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Page 1: Beta Laktoglobulin

DOI 10.1140/epje/i2009-10465-y

Regular Article

Eur. Phys. J. E 29, 173–182 (2009) THE EUROPEANPHYSICAL JOURNAL E

Investigating the inner structure of irregular β-lactoglobulinspherulites

K.R. Domike1,4,a, E. Hardin1,2, D.N. Armstead1,3, and A.M. Donald4

1 Department of Physics, The College of Wooster, Wooster, Ohio 44691, USA2 Department of Physics, Slippery Rock University, Slippery Rock, PA, 16057, USA3 Department of Physics, Westminster College, New Wilmington, PA, 16172, USA4 Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK

Received 10 July 2008 and Received in final form 31 December 2008Published online: 19 June 2009 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2009

Abstract. When β-lactoglobulin in low pH aqueous solutions is exposed to high temperature for extendedtime, spherulites composed of amyloid fibrils of the β-lactoglobulin protein form. Many of these spheruliteshave fibrils that radiate out from a centre and, under crossed polarisers, exhibit a symmetric MalteseCross structure. However, a significant fraction (50 of the 101 observed spherulites) of β-lactoglobulinspherulites formed under these conditions demonstrate various forms of irregularity in apparent structure.The irregularities of spherulites structures were qualitatively investigated by comparing optical microscopyimages observed under crossed polarisers to computationally produced images of various internal structures.In this way, inner spherulite structures are inferred from microscopy images. Modelled structures that werefound to produce computed images similar to some of the experimentally viewed images include fibrilscurving as they radiate from a single nucleation point; multiple spherulites nucleating in close proximityto one another; and fibrils curving in opposite directions above and below a single nucleation point.

PACS. 87.14.E- Proteins – 87.15.ad Analytical theories – 87.15.bk Structure of aggregates

1 Introduction

Protein aggregates from misfolded or partially unfoldedproteins are thought to be the cause of amyloidoses; dis-eases which include Alzheimer’s, type II diabetes, andParkinson’s [1,2]. Aggregates such as amyloid fibrils,which are characterized by a cross-beta sheet quater-nary structure, and spherulite aggregates, characterizedby spherical or near-spherical shape, have been shownto be present in some humans suffering from these dis-eases [3]. The exact role that the protein aggregates playin the diseases and details about the internal structure ofthese spherulites remains unknown [1,4]. The propensityto form spherulites is not unique to proteins but has alsobeen found in crystallisable synthetic polymers [2,5–7] andcrystallising metals [8,9].

The protein used in this study is bovine β-lactoglob-ulin (BLG). BLG has been found to form amyloid fibrilsand spherulites readily under low pH (most reproduciblyat pH 1.5–2.0) and elevated temperature (up to 90 ◦C)conditions [2,6,10–12]. Over time, typically 12–48 h de-pending on temperature, BLG spherulite aggregates canbe viewed directly by optical microscopy [2,5,11–14]. BLG

a e-mail: [email protected]

spherulites with diameters up to 200μm have been ob-served [3]. The spherulites have been shown to be com-posed of amyloid fibrils arranged radially outwards froma single centre [13].

The optical anisotropy of a spherulite (arising fromthe oriented amyloid fibrils within the spherulite) inter-acts with polarised light as it traverses the spherulite. Theresulting modulation of light of different polarisations en-ables analysis of the image generated in a light micro-scope under crossed polarisers to provide insight into theinternal structure of the spherulite. Specifically, the elec-tric field of the light that exits a polariser can be com-puted from the electric field of the entering light and thepolarization [15–17]. Optical microscopy with crossed po-larisers has been used in a wide variety of biological andpolymeric spherulite formation experiments to observe in-ternal structure and orientation [3,13,17,18].

The simplest internal spherulite structure is that inwhich the amyloid fibrils point radially away from a cen-tral point as shown in fig. 1. This symmetric structureproduces an image under crossed polarisers with a clearMaltese Cross, as shown from experiment in fig. 2 [3,13].In this research, the structures of BLG spherulites thatdo not form a regular Maltese Cross when imaged un-der crossed polarisers are analyzed and mathematically

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Fig. 1. Two-dimensional visualisation of spherulite structurewith amyloid fibrils or fibril bundles (shown as lines) pointingradially from a central point.

Fig. 2. Experimentally observed Maltese Cross pattern pro-duced by a BLG protein spherulite when imaged under crossedpolarisers in an optical microscope.

modelled. The sections of this paper include: the methodof obtaining the spherulite images, a description of the ob-servations of abnormal spherulites, the method of mathe-matical modelling, discussion of the results of relating themodelling to the observed spherulites, and relevant con-clusions.

2 Technique of monitoring BLG spherulites

The results in this paper are based on temperature ex-periments published previously [14]. The novelty of thisresearch is in developing and applying a mathematicalmodel of the spherulite inner structure to relate its struc-ture to the resulting microscopy images and thereby ob-taining insight into the structural sources of irregularitiesin the observed images.

The protein used in this research was analytical gradebovine β-lactoglobulin (BLG) (product number L0130;mixture of types A and B) obtained from Sigma-Aldrich(Gillingham, UK). Protein solutions were made by dissolv-ing the protein in distilled and de-ionized water. The de-sired pH was achieved by adding 1M HCl (Sigma-Aldrich)in 5–50μl increments. Glass slides with wells were filledwith 100μl of protein solution, covered with a glass cover-slip, and heated to the required temperature on a heatingstage placed within the Zeiss Axioplan optical microscope(Carl Zeiss Ltd., Welwyn Garden City, UK). Total mag-nification of either 50×, 100×, 200×, or 500× was usedfor the optical imaging. A polariser and analyzer wereput in fixed positions, orthogonal to one another for the

Fig. 3. (Colour on-line) Example images of characteristicshapes seen for BLG spherulites viewed as optical microscopyimages through crossed-polarisers: A) normal Maltese Cross;B) rotated Maltese Cross (compared to horizontal and verticaldashed lines which represent the orientation of the polariserand analyser); C) small wedge; D) extra lobes; E) other mis-shapen (two spherulites); F) other mis-shapen (two lobes). Thesolid scale bar in each image represents 50 μm.

crossed-polariser imaging. The size of spherulites seen inthe images was determined from calibration with a scalebar of known dimension, and subsequent quantificationof the spatial distance per image pixel. For example, theimages of BLG spherulites taken at 50× magnificationhad a resolution of 1.56μm/pixel; at this magnificationspherulites with radii less than 10μm could not be clearlyresolved. The time required to bring all of the BLG pro-tein solution to the temperature of the heating stage wasnot more than three minutes, as determined via finite ele-ment simulation published previously [14], and this offsetis ignored from here on as the incubation times were ofthe order of hours and days. The solution temperaturewas not measured experimentally.

3 Observations of BLG spherulites

When viewed through the crossed-polarisers, it was ob-served that β-lactoglobulin (BLG) spherulites did not al-ways have a Maltese Cross composed of four nearly iden-tical lobes without any rotation of the lobes or notice-able defects as depicted in fig. 3A and described here as a“normal cross.” Across the experiments performed, manyinstances of abnormal spherulite images were observed.Four types of abnormal crosses and shapes that were seenregularly in BLG incubation experiments are presented infig. 3B-E. In some spherulites, the Maltese Cross is signif-icantly rotated from the orthogonal axes as exemplified infig. 3B with the orthogonal axes depicted as dashed yel-low lines. In all experiments and modelling, the polariserswere strictly horizontal and vertical relative to the op-tical microscope and mounted camera. Maltese Crosseswith a deviation of 10◦ or greater from the orthogonalaxes were noted as abnormal. Within some of the images,such as fig. 3B, continuous strand shapes can be seen.

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Table 1. Summary of Maltese Cross shapes in BLG incubation experiments at pH 1.6 and 4wt%. Example shapes are shownin fig. 3. Aggregates demonstrating multiple abnormalities were counted in multiple columns. Percentages relative to the totalnumber of aggregates in the experiment are shown in parentheses.

SolutionTotal Normal Maltese

SmallExtra lobes Mis-shapen

temperaturenumber of Maltese Cross

wedgeaggregates Cross shape rotated

60 ◦C 6 3 (50%) 1 (17%) 2 (33%)

60 ◦C 4 1 (25%) 3 (75%)

60 ◦C 4 2 (50%) 2 (50%)

70 ◦C 15 6 (40%) 3 (20%) 6 (40%) 1 (7%)

70 ◦C 4 2 (50%) 2 (50%)

70 ◦C 7 3 (43%) 4 (57%)

80 ◦C 12 5 (42%) 4 (33%) 2 (17%) 5 (42%)

80 ◦C 7 5 (71%) 2 (29%)

80 ◦C 6 4 (67%) 2 (33%) 2 (33%)

85 ◦C 12 9 (75%) 2 (17%) 1 (8%)

85 ◦C 10 8 (80%) 2 (20%) 1 (10%)

85 ◦C 14 2 (14%) 12 (86%) 4 (29%) 6 (43%)

TOTAL 101 50 (50%) 16 (16%) 16 (16%) 17 (17%) 20 (20%)

Based on the large apparent resolved width of the strandshapes relative to the width of a fibril (8.5± 1.4 nm) [19],the shapes seen are not likely to be individual fibrils butmay be fibril bundles. In many of the images that demon-strate Maltese Cross rotation, the apparent fibril bundlescurl in a direction opposite to the rotation of the Mal-tese Cross. Some other spherulite images demonstrateddifferent abnormalities: some had a small wedge or minia-ture fifth lobe (“lobes” being the bright shapes in the im-ages) as shown in fig. 3C. Occasionally, spherulites ap-peared to have more than five lobes; six full lobes or hav-ing two of the four lobes that were slightly divided as infig. 3D. In addition to well-defined shapes amongst theabnormal spherulite images, some spherulites appeared tobe significantly mis-shapen with a representative exam-ple of a spherulite appearing to be formed from multiplespherulites presented in fig. 3E, and another appearing tohave two small and two larger lobes, shown in fig. 3F.

The number and relative frequency of each typeof spherulite observed for all BLG incubation experi-ments are shown in table 1 with the abnormality de-scriptors corresponding to those shown in fig. 3. Aggre-gates demonstrating multiple abnormalities were countedin all relevant columns (e.g. if a spherulite demon-strated both a rotated Maltese Cross and a small wedge,then it was counted for both of those abnormalities).In total, across all experiments, the four types of ab-normalities were viewed with similar frequency —eachpresent in approximately 17% of the observed spherulites.Temperature-dependent trends of frequency of abnormal-ities are not clear from the data presented in table 1,perhaps due to the relatively small number of spherulitesviewed in each experiment. The full table of data is in-cluded to demonstrate reproducibility of the results of re-peat experiments at a common temperature. One notice-able trend in the data is that abnormalities of a certain

type tend to either be present in relative abundance or ab-sent from a given experiment. For example, the MalteseCross rotation was present and abundant in one experi-ment at 80 ◦C and one experiment at 85 ◦C but not presentin any other experiments (including other experiments atthose temperatures). This suggests that these abnormal-ities may be due to some unidentified systemic detail inthe experiment. Hypotheses of causes of abnormalities areincluded in the Discussion section of this paper. Of addi-tional note, there is at least one normal spherulite with anormal Maltese Cross shape in each of the experiments.This suggests that, while conditions of an experiment maycause a propensity for certain abnormalities, the abnor-malities did not impact all spherulites in the solution andnormal spherulites were able to form in all experiments.

4 Modelling BLG spherulite abnormalities

In an attempt to understand what spherulite inner struc-tures could be resulting in abnormal spherulite images,mathematical modelling of the influence of spherulite in-ternal geometry on the final resulting optical microscopyimage viewed with crossed polarisers was completed.

4.1 The optical path in the microscope

The portion of the optical path in the microscope includedin the model contains three optically anisotropic pieces:the polarizer, the spherulite, and the analyzer. A cross-section, Ci, of the electric field associated with the incidentlight was determined after the light passes through each ofthese objects. The initial light source is assumed to haveno net polarization. The final intensity, I, follows directly

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Fig. 4. Schematic of the parts of a polarising light microscope.In addition to the lenses of an ordinary light microscope thereis a polariser, P , a sample solution containing spherulites, S,and an analyser, A (which is itself a polariser perpendicularto P ). Also shown are three locations, C1, C2, and C3, of thecalculated electric field perpendicular to the direction of thelight propagation. The resulting image is computed from thefinal intensity distribution I.

from the final cross-section. A schematic of the polarizinglight microscope is shown in fig. 4.

In computing the electric field associated with theincident light at the locations C1, C2, each opticallyanisotropic object in the beam path was modelled as aperfect linear polariser. When light passes through a po-lariser, the electric field leaving the polarizer, �C1, is theprojection of the electric field entering the polarizer, �Ei,in the polarization direction:

�C1 = ( �Ei · P ) P , (1)

where P is a unit vector in the polarization direction.The polarizer causes �C1 to be uniform in magnitude

and direction. As in the form of eq. (1), �C2, is computedusing the polarization field of the spherulite, S, and theprevious cross-section, �C1:

�C2 = (�C1 · S)S. (2)

Similarly, �C3, is computed using the analyzer, A, and theprevious cross-section, �C2:

�C3 = (�C2 · A) A. (3)

The final intensity, I, is then:

I = |�C3|2. (4)

4.2 Spherulites as 2D polarization field

Each BLG spherulite, as with the polariser and analyser,was mathematically represented by a two-dimensional vec-tor field perpendicular to the direction of light prop-agation. Each element in the vector field was repre-sented as a perfect linear polariser. In simplifying the full

three-dimensional spherulite structure to two dimensions,it has been assumed that the projection of the fibrils’ tan-gent vectors for the various z values into the x-y planedo not lead to a significant net alteration to the directionof light propagation or polarisation. The two-dimensionalspherulite is not expected to impart any net polarisationto the light field near the spherulite centre as polariza-tion axes of the fibrils in this region are radially oriented.The BLG spherulites do not appear to have a significantamorphous core (which has been observed in some otherprotein systems and would be expected to give a region ofno net polarisation).

In vector fields that represent fibril structures, the vec-tors are assumed to be perpendicular to the polarisationvectors. In a perfectly radial spherulite, the fibrils, or fib-ril bundles, are straight and radiating from the centre.In some cases, spherulites have abnormalities and are notperfectly radially oriented. The implementation and math-ematical details of a model of the internal spherulite struc-ture and resulting microscopy image are described in thefollowing sections.

4.3 Implementation of model

The mathematical model was implemented in Mathemat-ica. Each polarisation vector field was represented by anarray of vectors. The vectors represent the direction inwhich an electric field can travel through the polariserunchanged. The polarisation vectors are unit vectors andare perpendicular to the principle axis in the polariser.

Spherulites have been modelled as two-dimensionalplanar polarisation vector fields. Within a spherulite, po-larisation vectors are perpendicular to the amyloid fib-rils [3] because of the physically aligned orientation of thefibrils, and hence absorption of the light’s electric field isstronger along the direction of the anisotropy (i.e., fibrils)than perpendicular to it.

The two-dimensional arrays that represent the cross-sections of the light field, the two polarisers, thespherulite(s), and the final intensity each were modelled asa 200×200 square matrix. Spatially, for a BLG spherulitewith a radius of 100μm, this implies that each elementis separated from its nearest neighbour by approximately1μm. The dot products and multiplications were calcu-lated on an element by element basis to find cross-sectionsof light (eqs. (2) and (3)). The final intensity, I, for eachelement was then determined via eq. (4).

4.4 Details of the spherulite polarization field model

Optical microscopy images taken of BLG spherulites atlow pH and high temperatures using light microscopyshowed that in some cases the internal structure of sup-posed spherulite fibril bundles (fibrils are too small tobe seen individually with the resolution of optical mi-croscopy) curl at the edge of the spherulite. An exampledemonstrating this slight curvature within the spherulitecan be seen in fig. 3B. The curling within the spherulitic

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K.R. Domike et al.: Investigating the inner structure of irregular β-lactoglobulin spherulites 177

Fig. 5. Optical microscopy image taken under crossed-polarisers of BLG spherulites with arrows highlighting appar-ent fibril bundle curl.

structures can also be seen in fig. 5 and is highlighted witharrows.

To represent this curvature, or “curl”, the polarisa-tion vectors are written as a sum of a radial componentand a curled component. The curvature component can beachieved mathematically as depicted below, where i and jrepresent the directional vectors in the x and y directions,respectively:

�V1(x, y) =[(

1 − r

R

)

x − r

Ry]

i −[(

1 − r

R

)

y − r

Rx]

j.

(5)In eq. (5), the variables x and y represent the two-

dimensional locations of fibril directionality within aspherulite. The centre of the spherulite is taken to be theorigin. Every coefficient in eq. (5) contains the parameterr/R, where r is the distance the vector is away from thecentre and R is the radius of the spherulite. One boundarycondition is that at the centre of the spherulite, r/R = 0,there is no immediate curl component and the fibrils leavethe nucleation point radially. Additionally, it is assumedthat at the edge of the spherulite where r/R = 1, thereis no radial component. This allows the full radius of thespherulite to have normalised distance of 1. In order tokeep the fibrils at the edge of the spherulite from becom-ing purely tangent to the edge and to control the relativestrength of the curl throughout the spherulite, curvatureparameters a and b were added to �V1 to produce a newparameter �V2:

�V2(x, y) =[(

a − r

R

)

x − br

Ry]

i −[(

a − r

R

)

y − br

Rx]

j.

(6)The parameters a and b are dimensionless parame-

ters related to the relative rate of change of the curlwith respect to the radius. Varying parameter a affectsthe radial direction of the fibrils. The parameter b con-trols the strength of the curl relative to the radial com-ponent. The coefficient before the radial component of �V2

is a − r/R. This modelling equation requires that a is

Fig. 6. A) Array showing the fibril structure, �F , of a radialspherulite with no curl (b = 0). The fibril directional vectorsrepresent the direction of the fibrils, not the actual fibrils them-selves. B) Array showing the polarisation field of the spherulite,�S, that is produced by the radial fibril structure �F .

greater than or equal to unity to keep solutions realistic.The coefficient before the curl component of �V2 is equal tob(r/R). In the case of a perfectly radial spherulite, b = 0,and the spherulite has no curl component and becomes aspherulite with purely radial arrangement (i.e. a perfectly“normal” spherulite producing a “normal” Maltese Crosswhen under crossed-polarisers).

Finally, �V2 is normalised to produce the two-dimen-sional fibril directional vector structure of a spherulite

�F (x, y) =�V2

|�V2|. (7)

The fibril directional vectors do not represent individ-ual fibrils but instead indicate the net direction of thefibrils at that location. Rotating �F by 90◦ yields the po-larisation vector field, �S, of the spherulite:

�S(x, y) = Rot(�F ), (8)

where Rot is a 90◦ rotation matrix.The polarisation vectors in �S are perpendicular to the

vectors in �F because of the aligned orientation of the fib-rils. Assuming that the principal optic axis is along thefibril, the absorption of light’s electric field is strongeralong the direction of the fibrils than perpendicular to it.In this modelling, the microscope polarisers are assumedto be perfect, and it is assumed that the fibril structureacts as a perfect polariser. An example array for a radialspherulite and the resulting polarisation field are demon-strated in fig. 6.

5 Discussion of results

When the light intensity due to a perfectly radiallyoriented spherulite that is situated between crossed-polarisers (such as that presented in fig. 7A) is computedin the model, a resulting “computed image” is producedthat resembles the normal Maltese Cross as seen in fig. 7B.The influence of different types of variations of internalspherulite structure on the resulting computed image ispresented in the following sections.

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Fig. 7. A) Experimentally observed Maltese Cross patternproduced by a BLG spherulite of radially oriented fibrils with a50 μm scale bar shown. B) Computed image from the modellingof a perfectly radial spherulite structure.

Fig. 8. A schematic showing discrete points of fibril curl, θF .Solid black lines illustrate the direction of the fibril as it leavesthe nucleation point and the ending location and show the fibrilcurl angle between them.

5.1 Internal spherulite structure of slightly curledfibrils around a single nucleation point

Many spherulites with a visible rotation to their MalteseCross pattern were observed as exhibited in fig. 3B. Basedon the observation of slightly curving apparent fibril bun-dles in spherulite images with rotated Maltese Cross pat-terns, simulations were performed with the vectors repre-senting the fibril bundles within the spherulite having aslight curl. The extent of curvature, or curl, is quantifiedas an angle of curl, θF , as shown in fig. 8 for discrete pointswithin the simulation. The fibril curl is a function of thedirection of the fibril as it leaves the nucleation point, theending location of the fibril at the edge of the spherulite,and the size of the spherulite.

Two experimentally observed BLG spherulites along-side simulated internal spherulite structures and theresulting computed images are shown in fig. 9. Thespherulite in the top row of the figure has a relativelylarge rotation of its Maltese Cross and the spherulite in thebottom row of the figure has a relatively small rotation ofits Maltese Cross compared to other observed spherulites.The experimental images in fig. 9A and D are exampleswhere fibril bundles appear to be visible. The visualisationof the fibril bundles was not found in most images, but,qualitatively, was found to be common in the images withrotated Maltese Crosses. The reason for being able to view

Fig. 9. (Colour on-line) A) Microscopy image of BLGspherulite with orthogonal axes as yellow dashed lines anda 50 μm scale bar as a solid orange line. B) The vector fieldmodelling the inner structure of curled fibril bundles with arepresentative curl as a red dotted line with an angle of curl,θF , of 21.8◦. C) A computed spherulite image in which thefibril bundles curl based on the fibril directional vectors shownin B). D) Microscopy image of BLG spherulite with orthogonalaxes as yellow dashed lines and a 50 μm scale bar as a orangesolid line. E) The vector field modelling the inner structure ofslightly curled fibril bundles with an angle of curl, θF , of −9.3◦.F) A computed spherulite image based on the fibril directionalvectors shown in E) in which the Maltese Cross rotates slightlyfrom the orthogonal axes (yellow dashed lines).

the fibril bundles in these images and not in other imagesmay be that in these spherulites, the fibrils are curling andtherefore traverse the crossed polarisers and expose theirgeometry, or, alternatively, that the spatial packing of thefibrils is different and allows for visualisation of the fibrilbundle orientation. The inclusion of curled vectors in thesimulation results in a Maltese Cross rotated in the op-posite direction from the curl, showing clockwise rotationin fig. 9C and consistent with the experimental image infig. 9A. The same concept is shown with rotation in theopposite direction in fig. 9F and D.

In the far outer regions of the spherulites, the fibrilsmay be sufficiently separated that they do not significantlypolarise the light. Spatially separated fibrils do not scatterlight because of their small volume [20]. The effect of thiscan be seen in the images presented in fig. 10. The samespherulite is shown in both fig. 10A and B, but viewedwith a red/green phase contrast filter in fig. 10B. Looselypacked fibril shapes in the outer regions can be seen infig. 10B, beyond the region which appears as the edge ofthe spherulite in the crossed-polariser image of fig. 10A.

Due to the visible rotation of the Maltese Cross whenthere appeared to be fibril curling, an effort was madeto compute the generic relationship between fibril curland Maltese Cross rotation. Computed Maltese Cross im-ages were generated from inner vector structures of vari-ous curl angles. The correlation of the simulated internalspherulite fibril curl and the resulting computed MalteseCross rotation are plotted in fig. 11. The relationship be-tween simulated fibril curl and resulting Maltese Cross

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Fig. 10. Optical microscopy images of a BLG spherulite with25 μm scale bar where A) was imaged using crossed polarisersand B) was imaged using crossed polarisers and an additionalphase contrast filter.

Fig. 11. A plot of computed Maltese Cross rotation from curlof internal spherulite fibril orientation. Both Maltese Cross ro-tation and fibril curl have been taken to be positive althoughthey curl in opposite directions.

rotation appears linear. This computational result pro-vides insight into the inner structure of the spheruliteswith abnormally rotated Maltese Crosses observed ex-perimentally in the BLG solutions. However, the under-lying cause of this abnormal inner structure is as yetunidentified. As significant Maltese Cross rotation wasonly observed experimentally in some (but not all) high-temperature experiments (80 ◦C and 85 ◦C), one possiblehypothesis of the underlying cause of fibril curl is that athermal gradient in the slide well, from hot spots near theheating stage at the bottom of the well to cooler spotsnear the slide cover exposed to ambient conditions, causessmall convective flows in the solutions that swirl past thespherulites and cause curvature in the fibril bundles. Thepresence and magnitude of the convective flows could bespecific to the relative geometry of the slide and the heat-ing stage and to the location of microscopic observation,both of which could have varied across experiments.

5.2 Internal spherulite structure of multiple nucleationpoints

Structures were experimentally observed that appeared tobe composed of two halves or three sections, suggesting

Fig. 12. (Colour on-line) A) Optical microscopy image of afull grown BLG spherulite with a 50 μm scale bar shown asan orange solid line. B) Computed image produced by mod-elling two nucleation points, located at (−2.4 μm, +3.2 μm)and (+2.4 μm, −3.2 μm) relative to the centre of the image,in which the fibrils stop growth as they meet. C) Same asimage B) with nucleation points highlighted with circles andgrowth boundary highlighted with a straight line. D) Opticalmicroscopy image of a BLG spherulite with a 50 μm scale baras a solid orange line. E) Computed image of three spherulitenucleation points located at (+3.4 μm, +3.4 μm), (−0.7 μm,−2.7 μm), and (−3.4 μm, −1.4 μm) relative to the centre ofthe image, and each spherulite having an angle of curl, θF , of−7.6◦. F) The vector plot created of the internal fibril struc-ture of the spherulite in D). G) Optical microscopy image oftwo BLG spherulites with a 50 μm scale bar as a solid orangeline. H) Computed image produced by two spherulite nucle-ation points separated by 9.6 μm in each orthogonal direction.

that the image is composed of multiple spherulites thathave grown close to each other. This could be caused bymultiple, distinct but close nucleation points resulting inadjacent spherulites that form domain boundaries wherethey meet.

Multiple nucleation points in close proximity withspherulites that stop growing upon contact with eachother were investigated computationally. The number ofnucleation points, nucleation point separation distance,orientation of the nucleation points with respect to thepolarisers, and internal fibril curl were all varied. Threeexamples of spherulites with visible multiple nucleationpoints and associated computed images are shown infig. 12. In the first row of fig. 12, the computed imagecontains two spherulite nucleation points, one located at(−2.4μm, +3.2μm) relative to the centre of the imageand the other located at (+2.4μm, −3.2μm) relative tothe centre of the image. There are notable similarities be-tween the Maltese Cross properties of the image of a BLG

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spherulite and its simulated counterpart. In both fig. 12Aand B, there is a diagonal line where the two spherulitehalves meet, as explicitly drawn in fig. 12C. Both exper-imental and computed images also show the left half ofthe horizontal arm of the Maltese Cross as slightly higherthan the right half. An additional consistency between theexperimental and computed images is the presence of abright area in the centre of both crosses. These qualitativeagreements between the experimental image and the com-puted image suggest that, in this particular case, there aretwo spherulites with nucleation points close to each other,and the spherulites stop growth upon impinging uponone another. The result is the illusion of one spherulite.In the second row of fig. 12, the fibril structure vectorplot is based on the experimental image and modelledas three spherulite nucleation points located at (+3.4μm,+3.4μm), (−0.7μm, −2.7μm), and (−3.4μm, −1.4μm)relative to the centre of the image. Each spherulite is mod-elled with an angle of curl, θF , of −7.6◦. There is remark-able similarity between the experimental image and thecomputed image. Specifically, there is brightness in thecentre of the experimental image, fig. 12D, consistent withthe computed image, fig. 12E. Additionally, there is slightcurving at the ends of the arms within the Maltese Crossvisible in both images. There are boundaries visible infig. 12D that are less distinct than the arms of the Mal-tese Cross, but correlate with the boundaries going fromthe centre outward in northwest, southwest and south-east directions in the computed image. In the third rowof fig. 12, the two simulated nucleation points are locatedfarther apart (separated by 9.6μm in each orthogonal di-rection for a total separation distance of 13.6μm) fromeach other relative to the total spherulite radius. The re-sults from the computation performed with these two nu-cleation points matched well with the experimental imageof fig. 12G where two spherulites grew in close proximity.

It is unclear from the individual experimental imagesand the summary of temperature-dependent experimen-tal results recorded in table 1 why multiple nucleationpoints occur in close proximity to one another or whatenvironmental factors influence the likelihood of nucle-ation points in close proximity. The abnormalities asso-ciated with multiple nucleation points in proximity (extralobes, mis-shapen) were found in experiments at all testedtemperatures (as can be seen in table 1). Physically, nu-cleation points may occur with higher probability in somesub-volumes of the solution which would lead to a likeli-hood of multiple nucleation points in close proximity (e.g.,perhaps near an edge, boundary layer, or higher than nor-mal concentration of protein or impurities). The detailsof sources of BLG spherulite nucleation are not yet deter-mined from the observations in this research.

5.3 Internal spherulite structure of opposite curlsaround a single nucleation point

Of the five types of abnormal spherulite images intro-duced in fig. 3, only the small wedge shape (fig. 3C) isnot described by either curling fibril bundles and/or mul-

Fig. 13. A) Schematic looking down on a spherulite with asingle nucleation point where the top hemisphere (solid lines)is one half spherulite with fibrils curled counter-clockwise andthe lower hemisphere (dotted lines) is another half spherulitewith fibrils curled clock-wise. B) Isometric view of a hyperbolicaggregate structure. The lines represent the aggregate directionof fibrils at that point within the spherulite. The space in thefigure without lines does not indicate that there are no orientedfibrils in that space. Note that fibril orientation has an oppositecurl in the upper half compared to the lower half. C) Top viewof the hyperbolic aggregate structure.

tiple nucleation points in close proximity. One structurethat was computationally investigated that forms a smallwedge, but not a Maltese Cross, is one in which the fibrilbundles have opposite curl direction in the bottom hemi-sphere compared to the top hemisphere. The image ex-pected to result from this structure was computed usinga two-layered spherulite vector field: one layer with fib-ril directionality of counter-clockwise curl, and the sec-ond spherulite vector field composed using clockwise curl,placed one above the other (in the z-direction). Physi-cally, this would occur if two spherulites nucleated at thesame (or very close to the same) point and were on top ofeach other relative to the optical microscope observation,rather than next to each other in the plane of the field ofview, and the fibrils in each hemisphere curled in oppo-site directions from the other hemisphere. At the interfaceof the two hemispheres, the spherulites were assumed tostop growing. A schematic of this type of structure (i.e.,one curl direction on top, opposite direction on the bot-tom) is shown in fig. 13A. Another case in which this innerstructure could result is a single hyperbolic spherulite re-sembling the shape of an hour glass as shown in fig. 13Band C. Hyperbolic structures have been found in some liq-uid crystal structures [21]. Experiments were performedwhich attempted to rotate a single spherulite within anoptical microscope to view changes in the Maltese Crossstructure and determine if either of these structures werepresent. However, these attempts were unsuccessful andfurther research is necessary to refute or support the pres-ence of a hyperboloid.

In the Maltese Cross structure resulting from an in-ternal spherulite structure of either the opposite curl di-rections around a single nucleation point or a hyperbolicaggregate, one pair of lobes in the image is significantlysmaller than the other pair. Computationally, strongercurl within the spherulite results in the two small lobesdecreasing in size. This was determined by computing im-ages with variations in the degree of curl in top and bot-tom hemispheres independently. If the orientation of thepolarisers is known, then the spherulite structure can bedetermined by observing which lobes are small and large.

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Fig. 14. (Colour on-line) A) Experimental image of a BLGspherulite with a 50 μm scale bar shown as a solid orange line.B) A computed image from a spherulite structure in which thefibrils in the top hemisphere curl counter-clockwise and in thebottom hemisphere curl clockwise. C) Truncated experimentalimage of a BLG spherulite with a 50 μm scale bar shown asa solid orange line. D) A computed image from a spherulitestructure in which the fibrils in the top hemisphere curl clock-wise and in the bottom hemisphere curl counter-clockwise. Inthis case, the lobes (the bright areas) are small because thecurl is relatively strong.

Although the resulting simulated spherulite image doesproduce a small wedge in two of the four quadrants, it doesnot match very well with the small wedge type of abnor-mality introduced in fig. 3. There were some other exper-imental spherulite images that were matched well by thecomputed image. Two experimental images of spherulitesthat were classified as mis-shapen type and the associatedtwo-dimensional computed images are shown in the tworows of fig. 14. The experimental image in the second rowof the figure is slightly out of focus, but the spatial inten-sity of the light can be discerned. The primary differencebetween the experimental images in the two rows of fig. 14is which lobes are small and large. Computationally, thislobe difference is attributed to a difference in the directionof curl of the two hemispheres.

Experimental spherulite images that were well de-scribed by computed images utilizing internal spherulitestructures of opposite curls around a single nucleationpoint were not frequently observed. It is unclear from theindividual experimental images and from the temperaturedependent summary experimental results recorded in ta-ble 1 why or when this phenomenon occurs. As mentionedpreviously, it may be that the two semi-spheres with op-posite curl are two spherulites with vertically aligned (ornearly aligned) nucleation points. If this is the case, thenthese mis-shapen images may be a subset of images inwhich nucleation points are in close proximity to one an-other. This type of image, as indicated by multiple lobes in

table 1, were relatively common (found in approximately17% of the viewed spherulites). The rationale for the twohemispheres having opposite curls is not yet determinedfrom the observations in this research.

6 Conclusions

Many of the β-lactoglobulin spherulites observed experi-mentally under crossed polarisers had “normal” MalteseCross images. “Normal” Maltese Crosses are those withfour nearly identical quadrants without significant rota-tion to the cross. Some of the spherulites observed un-der crossed polarisers were not “normal.” These abnormalspherulite images were grouped into four types: those withsignificant rotation to the cross, those with small wedgesnear the centre, those with extra lobes (greater than four),and those with other mis-shapes. The number of each typeof spherulite was measured for each β-lactoglobulin ex-periment completed across a range of temperatures from60 ◦C to 85 ◦C. Across all 101 spherulites, normal MalteseCrosses were found in 50%, and abnormalities of the fourtypes were each found in 17% ± 3% of the spherulites.No clear dependence of abnormal frequency on tempera-ture was found, perhaps because of the small number ofdata points in each experiment. Two observations of notethat were found across the experiments are: in all exper-iments at least one normal spherulite was present, andabnormalities of a certain type were either absent from anexperiment or likely to be present in larger than averagefrequency. These observations lead to the conclusion thatunknown details of the solution conditions can cause sys-temic propensities for spherulite abnormalities, but thatin each observed solution normal spherulites can form.

A mathematical model was developed which allowedfor creation of computed images from defined innerspherulite structure and allowed for comparison to ex-perimentally observed images. This comparison enablessuggestions of the influence of the spherulite inner struc-ture on the resulting images of β-lactoglobulin spheruliteswhen viewed by optical microscopy with crossed polaris-ers. The types of inner spherulite structures investigated inorder to describe abnormal Maltese Cross images includespherulites with fibrils slightly curled outward from a sin-gle nucleation point (resulting in rotated Maltese Crossimages), the growth of spherulites from multiple nucle-ation points in close proximity that form growth bound-aries at their interfaces (resulting in Maltese Cross imageswith extra lobes), and two lobes of fibrils curled in oppo-site directions from a single nucleation point (resulting inimages similar to some mis-shapen Maltese Cross images).

The model was used to investigate the generic quanti-tative relationship between extent of spherulite fibril bun-dle curl and rotation of the resulting Maltese Cross shape.The model was also used to attribute specific abnormalexperimental Maltese Cross images to certain spheruliteinner structures. An example of this approach was themodelling of two spherulites growing from multiple, dis-tinct nucleation points in close proximity to explain Mal-tese Cross images that appear to have a solid line through

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the image (which the computation results suggest is at thedomain boundary of the two spherulites). In this way, themodel provided significant insight into the inner structureof β-lactoglobulin spherulites that was not possible withthe optical images alone.

This research was supported by the EPSRC, the NSF DMR-0649112 and The College of Wooster. We would like to thankDr. Shila Garg for her discussion about the existence of hyper-bolic liquid crystal structures.

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