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Article| Volume 110, ISSUE 3, P646-660, February 02, 2016

# Principles Governing the Self-Assembly of Coiled-Coil Protein Nanoparticles

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## Abstract

Self-assembly refers to the spontaneous organization of individual building blocks into higher order structures. It occurs in biological systems such as spherical viruses, which utilize icosahedral symmetry as a guiding principle for the assembly of coat proteins into a capsid shell. In this study, we characterize the self-assembling protein nanoparticle (SAPN) system, which was inspired by such viruses. To facilitate self-assembly, monomeric building blocks have been designed to contain two oligomerization domains. An N-terminal pentameric coiled-coil domain is linked to a C-terminal coiled-coil trimer by two glycine residues. By combining monomers with inherent propensity to form five- and threefold symmetries in higher order agglomerates, the supposition is that nanoparticles will form that exhibit local and global symmetry axes of order 3 and 5. This article explores the principles that govern the assembly of such a system. Specifically, we show that the system predominantly forms according to a spherical core-shell morphology using a combination of scanning transmission electron microscopy and small angle neutron scattering. We introduce a mathematical toolkit to provide a specific description of the possible SAPN morphologies, and we apply it to characterize all particles with maximal symmetry. In particular, we present schematics that define the relative positions of all individual chains in the symmetric SAPN particles, and provide a guide of how this approach can be generalized to nonspherical morphologies, hence providing unprecedented insights into their geometries that can be exploited in future applications.

## Introduction

### Self-assembling protein nanoparticle

Self-assembly is one of the mechanisms utilized by nature to foster the creation of diverse architectures from a limited toolset of building blocks. Striking examples are spherical viruses, which usually have a protective protein shell (capsid or nucleocapsid) that is formed from its constituent protein building blocks by such a process (
• Zandi R.
• Reguera D.
• Rudnick J.
• et al.
Origin of icosahedral symmetry in viruses.
,
• Crick F.H.
• Watson J.D.
Structure of small viruses.
,
• Caspar D.L.
• Klug A.
Physical principles in the construction of regular viruses.
). An intriguing feature of this assembly mechanism is that some of these viruses do not require more than one type of capsid protein to fully encase their nucleic acid genome. The building blocks arrange themselves according to geometric constraints, and can form stable shapes with a high level of efficiency, maximizing container volume while minimizing the length of the genomic sequence required to code for it. In the design of artificial self-assembling nanoparticles, such as those discussed in this article, the same principle can be exploited, and building blocks engineered to have specific geometric properties can guide the efficient formation of nanoscale structures, allowing a level of control that could otherwise not be achieved with tools developed at the micro level (
• Nikitin M.P.
• Zdobnova T.A.
• Deyev S.M.
• et al.
Protein-assisted self-assembly of multifunctional nanoparticles.
,
• Sun J.
• DuFort C.
• Dragnea B.
• et al.
Core-controlled polymorphism in virus-like particles.
,
• Kostiainen M.A.
• Hiekkataipale P.
• Ceci P.
• et al.
Electrostatic assembly of binary nanoparticle superlattices using protein cages.
). Using protein oligomers and applying symmetry principles, King et al. (
• King N.P.
• Sheffler W.
• Baker D.
• et al.
Computational design of self-assembling protein nanomaterials with atomic level accuracy.
,
• King N.P.
• Bale J.B.
• Baker D.
• et al.
Accurate design of co-assembling multi-component protein nanomaterials.
) have been able to engineer nanocages with atomic precision.
In this study, we analyze the self-assembly of a single protein chain that was inspired by the icosahedral symmetry governing viral capsid formation. The geometric foundation of the design is based on the concept of monomeric protein chains self-assembling into a larger nanoparticle architecture, as previously introduced (
• Caspar D.L.
• Klug A.
Physical principles in the construction of regular viruses.
). This self-assembling protein nanoparticle (SAPN) has successfully been utilized as a vaccine for the presentation of epitopes, resulting in high immunogenicity, even in the absence of adjuvants (
• Pimentel T.A.
• Yan Z.
• Burkhard P.
• et al.
Peptide nanoparticles as novel immunogens: design and analysis of a prototypic severe acute respiratory syndrome vaccine.
,
• Kaba S.A.
• Brando C.
• Lanar D.E.
• et al.
A nonadjuvanted polypeptide nanoparticle vaccine confers long-lasting protection against rodent malaria.
,
• Kaba S.A.
• McCoy M.E.
• Lanar D.E.
• et al.
Protective antibody and CD8+ T-cell responses to the Plasmodium falciparum circumsporozoite protein induced by a nanoparticle vaccine.
,
• Wahome N.
• Pfeiffer T.
• Burkhard P.
• et al.
Conformation-specific display of 4E10 and 2F5 epitopes on self-assembling protein nanoparticles as a potential HIV vaccine.
,
• Babapoor S.
• Neef T.
• Burkhard P.
• et al.
A novel vaccine using nanoparticle platform to present immunogenic M2e against avian influenza infection.
,
• El Bissati K.
• Zhou Y.
• McLeod R.
• et al.
Effectiveness of a novel immunogenic nanoparticle platform for Toxoplasma peptide vaccine in HLA transgenic mice.
), and also for the design of other tools for nanobiotechnology applications (
• Yang Y.
• Burkhard P.
Encapsulation of gold nanoparticles into self-assembling protein nanoparticles.
,
• Maalouli N.
• Gouget-Laemmel A.C.
• Szunerits S.
• et al.
Development of a metal-chelated plasmonic interface for the linking of His-peptides with a droplet-based surface plasmon resonance read-off scheme.
). The design resulting from the monomeric building blocks depends on the presence of two oligomerization domains in a single protein chain, one of which forms a pentameric coiled-coil domain and the other a trimeric coiled-coil domain (Fig. 1 A).
Specifically, two α-helical coiled-coil domains, an N-terminal peptide derived from cartilage oligomeric matrix protein that forms pentameric coiled coils, and a de novo designed C-terminal leucine zipper peptide that forms trimeric coiled coils, are joined by two glycine residues to drive the formation of higher order nanoparticles. The individual coiled-coil domains form long and thin structures that allow protein assembly around fivefold and the threefold local symmetry axes (Fig. 1 B). The original idea was that, in analogy to viral capsids, these domains should align along global five- and threefold symmetry axes, forming icosahedrally symmetric structures but maintaining a length that will not sterically hinder the formation of a spherical core-shell particle (Fig. 1 C). We will consider here, more generally, spherical particles formed with any kind of three-dimensional (3D) symmetry, which exhibit the characteristic local three- and fivefold symmetry axes that SAPN particles are designed to form, and we also discuss how our approach could be generalized to model nonspherical morphologies. The self-assembly of these domains is governed by a combination of hydrophobic and ionic interactions present within the individual coiled-coil domains, as well as the symmetry constraints that determine the growth of the nanoparticle (
• Lee D.L.
• Ivaninskii S.
• Hodges R.S.
• et al.
Unique stabilizing interactions identified in the two-stranded α-helical coiled-coil: crystal structure of a cortexillin I/GCN4 hybrid coiled-coil peptide.
,
• Meier M.
• Lustig A.
• Burkhard P.
• et al.
Removing an interhelical salt bridge abolishes coiled-coil formation in a de novo designed peptide.
,
• Meier M.
• Stetefeld J.
• Burkhard P.
The many types of interhelical ionic interactions in coiled coils—an overview.
).
What has not been addressed in previous studies is the need for a stringent treatment of why such a monomer, encoded by the two underlying coiled-coil domains, would self-assemble into a nanoparticle of definable properties. Therefore, this article is an attempt to describe some of the morphologies of these nanoparticles in more detail. We address the following question: if one were to place single chains of the protein into a buffer, what are the symmetrically spherical relevant possibilities that can occur? Because the SAPN oligomerization domains are limited to those that form pentamers and trimers, the surface structures introduced by Caspar and Klug (
• Crick F.H.
• Watson J.D.
Structure of small viruses.
) in the framework of quasi-equivalence theory are not sufficient to model these structures.
This article is organized as follows. First, we experimentally determine in which cases the nanoparticles have a spherical core-shell morphology. Second, we define the mathematical rules governing the arrangement of the coiled-coil building blocks on the nanoparticle surfaces. We show that in addition to an ensemble of polymorphous particles, there is a significant yield of spherical particles with well-defined, highly regular structures that could potentially be used in nanotechnology applications and provide structural blueprints for each.

## Materials and Methods

### Molecular cloning

The DNA strands encoding the protein sequences for SAPN-K (sequence: MGHHHHHHGDWKWDGGLVPRGSDEMLRELQETNAALQDVRELLRQQVKQITFLRALLMGGRLLARLEELERRLEELERRLEELERAINTVDLELAALRRRLEELAR), and SAPN-R (sequence: MGHHHHHHGDWKWDGGLVPRGSDEMLRELQETNAALQDVRELLRQQVKQITFLRALLMGGRLLARLEELERRLEELERRLEELERAINTVDLELAALRRRLEELAR) were obtained from Integrated DNA Technologies (Coralville, IA). Both sequences include histidine-tags to facilitate later purification. Standard annealing and ligation procedures were performed to place the oligonucleotides into a pPEP-T plasmid vector, which was transformed into an Escherichia coli BL21 strain.

### Protein expression and purification

The transformed cells were incubated at 37°C in Luria broth containing the antibiotics, ampicillin (100 μg/mL) and chloramphenicol (34 μg/mL). When the optical density of the cell culture had reached ∼0.8, isopropol β-D-thiogalactopyranoside (1 mM) was added to induce protein expression.
After incubation for 4 h, the excess Luria broth was decanted and the cell culture was centrifuged for 10 min at 4000 × g. The culture was then frozen at −20°C overnight, thawed at room temperature, and resuspended in a buffer containing 9 M urea, 100 mM NaH2PO4, and 10 mM Tris, at pH 8.0. Lysis of the pellet was achieved by sonication, with the excess cell waste being centrifuged at 30,600 × g for 45 min. The supernatant was decanted and placed in nickel-NTA beads (Qiagen, Madison, WI) at room temperature overnight. The supernatant/nickel-NTA slurry was then used to fill a column for purification. Purification occurred over a pH gradient (8.0, 6.3, 5.9, 5.1, and 4.5), with 20 mM sodium citrate replacing the 10 mM Tris in all buffers except at pH 8.0. Elution of the protein from the column was achieved using an imidazole gradient ranging from 50, 150, 300, and 500 mM to 1 M imidazole, at pH 8.0. Purity was ascertained by sodium dodecyl sulfate polyacrylamide gel electrophoresis. A molecular-weight ladder was used to correlate SAPN-R and SAPN-K with the expected sizes.
The purified fractions were filtered using a 0.1 μm polyvinylidene fluoride membrane (Cat. No. SLVVO33 RS; Millipore, Billerica, MA). SAPN-K was placed in a buffer containing 8 M urea, 150 mM NaCl, and 20 mM HEPES, at pH 7.5. SAPN-R was placed in the same buffer supplemented by 5% glycerol. Stepwise dialysis was performed to remove the urea, and promote SAPN self-assembly with the following dialysis steps: 8, 6, 4, 2, and 1 M urea, and finally dialysis into a 0 M urea buffer, with all other buffer constituents remaining constant. The SAPN particles in 0 M urea buffer were filtered through a 0.1 μm filter (Millipore).

### Scanning transmission electron microscopy

The nanoparticles were at a concentration of 400 μg/mL in 20 mM Tris pH 7.5, 150 mM NaCl. It was diluted 12× in buffer immediately before the scanning transmission electron microscopy (STEM) grid was prepared. Afterwards, a 5 μL sample aliquot was adsorbed for 1 min to a glow discharged STEM film (thin carbon film that spans a thick fenestrated carbon layer covering 200-mesh/inch, gold-plated copper grids). The grid was then blotted, washed on eight drops of quartz double-distilled water, blotting between each step, and plunge-frozen in liquid nitrogen. It was freeze-dried at −80°C and 5 × 10−8 Torr overnight in the microscope. Tobacco mosaic virus particles (kindly provided by R. Diaz Avalos) were used as a mass standard. These particles were similarly adsorbed to separate STEM grids, washed on 8 drops of 100 mM ammonium bicarbonate, and air-dried.
A Vacuum Generators HB-5 scanning transmission electron microscope interfaced to a modular computer system (Tietz Video and Image Processing System, Gauting, Germany) was employed. Series of 512 × 512 pixel, dark field images were recorded from the unstained sample at an acceleration voltage of 80 kV and a nominal magnification of 200,000×. The recording dose was 402 ± 18 electrons/nm2. Selected regions of the sample grid were also repeatedly scanned to determine the beam-induced mass-loss incurred by the nanoparticles. The digital images were evaluated using the program package MASDET (
• Krzyzánek V.
• Müller S.A.
• Reichelt R.
• et al.
MASDET-A fast and user-friendly multiplatform software for mass determination by dark-field electron microscopy.
). In short, projections were selected in circular boxes and the total scattering of each region was calculated. The average background scattering of the carbon support film was then subtracted and the mass calculated. The results were scaled according to the mass per length measured for tobacco mosaic virus, corrected for beam-induced mass-loss (
• Müller S.A.
• Goldie K.N.
• Engel A.
• et al.
Factors influencing the precision of quantitative scanning transmission electron microscopy.
), binned into histograms, and described by Gaussian curves. The number of particle masses within each peak (n) was estimated from the peak area. Nanoparticles with masses in operator-defined ranges centered at the peak maxima were sorted into galleries for inspection.

### Calculation of circularity

Previous analytical ultracentrifugation and STEM studies showed SAPN to be polymorphous. However, a majority of the species still had essentially circular projections. This was ascertained by measuring the circularity of the SAPN from the electron micrograph via the Analyze Particles module present in ImageJ (
• Abràmoff M.D.
• Magalhães P.J.
• Ram S.J.
Image processing with ImageJ.
), where the circularity is obtained from the equation 4π (area/perimeter2), with a 1.0 score indicating a perfect circle/roundness.

### Small angle neutron scattering

The small angle neutron scattering (SANS) experiment was performed at 30 m NG3-SANS located at the National Institute of Standards and Technology (NIST, Gaithersburg, MD). The scattered neutron intensity was obtained as a function of scattering vector, q, defined as $(4π/λ)sin(θ/2)$, where λ and θ are the neutron wavelength and scattering angle, respectively. The λ and Δλ/λ were set to be 6 Å and 15%, respectively. We chose three sample/detector distances 1, 4, and 13 m to cover the q range from 0.004 to 0.4 Å−1. The raw scattering intensity was then normalized by the incident neutron flux and exposed sample volume to yield absolute intensity (
• Wignall G.L.
• Bates F.
Absolute calibration of small-angle neutron scattering data.
), I(q), which for particulate systems can be mathematically expressed as ϕΔρ2P(q)S(q), where ϕ, Δρ, P(q), and S(q) are the volume fraction of the particles, the neutron scattering length density’s difference among the particles and the medium, the form factor, and the structure factor, respectively. Because the scattering lengths of hydrogen and deuterium are drastically different, to enhance contrast without changing the chemical properties, deuterium oxide (D2O) is commonly used to replace H2O. The form factor, P(q), is the square of the spatial Fourier transform of density function of the particle, containing the morphological information (e.g., size and shape) of the particles. The structure factor, S(q), is contributed by the interactions among particles. In this study, based on the self-assembly mechanism a polydisperse spherical shell (PSS) model derived by Hayter (
• Hayter J.B.
Physics of amphiphiles: micelles, vesicles and microemulsions.
) was proposed to be used as P(q), while the SAPN is presumably dilute enough (minimal interparticle interaction), thus S(q) is assumed to be 1.
Specifically, the PSS model describes hollow spheres with a constant shell thickness of t and a polydisperse core whose average radius is 〈Ri〉. The PSS model can be written as
$P(q)=∫0∞f(Ri)16π2q6[(sin(qζRi)−sin(qRi))−qRi(ζcos(qζRi)−cos(qRi))]2dRi,$
(1)

where ζ is the ratio of outer/inner radii, i.e., $t+〈Ri〉/〈Ri〉$. The inner radius, Ri, follows the Schultz distribution function, f(Ri) as expressed in Eq. 2, where p is the polydispersity of Ri, defined by $σR/〈Ri〉$, where σR is the standard deviation of Ri:
$f(Ri)=p−2p2〈Ri〉Γ(1p2)(Ri〈Ri〉)1−p2p2.$
(2)

This morphology is depicted later in Fig. 4 B.

### SAPN-R preparation for SANS

The sphericity of the nanoparticle SAPN-R in a buffer containing 20 mM HEPES, 150 mM NaCl, 450 mM Urea, and 5% glycerol, at a pH of 7.5 was investigated at three different protein concentrations: 0.119, 0.289, and 0.435 mg/mL. To increase the contrast ratio, samples were prepared with a 95% D2O to 5% H2O ratio. To transfer the samples into D2O buffer, SAPN-R protein in 9 M urea, 20 mM HEPES, 150 mM NaCl, at pH of 7.5, was concentrated up to an approximate concentration of 1 mg/mL by centrifugation. The concentrated protein was then diluted 20-fold to a concentration of 0.05 mg/mL via a dropwise method into a buffer containing 20 mM HEPES, 150 mM NaCl, at a pH of 7.5 in 100% D2O. This meant that there was a residual amount of urea (450 mM) and H2O (5%). The dilute protein was then concentrated using centrifugation filters (No. SLVV033 RS; Millipore) to give the three different concentrations. The final protein concentration was determined by UV absorption at 280 nm.

### Dynamic light scattering

The refolded protein (SAPN-R) was characterized at three different concentrations by dynamic light scattering (DLS) using a Malvern Zetasizer Nano S (Worcestershire, UK). An average of five DLS readings was obtained over a period of 10 min per run, at a temperature of 25°C.

### Transmission electron microscopy

For the transmission electron microscopy (TEM), a 5 μL sample aliquot of the SAPN-R sample used for STEM was adsorbed for 60 s to a glow discharged, carbon-coated STEM film. The grid was then blotted, washed on eight droplets of quartz double-distilled water, and negatively stained on two droplets of 2% (w/v) uranyl acetate. Bright field images were recorded on photographic film using a model No. 7000 electron microscope (Hitachi, Tokyo, Japan) at an accelerating voltage of 100 kV and a magnification of 39,000×. Negatives were digitized on a PrimescanD 7100 drum scanner (Heidelberg, Kennesaw, GA) yielding a final pixel size corresponding to 0.5 nm at the specimen level.

## Results and Discussion

### The SAPN system

Higher order SAPNs were generated from two different SAPN species, SAPN-K (during STEM runs), and SAPN-R (during SANS, DLS, and TEM runs). For the purpose of this study, the two species can be considered as virtually identical, because they differ only by a single point mutation to a very similar amino acid in the respective monomers (present in the region encoding the pentameric coiled-coil) that is not proximal to the flexible double glycine hinge/linker region (Fig. 1 B). The flexible linker region demarcates the separation of the pentamer encoding domain from the trimer encoding one. The linker region might also have considerable influence on the formation of higher order oligomers, its flexibility allowing the addition of extra monomers (through compaction) into any given particle.

### Morphology of the protein nanoparticles

While the self-assembly of the individual coiled-coil domains of the nanoparticle is well documented (
• Malashkevich V.N.
• Kammerer R.A.
• Engel J.
• et al.
The crystal structure of a five-stranded coiled coil in COMP: a prototype ion channel?.
,
• Burkhard P.
• Meier M.
• Lustig A.
Design of a minimal protein oligomerization domain by a structural approach.
), the polymorphism displayed by SAPNs has not been fully elucidated (
• Pimentel T.A.
• Yan Z.
• Burkhard P.
• et al.
Peptide nanoparticles as novel immunogens: design and analysis of a prototypic severe acute respiratory syndrome vaccine.
,
• Raman S.
• Machaidze G.
• Burkhard P.
• et al.
Structure-based design of peptides that self-assemble into regular polyhedral nanoparticles.
,
• Yang Y.
• Ringler P.
• Burkhard P.
• et al.
Optimizing the refolding conditions of self-assembling polypeptide nanoparticles that serve as repetitive antigen display systems.
). The data reported below and less detailed analyses reported in these earlier articles show that the molecular weight of the assembled SAPNs and accompanying polymorphism of the sample varies depending on the sequence and the buffer conditions.
Negative-stain TEM gave a visual impression of the SAPN-K heterogeneity (Fig. 2; Fig. S1 in the Supporting Material). The particle projections vary in size, and although some are circular others are clearly elongated. The irregularly spaced protrusions at the edges of many projections indicate a degree of particle flexibility.
A quantitative measure of particle heterogeneity was obtained by STEM. The single molecule mass measurement technique offered by STEM calculates the mass of freeze-dried unstained single particles (molecules) from their electron scattering power. Each mass value is directly linked to the projected image of the particular particle, allowing the projections to be classified according to the mass histogram and inspected (Figs. 3 A and S2). For SAPN-K the analysis confirmed the presence of different molecular species (Figs. 3 A and S2), allowed the average number of peptide chains present in each to be estimated (Fig. 3 B), and showed most particles to have essentially circular projections at the magnification employed (Figs. S2 and 3 B) compatible with, but not exclusive to, a spherical nature. The results in Fig. 3 C show that the measured particles maintain a high degree of circularity until a size of ∼360 chains, which indicates that they are most likely spherical. The data imply that there is a molecular-weight cutoff for the formation of spherical particles, and set the upper limit of possible symmetrically defined core-shell species to this particle size. This is consistent with viral shell morphologies: up to that number of capsid proteins (e.g., the T = 7d shell of polyoma virus) such shells appear mostly spherical, while for larger structures buckling leads to the occurrence of flat faces and an overall icosahedral shape, i.e., a change in the qualitative nature of the capsid architecture. In general, larger capsids require auxiliary components (such as scaffold proteins) to aid shell formation; because such additional features are absent here, it is perhaps not surprising that spherical shell morphologies cease to exist for larger chain numbers. The higher mass values (>5.6 MDa) in the histogram arose from some higher order molecular-weight malformations and asymmetric aggregates (possibly tubular in nature), as well as simply from the close proximity of two lower mass nanoparticles (Fig. S2).
The spherical particles are candidates for geometries in which domains are aligned along rotational symmetry axes, such as those of icosahedral symmetry, mimicking the core-shell architectures reminiscent of symmetry-constrained viruses. This means that each such particle should have a cavity or a central core (based on computer models, 5–10 nm in diameter), as well as an outer protein layer or shell encompassing the individual length of the trimer or pentamer; in other words, the particles should have a core-shell form. Because the STEM results only provide a two-dimensional (2D) perspective of the nanoparticles, SANS experiments were performed to gain an understanding of their 3D shape. This technique utilizes nondestructive neutron scattering events to determine the structural properties of samples (
• Chen S.-H.
Small angle neutron scattering studies of the structure and interaction in micellar and microemulsion systems.
). The SANS data were best fitted using the PSS model (Fig. 4 B) as shown in Fig. 4 A (
• Nieh M.-P.
• Raghunathan V.A.
• Katsaras J.
• et al.
Spontaneously formed unilamellar vesicles with path-dependent size distribution.
,
• Yue B.
• Huang C.-Y.
• Katsaras J.
• et al.
Highly stable phospholipid unilamellar vesicles from spontaneous vesiculation: a DLS and SANS study.
).
The three samples with different SAPN-R concentrations were best fitted to this model, yielding an adequate agreement between each other for the two higher concentrations. However, the model does not describe the SANS data sufficiently well at q < 0.007 Å−1 for the lowest concentration (0.119 mg/mL, Fig. 4 A), presumably due to some large aggregates that dominate the intensity contribution in this q range. Table 1 illustrates the best fitting parameters of the core-shell structure, revealing cavities (i.e., core radius) with consistent sizes from 35 to 37 Å (i.e., 3.5∼3.7 nm), and the overall diameter of the SAPN-R is ∼240 Å (i.e., 24 nm).
Table 1The Best Fitting Parameters for Three Concentrations of SAPN-R
Smear ParametersPeptide Concentration
0.435 mg/mL0.289 mg/mL0.119 mg/mL
Average shell thickness (Å)878792
Overall polydispersity (0:1)0.160.170.18
The volume/size distributions measured by DLS did not depend on sample concentration; the peak widths measured for the three protein formulations were 32.97 nm (0.119 mg/mL), 33.58 nm (0.289 mg/mL) and 32.80 nm (0.435 mg/mL). The electron microscopy showed the effects of increased concentration, with higher particle density per respective grid. However, the core morphology of the particles did not vary much between the different species (Fig. 4, CE).
The combined results from the DLS, TEM, STEM, and SANS experiments suggest that SAPN species with up to 360 chains follow a spherical core-shell architecture. Together with the fact that these particles are designed to form clusters of three- and fivefold symmetry, it should therefore be possible to provide blueprints for their organization using symmetry techniques.

### Describing SAPNs via generalized symmetry techniques

The structures of small SAPNs whose constituent peptide is made of two oligomerizing domains, a pentamer and a trimer (Fig. 1 A), have already been described by Raman et al. (
• Raman S.
• Machaidze G.
• Burkhard P.
• et al.
Structure-based design of peptides that self-assemble into regular polyhedral nanoparticles.
) assuming icosahedral symmetry. In fact, these two domains have the ability to assemble into oligomers known as least common multiple (LCM) subunits comprising 15 monomers each. This 15-monomer threshold is required to allow each building block to participate in precisely one three- and one fivefold cluster. These LCM building blocks are expected to self-assemble at varying ratios, for example, in groups of four to create a 60-chain species that satisfies the packing geometry of icosahedral symmetry (see Figs. 1 C and 5). The properties of the resulting nanoparticle could be tuned according to the amino acid sequence present in each oligomerization domain, the length of the linker region between each domain, and the geometric properties of each domain.
However, the icosahedral packing geometry described in Caspar and Klug (
• Caspar D.L.
• Klug A.
Physical principles in the construction of regular viruses.
) does not adequately explain species of SAPNs that are larger than 60 chains, such as those formed by SAPN-K according to STEM mass analysis (Fig. 3). Moreover, because SAPNs are designed to form clusters of 5 s and 3 s throughout, the geometry of the higher order assemblies cannot be modeled according to Caspar-Klug theory (
• Caspar D.L.
• Klug A.
Physical principles in the construction of regular viruses.
). In particular, these structures cannot be quasi-equivalent, and standard approaches for the modeling of higher order protein complexes, such as representations as Buckminster-Fuller architectures, do not apply.
Therefore, we introduce here a classification scheme that allows us to determine the architectures of the spherical SAPN particles with more than 60 oligomers, focusing on particles corresponding to the lowest three peaks (Fig. 3 A). We first study the connectivity of the network of polypeptides, using only the fact that they tend to aggregate in pentamers and trimers. Second, we focus on particles of maximal symmetry (tetrahedral, octahedral, and icosahedral), because these are likely to have almost-spherical shapes, and derive constraints for the number of polypeptides in each case. Strikingly, it turns out that particles with maximal symmetry lie within the first three peaks of the mass distribution function (Fig. 3 B). Finally, we mimic the Caspar-Klug approach to find the optimal geometry of the high-symmetry particles. It is possible to extend this approach to describe particles with no or low symmetry, and we also discuss a nonspherical example, but the full classification of such nonspherical morphologies lies outside the scope of this article and will be discussed elsewhere.

### Connectivity

All SAPN particles have the property of being formed entirely from pentagonal and triangular clusters. We analyze here the connectivity between these clusters. For this, we represent the surface of the particle as a graph, in which vertices correspond to the positions of the protein clusters, and edges correspond to individual SAPN chains (Figs. 6 A, 7 A, S3, D and E, and S4). By construction, the graph is bipartite with two types of vertices, one with degree three (corresponding to trimers) and one with degree five (corresponding to pentamers). Because edges correspond to SAPN chains, the graph is characterized by the fact that degree-3 vertices must always be connected with degree-5 vertices and vice versa.
Consider from now on a particle made of N oligomers, which, as discussed above, can be represented by a graph with N edges on the sphere. In general, for every graph on a sphere, Euler’s formula constrains the number of vertices v, edges e, and faces f according to
$v−e+f=2.$

Now, let p be the number of vertices of degree five, and t the number of vertices of degree three, so that $v=p+t.$ Then,
$e=5p=3t=N$

is the total number of edges. Note that by construction every edge represents a SAPN chain, so e equals the overall number of chains (N) in the particle. From this formula it follows that 3 divides p and 5 divides t, so that we can write
$p=3m,t=5m,e=N=15m,$

where m is an integer. Therefore, the number of chains in any SAPN particle must be a multiple of 15, i.e., a so-called LCM unit. Using Euler’s formula, we hence obtain that the number of faces f of the graph must be
$f=7m+2.$

Because three- and fivefold vertices must alternate on the boundary of any face, every face must have an even number of edges (Fig. 6 A). The two options with the smallest numbers of vertices are rhomb-shaped and hexagonal faces. Because the edges of the faces encode the positions of the monomers underlying the SAPNs, larger faces (e.g., octagons) would impose a larger disparity in angles between the individual monomers within the same cluster. Therefore, here we only consider the case in which the faces are rhombs and hexagons, because we are looking for particle morphologies in which pentamers are uniformly distributed. Let r and x be respectively the number of rhombs and hexagons in the tessellation describing the particle surface, so that $f=r+x$. Because each edge belongs to two faces, each rhombic face contributes four edges, and each hexagonal face contributes six edges, the following identity holds:
$4r+6x=2e=2N.$

Because there are no spherical graphs with the above characteristics for m = 1,2,3, we assume m ≥ 4 and obtain
$r=6(m+1),x=m−4.$

The case m = 4 corresponds to a unique graph, which is the graph of an icosahedrally symmetric polyhedron, the rhombic triacontahedron, already described by Raman et al. (
• Raman S.
• Machaidze G.
• Burkhard P.
• et al.
Structure-based design of peptides that self-assemble into regular polyhedral nanoparticles.
). It is a polyhedron with 30 rhombic faces, 60 edges, 12 fivefold vertices, and 20 threefold vertices. All SAPNs must correspond to higher m values and the formulas above will be used to determine the number of hexagonal and rhombic faces in the graphs describing their structures.

### Possible symmetries of the particles

We now focus on particles with maximal symmetry, i.e., tetrahedral, octahedral, and icosahedral. When acting on the sphere, these groups have 12, 24, and 60 asymmetric units that tessellate the sphere, respectively.
Let pd be the number of pentagonal clusters and td the number of triangular clusters in the interior (i.e., not lying on any global symmetry axis) of the asymmetric unit of a given symmetry group. Then we have the following symmetry conditions:

#### Sym 1

For the particle to have icosahedral symmetry (order of symmetry group, 60), the following relation has to be fulfilled:
$N=α60+180td=γ60+300pd,withα,γin{0,1},$

where α in {0,1} indicates whether there are trimers on the threefold axes of icosahedral symmetry (α =1) or not (α = 0), and γ in {0,1} indicates whether there are pentamers on the fivefold axes of icosahedral symmetry (γ = 1) or not (γ = 0).
In particular, this relation implies the condition
$α+3td=γ+5pd,withα,γin{0,1}.$

As a consequence, only the following chain numbers are possible for SAPN particles with icosahedral symmetry with up to 360 chains: 60 (α = γ = 1, td = pd = 0); 360 (α = 0, γ = 1, td = 2, pd = 1).

#### Sym 2

For the particle to have octahedral symmetry (order of symmetry group, 24), the following condition has to be fulfilled:
$N=3(8α+24td)=120pd,$

where α in {0,1} indicates whether trimeric clusters are situated on the particle threefold axes.
In particular, this implies the condition
$α+3td=5pd,αin{0,1}.$

As a consequence, only the following chain numbers are possible for octahedral SAPN particles with up to 360 chains: 240 (α = 1, td = 3, pd = 2); 360 (α = 0, td = 5, pd = 3).

#### Sym 3

For the particle to have tetrahedral symmetry (order of symmetry group, 12), the following condition has to be fulfilled:
$N=3(4α+4β+12td)=60pd,$

where α, β in {0,1} indicate whether there are trimeric clusters on the two types of threefold sites (corners and centers of faces of the tetrahedron). In particular, this implies the condition
$α+β+3td=5pd,α,βin{0,1}.$

As a consequence, only the following chain numbers are possible for tetrahedral SAPN particles with up to 360 chains: 60 (α = β = td = pd = 1); 120 (α = 0, β = 1, td = 3, pd = 2 or α = 1, β = 0, td = 3, pd = 2); 180 (α = β = 0, td = 5, pd = 3); 240 (α = β = 1, td = 6, pd = 4); 300 (α = 0, β = 1, td = 8, pd = 5 or α =1, β = 0, td = 8, pd = 5); and 360 (α = β = 0, td = 10, pd = 6).
Note that if a certain chain number N occurs for different symmetries, it is likely that the higher order symmetry is realized by the particle, as long as it is compatible with the other requirements, such as the uniform distribution of pentamer clusters (see below).

### Determine possible geometries of the particles

To determine the actual positions of the chains in the particles, we generalize the Caspar-Klug approach as follows. First, we place the pentameric clusters onto a hexagonal tiling, which guarantees uniform spacing between the clusters. Then, depending on the symmetry, we superimpose onto this lattice the planar development (i.e., a planar representation of the surface) of one of the three polyhedra with maximal symmetry: a tetrahedron (made of four equilateral triangles), an octahedron (made of eight equilateral triangles), and an icosahedron (made of 20 equilateral triangles) (Fig. S3, A–C). Notice that each triangle is made of three copies of the asymmetric unit of the corresponding symmetry group. Next we place the trimeric clusters onto the hexagonal tiling according to the symmetry of the particle, ensuring that the correct connectivity is satisfied (Fig. S3, D and E). A 3D representation of the particle is then obtained by folding these planar developments into a 3D polyhedron, projecting it onto a sphere, and deforming the corresponding spherical graph to recover uniform spacing.
We discuss below a rule to superimpose the planar representation of the polyhedra onto the hexagonal tiling, which is inspired by the classical Caspar-Klug construction. We recall that in the Caspar-Klug approach (which is based on icosahedral symmetry and thus 20 triangular faces), each hexagon of the tiling represents a hexamer, and thus contains six proteins. Hence, every lattice point (center of hexagon) contributes six proteins, with the exception of the points at the corners of the triangles that, due to the folding requirements, contribute only five proteins. Therefore, if p is the number of lattice points not lying at vertices of the triangles, then 6p is the total number of proteins contributed by hexamers. To this number we must add the proteins contributed by pentamers: in the icosahedral case, there are 12 fivefold axes, which amount to 60 = ord(G) additional proteins. On the other hand, this number must equal 3 × 20 T = ord(G) T = 60 T, where T = (h2 + hk +k2) is the triangulation number.
Therefore, we reformulate the icosahedral Caspar-Klug counting rule as
$6p+ord(G)=(h2+hk+k2)ord(G).$

We generalize the idea of placing the pentamers at the centers of each hexagon, i.e., constructing a triangular lattice, to other symmetries. In particular, in the tetrahedral and octahedral case, p = N/5 is the number of pentamers, corresponding to the total number of lattice points positioned at the centers of the hexagons. To use the Caspar-Klug rule, we multiply this number by 6, thus obtaining the number of protein chains corresponding to the scenario that all pentagonal clusters are replaced by hexamers. Notice now that, in the octahedral and tetrahedral cases, the vertices of the triangles in the planar development are axes of order 4 and 3, respectively. Therefore the putative positions of the proteins have to be counted in multiples of 3 and 4, rather than 5 as in Caspar-Klug theory, around the corners of the triangular faces. In particular, in the tetrahedral case, there are four corners corresponding to axes of order 3, so that we must associate three proteins to each corner, obtaining 4 × 3 = 12 = ord(G) proteins. In the octahedral case, there are six corners corresponding to axes of order 4, so that we must associate four proteins to each corner, obtaining 6 × 4 = 24 = ord(G) proteins. Therefore, the analog of the counting rule is now
$(6/5)N+ord(G)=(h2+hk+k2)ord(G),$

where h and k are integers. In the icosahedral case, a similar argument yields
$(6/5)N+(1−γ)ord(G)−12γ=(h2+hk+k2)ord(G),$

where ord(G) = 60, and γ = 1 if there are pentamers at the fivefold axes, and zero otherwise. The geometric interpretation of h and k is as in Caspar and Klug’s approach; it indicates the position of one edge of one copy of the equilateral triangles needed to construct the particle surface. In particular, for each particle with chain number N we determine h and k, given the symmetry possible for such a particle. We then draw the corresponding equilateral triangle on a hexagonal grid, indicate the positions of the pentagonal clusters in the centers of the hexagons, and then place trimers on five of the six vertices of the hexagonal grid surrounding each pentamer (Fig. S3, D and E). The latter is done such that the rhombic and hexagonal tiles generated in this way correspond to the numbers determined above.

### Application to the observed SAPN species

While particles with chain numbers larger than 360 do not appear to be isometric in the micrographs (see Fig. 3) and therefore cannot be modeled as symmetric particles, particles with smaller chain numbers corresponding to the first three peaks of the mass histogram appear to be spherical, to a good approximation (as discussed above). They are therefore candidates for symmetric particles, and we investigate their possible blueprints using the method above.

### Spherical particles corresponding to peak 1

The first mass peak corresponds to the presence of 234 chains (plus or minus 30 chains). There is only one chain number for symmetric particles that falls into this range. It is a 240-chain particle with either tetrahedral or octahedral symmetry. This particle must correspond to a tessellation with 102 rhombs and 12 hexagons with tetrahedral symmetry. According to the previous section, it can be modeled via a hexagonal grid and an equilateral triangle satisfying
$h2+hk+k2=25.$

The tessellation superimposed onto a triangular face corresponding to (h,k) = (5,0) in the Caspar and Klug approach is shown in Fig. 6 B. Note, however, that the vertices of the triangle correspond to threefold positions on which trimers are located, rather than fivefold vertices as in Caspar-Klug theory. The shape formed from four copies of this triangle has three twofold symmetry axes and four threefold symmetry axes, and corresponds to a spherical particle formed from 240 chains that are organized as 48 pentamers and 80 trimers (Fig. 6).

### Spherical particles corresponding to peak 2

The second mass peak represents 312 chains (plus or minus 30 chains), and therefore is likely to correspond to the 300-chain particle with tetrahedral symmetry, i.e., a graph with 300 edges. There are 60 vertices of degree five (p = 60) and 100 vertices of degree three (t = 100), and m = 20. From the formulas r = 6 (m +1) and x = m − 4 we deduce that the graph has 16 hexagonal faces and 126 rhombic faces. We obtain the condition
$h2+hk+k2=31.$

We therefore represent the particle morphology with respect to a (h,k) = (5,1), or (h,k) = (1,5) triangle, with corners marking tetrahedral threefold axes. Note that the two solutions differ in helicity only. The surface structure is shown in Fig. 6.

### Spherical particles corresponding to peak 3

This peak, 367 chains (plus or minus 30 chains), can be attributed to an icosahedral SAPN species formed from 360 chains. There should be 20 hexagonal and 150 rhombic tiles in the corresponding surface tessellation. The counting rule yields
$h2+hk+k2=7.$

There are two solutions, (h,k) = (2,1) and (h,k) = (1,2), which are related by handedness. We then place trimers on five of the six vertices of the hexagonal lattice, and connect pentamers and trimers so as to obtain a surface tiling in terms of rhombs and hexagons (see Fig. 6).

### Particles morphologies with lower symmetry or corresponding to higher chain numbers

In addition to the spherical particle morphologies determined above, nonspherical particles may arise. In particular, particles corresponding to peaks with higher chain numbers appear to correspond to aggregates of more than one smaller-sized particle and hence do not exhibit any apparent regularity in their organization. In general, the graph-theoretical approach introduced here allows one to determine the connectivity of all possible nondefective arrangements of polypeptides, i.e., those in which all helices of the SAPN particles either participate in a three- or fivefold local cluster, and all clusters are of one of these types. In particular, this also includes tubular structures with low symmetry. One such example is shown in Fig. S4 in which an oblong particle formed by 315 chains is represented; by the connectivity restrictions the graph associated to the particle has 17 hexagonal faces and 132 rhombic faces. However, such particles would have a nonuniform curvature, and would be likely to occur only if the flexible regions of the oligomers are highly deformable. Symmetric particles would have a smaller concentration of stress due to the uniform distribution of links. For this reason, and because the complete classification of all possible particle morphologies with low or no symmetry, as well as that of higher chain numbers, requires further developments of the mathematical approach adopted here, it will be discussed elsewhere.

## Conclusions

### Symmetric self-assembly of SAPNs

We have elucidated here how and why the SAPNs form larger oligomers beyond the 60 chains of the classical icosahedron. From a geometric perspective, the only factors limiting the assembly of non-epitope-bearing SAPN species are the oligomerization domains. This means that a matching rule exists, restricting the particle formation to include a packing geometry that has to account for the presence of pentameric and trimeric vertices. This connectivity constraint is compatible with a large variety of possible particles, but electron microscopy and small angle neutron scattering show that these species do indeed mainly exist as pseudo-spheres, so that we can restrict the analysis to particles with maximal symmetry, i.e., icosahedral, octahedral, and tetrahedral. In principle, the necessity for symmetry is based on the fact that SAPNs consist of homo-oligomers, which ideally align in an equidistant manner, with no one set of interactions between monomers being favored over another, resulting in a (nearly) uniform distribution of pentamers.

### Biophysical examples of symmetric SAPN assembly

According to STEM mass measurements, the monomeric SAPN building blocks assemble into a polymorphous range of species (Fig. 3). The three populations with the lowest molecular weights, i.e., the majority of the particles, arose from SAPNs with essentially circular projections (Fig. 3 C). Some elongation or aggregation occurred at higher molecular weights beyond 360 chains. In agreement, the SANS behavior of the SAPNs could be described by a spherical core-shell model (Fig. 4 A). The possible spherical core-shell morphologies were determined using symmetry techniques and surface lattices, and it was shown that they are consistent with high-symmetry particles formed from 240, 300, and 360 chains, respectively, in accordance with the STEM data (Fig. 3).
By graph-theoretical techniques and a generalization of the Caspar-Klug construction to the case of SAPNs with different symmetries, the putative locations of the pentameric and trimeric domains on the nanoparticle surfaces were identified, and it was shown that both the 240- and the 300-chain particle are organized with tetrahedral symmetry, while the 360-chain particle displays icosahedral symmetry.
It is interesting to note that the 360-chain particle can also be described by an alternative technique related to that used for the modeling of the all-pentamer viral capsids of polyoma virus, papilloma virus, and Simian virus 40 (
• Keef T.
• Twarock R.
• Elsawy K.M.
Blueprints for viral capsids in the family of polyomaviridae.
,
• Twarock R.
A tiling approach to virus capsid assembly explaining a structural puzzle in virology.
,
• Twarock R.
Mathematical virology: a novel approach to the structure and assembly of viruses.
). In these cases the positions of the pentamers were determined with reference to a T = 7d icosadeltahedral geometry, but in the case of the SAPNs, there are trimeric clusters in addition. This is reflected in the choice of tiles in the surface tessellation; to represent SAPNs, we use rhombs and hexagons with vertices indicating alternatingly pentagonal and trimeric clusters and edges representing the links between them. By contrast, in the applications of viral tiling theory to viruses of that type, rhomb- and kite shapes were used to indicate the positions of individual proteins in the corners meeting at angles sustaining fivefold vertices, akin to protein positions in the corners of triangular faces in Caspar-Klug theory. We illustrate this in Fig. 5 B, where the tiling corresponding to the virus case is shown. Three juxtaposed rhombs highlight a global threefold (trimeric) axis, while five adjacent kites mark the positions of the global fivefold axes (pentamers). Local fivefold symmetry occurs at the intersections of two kites and three rhombs (shown in cyan-green). Hence, the trimer locations can be found at positions equidistant between three adjacent pentamers.
Finally, we note that our geometric classification in this article implies that tetrahedral particles with 120 and 180 chains can potentially also be formed with this assembly system. Their surface structures are shown in Fig. 7. While these particles have not been seen in the experiments performed here, they may correspond to particles determined earlier. For example, Pimentel et al. (
• Pimentel T.A.
• Yan Z.
• Burkhard P.
• et al.
Peptide nanoparticles as novel immunogens: design and analysis of a prototypic severe acute respiratory syndrome vaccine.
) reported a trimeric SARS epitope bearing SAPN species with 120 chains (determined using analytical ultracentrifugation; see Fig. 5 in that reference, which corresponds to a species of ∼110 monomers), and Yang et al. (
• Yang Y.
• Ringler P.
• Burkhard P.
• et al.
Optimizing the refolding conditions of self-assembling polypeptide nanoparticles that serve as repetitive antigen display systems.
) reported a SAPN-R species with 180 chains (determined using STEM; see Fig. 3 in that reference, which corresponds to a species of ∼176 monomers). Both of the SAPN species were measured in the presence of 5% glycerol, which was utilized to increase buffer viscosity and reduce the size of aggregates. Therefore, it is hard to determine the influence of confounding factors such as epitopes or cosolutes (e.g., glycerol), on the native self-assembly equilibrium. The SAPN-K species used in this study was measured in the absence of epitope or glycerol, and should be a more accurate reflection of the assembly tendencies of covalently linked pentamer and trimer oligomerization domains.
Larger molecular weight species, beyond 360 chains, appear to be aggregates of lower species, suggesting that formation of larger spherical particles is energetically unfavorable. Moreover, potentially, similar lattices to those used in the construction of the blueprints of the spherical SAPNs could also be used to form tubular structures, as, e.g., seen in in vitro studies of polyoma virus VP1 assembly. However, the micrographs suggest that in the experiment carried out here, these higher order species are likely to be aggregates, rather than cylindrical lattice morphologies.
Our analysis implies that the self-assembly of SAPNs can be described as one that is driven by local connectivity and the need for a (nearly) uniform distribution of pentamers across the particle surface. The results determined here provide, to our knowledge, new insights into the structures of the nanoparticles formed on the self-assembly of the SAPN system, which paves the way for their exploitation in nanotechnology and vaccine design.

## Author Contributions

P.B., R.T., and N.W. contributed to the design of the experiments; N.W. (bio-production, TEM, DLS), G.I. and R.T. (modeling), P.R. and S.A.M. (TEM, STEM), and M.-P.N. (SANS) performed the experiments; and N.W., G.I., R.T., and P.B. wrote the manuscript. All authors read and modified the text and were involved in interpretation of results and approved the final manuscript.

## Acknowledgments

P.R. and S.A.M. thank Andreas Engel and Henning Stahlberg (C-CINA, Biozentrum, University of Basel. Switzerland) for supporting this STEM project. Thanks to Alexandra Graff and Françoise Erne-Brand for assisting with TEM and STEM.
Support by the National Institutes of Health/National Institute on Drug Abuse (award No. 1DP1DA033524) to P.B. and a Royal Society Leverhulme Trust Senior Research Fellowship (No. LT130088) to R.T. for this work are gratefully acknowledged. STEM was funded by the Maurice E. Müller Foundation of Switzerland and the Swiss National Foundation under grant No. 3100A0-108299 to Andreas Engel and by the Swiss Systems Biology Initiative SystemsX.ch (Grant CINA to Andreas Engel and Henning Stahlberg).

## Supporting Material

• Document S1. Figs. S1–S4

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text:
Lemma 10.59.3. Suppose that $0 \to M' \to M \to M'' \to 0$ is a short exact sequence of finite $R$-modules. Then there exists a submodule $N \subset M'$ with finite colength $l$ and $c \geq 0$ such that

$\chi _{I, M}(n) = \chi _{I, M''}(n) + \chi _{I, N}(n - c) + l$

and

$\varphi _{I, M}(n) = \varphi _{I, M''}(n) + \varphi _{I, N}(n - c)$

for all $n \geq c$.

Proof. Note that $M/I^ nM \to M''/I^ nM''$ is surjective with kernel $M' / M' \cap I^ nM$. By the Artin-Rees Lemma 10.51.2 there exists a constant $c$ such that $M' \cap I^ nM = I^{n - c}(M' \cap I^ cM)$. Denote $N = M' \cap I^ cM$. Note that $I^ c M' \subset N \subset M'$. Hence $\text{length}_ R(M' / M' \cap I^ nM) = \text{length}_ R(M'/N) + \text{length}_ R(N/I^{n - c}N)$ for $n \geq c$. From the short exact sequence

$0 \to M' / M' \cap I^ nM \to M/I^ nM \to M''/I^ nM'' \to 0$

and additivity of lengths (Lemma 10.52.3) we obtain the equality

$\chi _{I, M}(n - 1) = \chi _{I, M''}(n - 1) + \chi _{I, N}(n - c - 1) + \text{length}_ R(M'/N)$

for $n \geq c$. We have $\varphi _{I, M}(n) = \chi _{I, M}(n) - \chi _{I, M}(n - 1)$ and similarly for the modules $M''$ and $N$. Hence we get $\varphi _{I, M}(n) = \varphi _{I, M''}(n) + \varphi _{I, N}(n-c)$ for $n \geq c$. $\square$

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# Probability that all patients get the right medicine

One nurse missed the doctor's instructions with the medication she has to give to $$10$$ patients. $$4$$ patients are taking a pill for hypertension, $$3$$ patients are taking a pill for diabetes, $$2$$ patients are taking a pill for heart arrhythmia and $$1$$ patient is taking a pill for hypotension. The nurse randomly gives the pills to the patients. Coincidentally, everyone receives medication in the right way. What is the probability that this will happen?

I think that all the cases are $$\dfrac{10!}{4! \cdot 3! \cdot 2! \cdot 1!}$$ but I face difficulty in finding the sum of cases that each patient takes the right medicine.I would appreciate for your quidance. Thank you very much in advance.

• Consider permutations of $(a,a,b)$ amongst three patients. The number of cases is $\frac{3!}{2!}=3$: $\{(a,a,b),(a,b,a),(b,a,a)\}$. How many are correct? is it not just 1? – Rahul Madhavan Apr 17 at 17:27
• You helped me a lot, Thank you very much! – Magda Apr 17 at 19:34

Suppose the patients are given the pills in the order $$H_1H_2H_3H_4D_1D_2D_3A_1A_2O_1$$ where $$H_i$$ represents the person who has to take the pill for hypertension, $$D_i$$ for diabetes, $$A_i$$ for arrhythmia and $$O_1$$ for hypotension.
Now, probability that $$H_1$$ gets the right pill is $$4/10$$, then $$H_2$$ gets the right pill is $$3/9$$ so in this way, the probability that all get the right pill would be $$\dfrac{4}{10} \cdot \dfrac{3}{9} \cdot \dfrac{2}{8} \cdot \dfrac{1}{7} \cdot \dfrac{3}{6} \cdot \dfrac{2}{5} \cdot \dfrac{1}{4} \cdot \dfrac23 \cdot \dfrac12 \cdot 1$$


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We have triangle $\triangle ABC$ where $AB = AC$ and $AD$ is an altitude. Meanwhile, $E$ is a point on $AC$ such that $AB \parallel DE.$ If $BC = 12$ and the area of $\triangle ABC$ is $180,$ what is the area of $ABDE$?

Aug 4, 2019

#1
+25226
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We have triangle $$\triangle ABC$$ where $$AB = AC$$ and $$AD$$ is an altitude.
Meanwhile, $$E$$ is a point on $$AC$$ such that $$AB \parallel DE$$.
If $$BC = 12$$ and the area of  $$\triangle ABC$$ is $$180$$, what is the area of $$ABDE$$?

$$\text{Let AB = AC  } \\ \text{Let DB = CD =\dfrac{BC}{2} = 6 }$$

$$\begin{array}{|rcll|} \hline \text{area of } \triangle ABC =180 &=& \dfrac{BC*AD}{2} \\ 180 &=& \dfrac{12*AD}{2} \\ 180 &=& 6AD \\ \mathbf{ AD } &=& \mathbf{30} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \dfrac{CD}{ED} &=& \dfrac{BC}{AB} \\\\ \dfrac{6}{ED} &=& \dfrac{12}{AB} \\\\ \dfrac{ED}{6} &=& \dfrac{AB}{12} \\\\ \mathbf{ED }&=& \mathbf{\dfrac{AB}{2}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \text{area of } ABDE &=& \dfrac{AB+ED}{2}\times H \quad| \quad H=\dfrac{DB*AD}{AB}=\dfrac{6*30}{AB} \\\\ \text{area of } ABDE &=& \left(\dfrac{AB+ED}{2}\right)\times \dfrac{6*30}{AB} \\\\ \text{area of } ABDE &=& \left(\dfrac{AB+\dfrac{AB}{2}}{2}\right)\times \dfrac{6*30}{AB} \\\\ \text{area of } ABDE &=& \dfrac{3}{4}AB\times \dfrac{6*30}{AB} \\\\ \text{area of } ABDE &=& 3* 3*15 \\\\ \mathbf{\text{area of } ABDE} &=& \mathbf{135} \\ \hline \end{array}$$

Aug 5, 2019


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# Proof of a corollary for the Strong Law of Large Number

Here is Theorem 1.13 from Mathematical Statistics Jun Shao.

$$X_i$$ are i.i.d. A necessary and sufficient condition for the existence of constant $$c$$ for which \begin{align} \frac{1}{n}\sum_{i=1}^n X_i \longrightarrow c\quad a.s. \end{align} is that $$E|X_1|<\infty$$ in which case $$c=EX_1$$. And \begin{align} \frac{1}{n}\sum_{i=1}^nc_i(X_i-EX_1)\longrightarrow 0 \quad a.s. \end{align} for every bounded real $$\{c_i\}$$

The first part is just the SLLN. The proof of the second part is marked as an exercise. However, I am having trouble to see this. My attempt is the following: To prove the result, it is sufficient to prove $$P(|c_i(X_i-EX_1)|\geq \epsilon \ i.o.)=0$$ for every $$\epsilon$$. And this is equivalent to prove $$P(|X_i-EX_1|\geq \epsilon \ i.o.) = 0$$ since $$c_i$$ is bounded. At this step, I try to use Borel-Cantelli, but it seems does not work. Does anyone have any idea for this?

• First, we can assume that $X_i$ has 0 mean by replacing it by $X_i-\mathbb{E}X_i$. Then I think that you can simply adapt the proof of the LLN, that is: let $Y_i=c_iX_i\mathbf{1}_{\lvert c_iX_i\rvert\leq i}$ and show that (1) $\lim\mathbb{E}Y_n=0$, (2) a.s. there exists $J\geq 1$ such that for all $i\geq J$, $Y_i=c_iX_i$ (3) $\sum_{i\geq 1}\operatorname{Var}(Y_i)/i^2<\infty$. Jan 4 at 11:35
Expanding a bit on my comment: replace $$X_i$$ by $$X_i-\mathbb{E}X_i$$. Let $$X$$ be distributed as the $$(X_i)_{i\geq 1}$$ and $$Y_i=c_iX_i\mathbf{1}_{\lvert c_iX_i\rvert\leq i}$$. By dominating convergence, $$\lim\mathbb{E}Y_i=0$$. Denoting $$M=\sup_{i\geq 1}\lvert c_i\rvert$$, $$\sum_{i=1}^\infty\mathbb{P}(c_iX_i\neq Y_i)=\sum_{i=1}^\infty\mathbb{P}(\lvert c_iX\rvert>i)\leq\sum_{i=1}^n\mathbb{P}(M\lvert X\rvert>i)\leq\int_0^\infty\mathbb{P}(M\lvert X\rvert>t)dt=\mathbb{E}M\lvert X\rvert<\infty$$ So the Borel-Cantelli lemma applies and a.s., there exists $$J\geq 1$$ such that for all $$i\geq J$$ the equality $$c_iX_i=Y_i$$ holds. Finally, $$\sum_{i=1}^\infty\frac{\operatorname{Var}(Y_i)}{i^2}\leq\sum_{i=1}^n\frac{\mathbb{E}Y_i^2}{i^2}=\mathbb{E}\left[\lvert X\rvert^2\sum_{i=1}^\infty\frac{\mathbf{1}_{\lvert c_iX\rvert>i}}{i^2}\right]\leq c\mathbb{E}\lvert X\rvert<\infty$$ for some constant $$c\geq 0$$. We can now apply the following lemma with $$Z_n=Y_n-\mathbb{E}Y_n$$.
Let $$(Z_n)_{n\geq 1}$$ be a sequence of independent r.v. with $$\mathbb{E}Z_n=0$$ and $$\sum_{n\geq 1}\operatorname{Var}(Z_n)/n^2<\infty$$. Denoting $$S_n=Z_1+...+Z_n$$, the sequence $$(S_n/n)_{n\geq 1}$$ converges a.s. to $$0$$.
Since $$(\mathbb{E}Y_i)_{i\geq 1}$$ converges to $$0$$ $$\frac{Y_1+...+Y_n}{n}=\frac{Z_1+...+Z_n}{n}+\frac{\mathbb{E}Y_1+...+\mathbb{E}Y_n}{n}$$ converges to $$0$$ as well a.s. Finally, a.s., for all $$n\geq J$$, $$\frac{c_1X_1+...+c_nX_n}{n}=\frac{c_1X_1+...+c_{J-1}X_{J-1}}{n}+\frac{Y_J+...+Y_n}{n}\longrightarrow 0$$


text:
1. Find derivative.

$f(x)=(\sqrt{x}-3x^{3})(2\sqrt{x}+5)$.

How I do...
$f'(x)=((\sqrt{x}-3x^{3})(2\sqrt{x}+5))'=$ $(\sqrt{x}-3x^{3})'\cdot (2\sqrt{x}+5)+(\sqrt{x}-3x^{3})\cdot(2\sqrt{x}+5)'=$ $(\frac{1}{2\sqrt{x}}-9x^{2})(2\sqrt{x}+5)+(\sqrt{x}-3x^{3})(\frac{2}{2\sqrt{x}}+0)=$ $\frac{2\sqrt{x}}{2\sqrt{x}}+\frac{5}{2\sqrt{x}}-18x^{2}\sqrt{x}-45x^{2}+\frac{\sqrt{x}}{\sqrt{x}}-\frac{3x^{3}}{\sqrt{x}}=$
$\frac{4\sqrt{x}+5-36x-90x^{2}\sqrt{x}-6x^3}{2\sqrt{x}}$

I don't know it's good or not. This answer looks strange. Maybe I do mistake? Hmm

2. Almost. $18x^2\sqrt{x}\frac{2\sqrt{x}}{2\sqrt{x}}= \frac{32x^3}{2\sqrt{x}}$, not $\frac{36x}{2
sqrt{x}}$


text:
## 507- Plünnecke inequalities and sumset estimates

George Petridis, a student of Gowers, has found very nice new arguments for the Plünnecke-Ruzsa sumset inequality (If $A$ is a finite subset of an Abelian group $G$, and ${}|A+A|\le C|A|$, then ${}|kA-lA|\le C^{k+l}|A|$ for any $k,l\in{\mathbb N}$) and for the Plünnecke graph inequalities.

In lecture we went through the nice standard argument. But the new proofs are significantly simpler. For example, the graph inequalities are no longer needed for the sumset ones, and Menger’s theorem is no longer needed fro the graph inequalities. Gowers has posted the nice proof in his blog, with links to Petridis’s papers on the ArXiv.


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Scientific Notation to Decimal
2
2
Entering edit mode
7 weeks ago
am29 ▴ 30

Hi,

I have a huge list of p-values ranging from E-05 to E-324. I want to use a toll (Cropper) but it requires decimal numbers as input. How can I transform scientific notations to decimals, all at once?

notation decimal scientific • 698 views
0
Entering edit mode

What is toll? What format are the numbers in (i.e. text file? excel? etc.)? What sort of tools are you familiar with? (i.e. linux, R, python, shell?)

0
Entering edit mode

I have txt file, and I work in both Linux and R. The tool for which I need a decimal format numbers is Cropper: https://genomics.ut.ee/en/tools

5
Entering edit mode
7 weeks ago

I'd first question if your program really doesn't accept scientific notation, if so that creates problems for working with these values and you should consider using a properly implemented program. The following example shows what I mean: In principle, the following bash code does the conversion just fine:

while read -r line;
do
printf "%.325f\n" \$line ; ## change format to %.308f as minimum sensible representation
done < testfile.txt

You need to decide for some sensible cutoff of precision, I am using 325 just to show the problem:

1E-5
1E-100
1E-324

Output:

0.00001000000000000000081803053914031309545862313825637102127075195312500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000199918998026028836196477607885341594201826030059365956992555434676176762886132929895827460748109118507985282705397496540222684360419612636083562831412787179427249289424690806658916305930004345786023014502507945

0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Note: 1E-324 is effectively zero in this notation but maybe smaller than the smallest positive non-zero double precision value. On 64bit platforms that is 2e-308 **

Of course you could do some trickery by having variable precision floating point numbers, but you rather don't want that.

** checked with the c program:

#include <float.h>
#include <stdio.h>
int main()
{
printf("%.e\n", DBL_MIN);
}

2
Entering edit mode

Something to consider: Cropper makes manhattan plots - so it's plotting the -log transformed p-values on the y-axis. It could be that the required input is transformed values (I don't know), which would be much more reasonable than a decimal value with 324 zeros! :)

2
Entering edit mode

Makes perfect sense, but I have no clue about that software. One can make the plots in R which would be my first choice, including the steps number parsing and transformation it will likely be much less of a hassle.

0
Entering edit mode

Thank you all for the quick and useful answers!

1
Entering edit mode
7 weeks ago

Microsoft excel can be useful. You can import data into it and change the number format

0
Entering edit mode

Unfortunately, I have tens of thousand of rows and it exceeds Excel limit

2
Entering edit mode

It's easy to do with awk's printf, see e.g. here


text:
# IEEE Transactions on Pattern Analysis and Machine Intelligence

## Filter Results

Displaying Results 1 - 20 of 20
• ### [Front cover]

Publication Year: 1986, Page(s): c1
| PDF (772 KB)
• ### List of Contributors

Publication Year: 1986, Page(s): nil1
| PDF (172 KB)
• ### IEEE Computer Society publications [inside front cover]

Publication Year: 1986, Page(s): nil1
| PDF (172 KB)
• ### Image Analysis Using Multigrid Relaxation Methods

Publication Year: 1986, Page(s):129 - 139
Cited by:  Papers (209)  |  Patents (1)
| | PDF (6258 KB)

Image analysis problems, posed mathematically as variational principles or as partial differential equations, are amenable to numerical solution by relaxation algorithms that are local, iterative, and often parallel. Although they are well suited structurally for implementation on massively parallel, locally interconnected computational architectures, such distributed algorithms are seriously hand... View full abstract»

• ### Image Structure Representation and Processing: A Discussion of Some Segmentation Methods in Cytology

Publication Year: 1986, Page(s):140 - 146
Cited by:  Papers (43)
| | PDF (5879 KB)

Image processing methods (segmentation) are presented in connection with a modeling of image structure. An image is represented as a set of primitives, characterized by their type, abstraction level, and a list of attributes. Entities (regions for example) are then described as a subset of primitives obeying particular rules. Image segmentation methods are discussed, according to the associated im... View full abstract»

• ### On Edge Detection

Publication Year: 1986, Page(s):147 - 163
Cited by:  Papers (486)  |  Patents (2)
| | PDF (4123 KB)

Edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them. A critical, intermediate goal of edge detection is the detection and characterization of significant intensity changes. This paper discusses this part of the edge detection problem. To characterize the types of intensity changes derivatives of... View full abstract»

• ### Optimal Edge Detector Design I: Parameter Selection and Noise Effects

Publication Year: 1986, Page(s):164 - 177
Cited by:  Papers (51)  |  Patents (1)
| | PDF (3323 KB)

This paper comments on the optimality of the Laplacian of a Gaussian edge detection filter which localizes edges through zero crossings in the filtered image. The arguments of both Marr and Hildreth, and Dickey and Shanmugam are reviewed to establish that the filter is optimal in the sense of maximizing output image energy near edge features. This filter's principal advantage over other edge detec... View full abstract»

• ### Optimal Edge Detector Design II: Coefficient Quantization

Publication Year: 1986, Page(s):178 - 187
Cited by:  Papers (19)
| | PDF (1323 KB)

A companion paper describes the design and implementation of the Laplacian of a Gaussian edge detection filter which localizes edges through zero crossings in the filtered image. Accuracy in the presence of noise has been found to be proportional to the square root of the filter's standard deviation. Digital implementation of any continuous filter requires sampling and coefficient quantization. Th... View full abstract»

• ### ANGY: A Rule-Based Expert System for Automatic Segmentation of Coronary Vessels From Digital Subtracted Angiograms

Publication Year: 1986, Page(s):188 - 199
Cited by:  Papers (101)  |  Patents (3)
| | PDF (6661 KB)

This paper details the design and implementation of ANGY, a rule-based expert system in the domain of medical image processing. Given a subtracted digital angiogram of the chest, ANGY identifies and isolates the coronary vessels, while ignoring any nonvessel structures which may have arisen from noise, variations in background contrast, imperfect subtraction, and irrelevent anatomical detail. The ... View full abstract»

• ### Collision Detection for Moving Polyhedra

Publication Year: 1986, Page(s):200 - 209
Cited by:  Papers (144)  |  Patents (4)
| | PDF (2901 KB)

We consider the collision-detection problem for a three-dimensional solid object moving among polyhedral obstacles. The configuration space for this problem is six-dimensional, and the traditional representation of the space uses three translational parameters and three angles (typically Euler angles). The constraints between the object and obstacles then involve trigonometric functions. We show t... View full abstract»

• ### Mesh-Oriented Line Drawings Theory (MOLD Theory)

Publication Year: 1986, Page(s):210 - 221
Cited by:  Papers (4)  |  Patents (1)
| | PDF (4128 KB)

In this paper, we describe a new model of bilevel digital images of line drawings. The key idea is that the constraints of generating the line drawings are available to select patterns which are considered to be a certain local appearance of line drawings. Concretely, when a bilevel digital image of line drawings is observed by 3 Ã 3 pixels' unit mesh, the patterns occurring in the unit mesh are ... View full abstract»

• ### Digital Image Registration Using Projections

Publication Year: 1986, Page(s):222 - 233
Cited by:  Papers (68)  |  Patents (8)
| | PDF (5387 KB)

In many application fields (e.g., aerospace and biomedical image processing), one has to deal with a sequence of images whose observation is made difficult by unpredictable relative movements of the camera and the scene. In order to obtain a stable display, the displacement of each image in the sequence with respect to one chosen as a reference must be preliminarily determined. This would usually ... View full abstract»

• ### Efficient Synthesis of Gaussian Filters by Cascaded Uniform Filters

Publication Year: 1986, Page(s):234 - 239
Cited by:  Papers (67)  |  Patents (5)
| | PDF (1763 KB)

Gaussian filtering is an important tool in image processing and computer vision. In this paper we discuss the background of Gaussian filtering and look at some methods for implementing it. Consideration of the central limit theorem suggests using a cascade of simple'' filters as a means of computing Gaussian filters. Among simple'' filters, uniform-coefficient finite-impulse-response digital f... View full abstract»

• ### A Test of the Gaussian-ness of a Data Set Using Clustering

Publication Year: 1986, Page(s):240 - 247
Cited by:  Papers (10)
| | PDF (1842 KB)

The properties of the magnifying glass'' method of clustering are discussed. These properties, which include unbiased and consistent estimation of the mean for Gaussian distributions and biased and inconsistent estimation of the mean for non-Gaussian distributions, lead to the development of a technique for testing data to determine whether or not it is Gaussian. An example of a non-Gaussian dis... View full abstract»

• ### Efficient Implementation of the Fuzzy c-Means Clustering Algorithms

Publication Year: 1986, Page(s):248 - 255
Cited by:  Papers (267)  |  Patents (2)
| | PDF (2537 KB)

This paper reports the results of a numerical comparison of two versions of the fuzzy c-means (FCM) clustering algorithms. In particular, we propose and exemplify an approximate fuzzy c-means (AFCM) implementation based upon replacing the necessary exact'' variates in the FCM equation with integer-valued or real-valued estimates. This approximation enables AFCM to exploit a lookup table approach... View full abstract»

• ### Relaxation Labeling with Learning Automata

Publication Year: 1986, Page(s):256 - 268
Cited by:  Papers (30)
| | PDF (2533 KB)

Relaxation labeling processes are a class of mechanisms that solve the problem of assigning labels to objects in a manner that is consistent with respect to some domain-specific constraints. We reformulate this using the model of a team of learning automata interacting with an environment or a high-level critic that gives noisy responses as to the consistency of a tentative labeling selected by th... View full abstract»

• ### Encoding of Line Drawings with a Multiple Grid Chain Code

Publication Year: 1986, Page(s):269 - 276
Cited by:  Papers (18)  |  Patents (6)
| | PDF (1190 KB)

The multiple grid(MG) chain code which uses four different square grids is proposed to encode line drawings. The main processes adopted in the code are: 1) a grid selection algorithm which allocates quantization points only to the vicinity of the course of a line drawing, 2) labeling rule on quantization points which makes the frequency of some codes larger than that of other codes, and 3) quantiz... View full abstract»

• ### Best Linear Unbiased Estimators for Properties of Digitized Straight Lines

Publication Year: 1986, Page(s):276 - 282
Cited by:  Papers (26)
| | PDF (1343 KB)

This paper considers the problem of measuring properties of digitized straight lines from the viewpoint of measurement methodology. The measurement and estimation process is described in detail, revealing the importance of a step called characterization'' which was not recognized explicitly before. Using this new concept, BLUE (Best Linear Unbiased) estimators are found. These are calculated for... View full abstract»

• ### One-Dimensional Scan Selection for Two-Dimensional Signal Restoration

Publication Year: 1986, Page(s):282 - 284
Cited by:  Papers (2)
| | PDF (606 KB)

The problem of m-D filtering using sequential scanning is considered. It is shown that the optimal causal filter and the performance measure depend on the scan selected. Examples show that this effect can be significant. Possible techniques to select a suitable scan are analyzed. View full abstract»

• ### On the Local Optimality of the Fuzzy Isodata Clustering Algorithm

Publication Year: 1986, Page(s):284 - 288
Cited by:  Papers (31)
| | PDF (715 KB)

The convergence of the fuzzy ISODATA clustering algorithm was proved by Bezdek [3]. Two sets of conditions were derived and it was conjectured that they are necessary and sufficient for a local minimum point. In this paper, we address this conjecture and explore the properties of the underlying optimization problem. The notions of reduced objective function and improving and feasible directions ar... View full abstract»

## Aims & Scope

The IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI) is published monthly. Its editorial board strives to present most important research results in areas within TPAMI's scope.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
Sven Dickinson
University of Toronto
e-mail: sven@cs.toronto.edu