_id":"937293154","term_text":"ALH84001"}}ALH84001, collected in Antarctica. The boxed area shows a segmented construction that resembles a (highly shrunken) bacterium from the meteorite. (Scale bar, 100 µm; estimated from [63]). (b) Scanning electron micrographs of silica-barium carbonate 'biomorphs' grown in the laboratory under sterile conditions. (Scale bar, 30 µm. Epitome courtesy of Anna Carnerup).
Still, work initiated by Juan Manuel Garcia Ruiz in Granada, has demonstrated that precipitates of carbonate microcrystals in the presence of silica in a test tube at high pH and room temperature also give very similar forms; so life-like that Garcia Ruiz christened them 'biomorphs' [65]. Subsequent piece of work confirmed that this synthesis readily produces such 'microfossils', complete with an enclosing membrane reminiscent of aged cell walls, in the absence of any biology (due east.g. figure 8 b) [66]! Given those findings, the claims of biogenicity for the Martian meteorite, and indeed the world's 'oldest known microfossils', dated to 3.four Gyr, must remain speculative, at best.
Today, supposed fossilized stromatolites, also from NW Australia, are at present trumpeted as the oldest fossils on the Earth [67]. Again, the arguments are complex, potent prove for their biogenicity is adduced from the regular curvilinear forms of these putative fossils. In my view, the evidence is weak and motivated largely past a prevailing prejudice that curved forms are biological, rather than inanimate. Once more, palaeontologists must piece of work mitt-in-mitt with materials physicists, in social club to explore potential abiotic explanations for these forms. For example, recent work demonstrates that extraordinarily 'life-like' forms can exist generated by circuitous rubberband composites [68–lxx], and those concerned past agreement the genesis of biomorphologies ignore the findings of physical scientists at their peril.
These examples point to just 1 conclusion: the supposed disjunction between Animal or Vegetable and Mineral is, on shut inspection, fiction rather than fact. We demand not return to the extreme viewpoint of Newton, who suggested the 'vegetation of metals'. Still, once we admit the possibility generalized crystals and curvature as a language of forms shared past the breathing and inanimate worlds, Linnaeus' rigid classification must be rejected. Every bit Astbury's comment higher up reminds us, still, biology is function every bit well as form. The issue of biological role is some other chapter, itself worth exploration in detail. Conspicuously, the boggling adaptability and functioning of biological matter far exceeds current biomimetic functioning materials. Here likewise, however, given the scientific efforts directed in the areas of synthetic biological science [71,72] and active matter [73,74], one can predict convergence with some confidence.
The moral of this tale is simply expressed though challenging to implement: biologists, communicate with physical scientists and physical scientists, learn the language of biology!
Acknowledgements
I give thanks Prof. JuanManuel Garcia Ruiz (Granada University), Ms Eva Hild, Dr Myfanwy Evans, Mr Stuart Ramsden (Canberra) and Dr Anna Carnerup (Canberra) for permission to utilise their images, (figures 1, 2, four, 5(b–d), respectively). I am also very grateful to the Astbury and Beighton families, particularly, Mr William Astbury and Ms Gemma Sanderson, for permission to reproduce Mozart's hair sample (figure three(c,d)) and to Dr Kersten Hall (Leeds) for generous assistance in uncovering the diffraction image, and much background information, derived in office from the efforts of Prof Tony Due north and Oliver Pickering and the late Chris Sheppard of Special Collections, Brotherton Library in Leeds.
Endnotes
1One of a series of extraordinarily talented and influential women in British crystallography, and later described by Bernal every bit 'the presiding genius of the place' [5].
2It is indeed a romantic tale, and one may well wonder how a lock of Mozart's hair ended upwards in Leeds Academy library! An abbreviated chronology runs thus. The hair was contained within a drove of diaries and papers, presented to Leeds by Donna Nerina Medici di Marignano Gigliucci, a descendent of Vincent Novello (the founder of the British sheet music company, Novello & Co, Ltd). Vincent, who was of Italian origin, visited Italia in 1829 to run into Nannerl Mozart, Wolfgang'due south elder and (likely) as talented sister. His diaries of the trip remained in family hands, secreted in Villa Novello, Genoa, after his death. This Villa was occupied by the British Ground forces in the Second Earth State of war. The diaries were institute past i Major Edward Croft Murray, then quartered in the house, and he passed them on to Donna Medici, who donated them the library.
iiiIn fact, this lack of crystalline social club was already well known to the Braggs. According to Astbury, the first effort to image Ten-ray scattering from wool was prepared for his then dominate, William Bragg at the Purple Institution, equally a Fri evening lecture, entitled The imperfect crystallisation of mutual things [8].
ivAn incisive phrase due to Barry Ninham.
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