# New Aspects of Analyzing Amyloid Fibrils

Xiaoxi Lin, Yunpeng Zi, Jie Wu

###### Abstract.

This is a summary of mathematical tools we used in research of analyzing the
structure of proteins with amyloid form [37]. We defined several geometry
indicators on the discrete curve namely the hop distance, the discrete
curvature and the discrete torsion. Then, we used these indicators to analyze
the structure of amyloid fibrils by regarding its peptide chains as discrete
curves in $\mathds{R}^{3}$. We gave examples to show that these indicators
give novel insights in the characterization analysis of the structure of
amyloid fibrils, for example the discrete torsion can detect the hydrogen
bonds interactions between layers of amyloid fibril. Moreover, the topological
tool performs better than the root mean square deviation (RMSD) in quantifying
the difference of the structure of amyloid fibrils, etc.

##  1\. Introduction

This is a summary of mathematical tools we used in research of analyzing the
structure of proteins with amyloid form [37]. Amyloids are aggregates of
proteins with a fibrillar morphology, which results from the protein
misfolding. This misfolding may cause the protein lose its original structure
and physiological functions, and get new structure and physiological
functions. Amyloids have been implicated in both physiological processes and
pathological diseases, which can be termed as pathological amyloids and
functional amyloids[31], respectively. The abnormal depositions of
pathological amyloids in various tissues are associated with kinds of diseases
known as amyloidosis. To date, there are more than 30 human proteins have been
found to form amyloidosis[13], such as transthyretin (TTR),
$\alpha$-synuclein[32] and $\beta$ amyloid peptide[30].

Studies over the past 2 decades have made a great contribution to explore the
molecular and physiochemical mechanisms of amyloidogenesis and its role in
biology, including the overall architecture of amyloid fibril [21, 10], the
interactions related to the stability of fibril structures [10], the process
of amyloid formation [45, 74], the amyloid core and flanking regions of fibril
structures [47, 69], the conformational dynamics [43, 44], etc. However,
challenges faced by studies on amyloids are still multiple and significant
[67]. One of them is that how the structure of amyloid fibril determines its
new biological functions. It is inevitable to analyze the structure of amyloid
fibrils, and then to explore the relationship between the structure and its
corresponding functions. The rapid development of structural biology methods,
including cryo-electron microscopy (cryo-EM) and solid-state nuclear magnetic
resonance (ssNMR), makes it possible to analyze the structure of amyloid
fibrils at atomic level[62].

In this paper, we gave several ways of analyzing the structure of amyloid
fibrils using the tools from topology and geometry, which could give insights
into further studies on amyloid fibrils. Examples in this paper are all
related to the structure of amyloid fibrils formed from TTR (ATTR). TTR is a
homotetramer with a dimer of dimers and high $\beta$-sheet content. The native
TTR transports the thyroxin hormone thyroxine and retinol to the liver. The
misfolding and aggregation of TTR result in the formation of ATTR and
amyloidosis [76]. The extracellular despositions of misfolded wild type TTR
could lead to senile systemic amyloidosis, while the extracellular
despositions of misfolded mutational variants, such as V30M and V122I, are
associated with hereditary ATTR amyloidosis, familial amyloid polyneuropathy
and familial amyloid cardiomyopathy. Before introducing our study, let us give
some biological and chemical knowledges about the structure of protein and a
historical introduction about the intersection of geometry, topology and
biology.

###  1.1. Primary Structures of Proteins

Protein is a class of large biomolecules, which plays important role in
biochemical reactions. Amino acids (AA) are the constituent units of proteins.
Specifically, each AA consists of an amino group (-$\text{NH}_{2}$), a
carboxyl group (-COOH) and a side chain group (-R). These three parts are
connected through a carbon atom, which we call it the $\alpha$-Carbon and
denote it as $C_{\alpha}$. The type of AA is entirely determined by its side
chain group. AAs form peptide bonds through dehydraton condensation, and then
determine the primary structure of proteins, called the peptide chain which is
shown in Fig.1. An individual AA in the peptide chain is called a residue.
Each peptide chain has its backbone, which consists of amino nitrogen,
$C_{\alpha}$, carboxyl carbon and carboxyl oxygen of each residue. Note that a
peptide chain is naturally oriented by its formation, that is, a peptide chain
always begin with a amino nitrogen and end with a carbonyl carbon.

Figure 1. A peptide chain. The backbone of this peptide chain is the set
$\\{\dots,C_{\alpha_{i-1}},C_{i-1},O_{i-1},N_{i},C_{\alpha_{i}},C_{i},O_{i},\dots\\}.$
Blue rectangles display the peptide bonds.

###  1.2. Geometry in Biology

The geometry plays important role in the study of the structure of proteins
and even other biomolecules. Scientists found that three-dimensional atomic
structures of these molecules provide rich information for understanding how
these molecules carry out their biological functions. Among all these
research, we found several cutting points that geometry can enter the study of
the structures of proteins or biomolecules.

The geometry could serve as a way to represent a molecule as a solid figure.
As we all know, a molecule is a system of moving particles (nuclei and
electrons) which are held together by electrostatic and magnetic forces in
nature. And it does not possess any definitive boundaries. However,
visualizing a molecule is still a very useful chemical concept and has proved
its value in medicinal chemistry.

The van der Waals radius of an atom, denoted as $r_{w}$, is the radius of an
imaginary hard sphere representing the distance of closest approach for
another atom. The van der Waals surface of a molecule is an abstract
representation or model of that molecule by treating each atom as a sphere
with the van der Waals radius. This model is named for Johannes Diderik van
der Waals, a Dutch theoretical physicist and thermodynamicist. Several models
have made improvements towards the van der Waals surfaces. The CPK models is
developed by and named for Robert Corey, Linus Pauling, and Walter Koltun in
the paper [16]. In this model, atoms were represented by spheres whose radii
were proportional to the van der Waals radii of the atoms, and whose center-
to-center distances were proportional to the distances between the atomic
nuclei in the same scale. The Lee-Richards surface model, raised by B.Lee and
F.M.Richards in the paper [36], was given by the center of a radius $r_{w}$
sphere when this ball was rolling everywhere along the van der Waals surface
of the molecule. While Lee-Richards surface might not be smooth, the
Connolly’s surface model, introduced by M.L.Connolly in [15], was constructed
by replacing the center by the ‘front’ of the same sphere.

On the other hand, it was found in biology that low-energy elements occur more
frequently than others in the 3D structure of proteins. This interesting
phenomenon was summarized as the Boltzmann like statistics for proteins, i.e.,

(1) $\hbox{OBSERVED OCCURRENCE}\sim exp(-\frac{\hbox{ENERGY}}{RT^{\ast}})$

where $T^{\ast}$ is the ‘conformational temperature’ of protein statistics
which is close to room temperature, and $R$ is the Boltzmann constant [24,
25]. From this point of view, geometry objects could enter as representations
of biomolecules. For example, the most commonly known theory in this side is
the Ramachandran plot for proteins raised in [59]. In this theory, people
focused on two backbone dihedral angles $\psi$ and $\phi$ of residues in
proteins. The Ramachandran plot visualized energetically allowed regions for
$\psi$ against $\phi$.

The existence of hydrogen bonds, especially the backbone hydrogen bond (BHB),
is also important in the formation of the structure of proteins and the
realization of the functions of proteins. Following the same philosophy of
Equation 1, the Backbone Free Energy Estimator was introduced by R.Penner in
[56, 55]. In this theory, he constructed the unit 3D frames on both sides of
BHB with $x-y$ plane is the peptide plane and $z$ axis is the associated cross
product. Hence, he could represent a BHB as an element on the Lie group
$\operatorname{SO}_{3}(\mathds{R})$. The distribution of BHBs on
$\operatorname{SO}_{3}(\mathds{R})$ could be computed from a well-selected
data set.

The energy also appears in the studies and simulations of variations of
protein conformations. This is a joint field of differential geometry, the
shape analysis and biology. The shape space is somehow like the moduli space
in mathematics. Since the energy usually appears as a sum of the potential of
interactions that only depend on the Euclidean distance between pairs of atoms
in the protein. Two conformations can only be distinguished up to rigid-body
transformations. This observation gives the idea of the shape space. The shape
space is a quotient manifold by identifying each protein conformation with the
orbit space of configuration space of all protein conformations by its rigid-
body transformations. The dynamics of the protein conformation could be
treated as the flow or the deformation on the shape space. There are several
ways of constructing the shape spaces, for example using the backbone dihedral
angles, using the point cloud and the Normal mode analysis (NMA) [6, 27, 7].
The Backbone Free Energy Estimator [56, 55] introduced above with the shape
space $\operatorname{SO}_{3}(\mathds{R})$ is another example. Shape space is
an important object in shape analysis and computer vision, we refer [9, 35,
75] for more detailed introduction.

Finally, let us mention a recent work [18] trying to bring the Riemannian
geometry into the study of the protein conformations. In this paper, they
constructed a smooth manifold from the quotient of the configuration space of
non-planer atom coordinates by the group of affine translations. The different
conformations of the proteins were represented as points on this manifold. An
energy motivated and efficiently evaluated metric could be defined on this
manifold such that making this manifold a Riemannian manifold. Then the
dynamics of changing of protein conformations could be efficiently
interpolated and extrapolated along non-linear geodesic curves.

###  1.3. Discrete Curve Model

Many topics in applied geometry like computer graphics and computer vision
have close connection with discrete geometry, especially the discrete curves
and discrete surfaces. For example, the tasks like analysis of geometric data,
its reconstruction, and further its manipulation and simulation.
Approximations of 3D-geometric notions play a crucial part in creating
algorithms that can handle such tasks. Similarly as the smooth realm where the
curvature and torsion play important roles in parametrizing a smooth curve or
surface, the estimation of curvatures and torsions of curves and surfaces is
also important in applied geometry.

Recall that the curvature of a smooth curve $\gamma$ at a fixed point $p$ is
defined as the inverse of the radius of the osculating circle of $\gamma$ at
point $p$. It quantifies the extent of the bending of $\gamma$ at $p$. The
torsion of the curve $\gamma$ is given by the ’percentage’ between the normal
bundle of the curve and the projection of the differential of the binormal
bundle on the normal bundle. It measures the speed of a smooth curve moves
away from it osculating plane. In the discrete case, we applied the ideas and
methods from classical differential geometry instead of “simply” discretizing
equations or using classical differential calculus. There are several versions
of discretized curvature and torsion based on using different starting point
of the classical differential geometry like from Frenet-frame [8] and from
geometric knot theory [68].

In this paper, we mentioned two ways of applying properties of the discrete
curve. The first, which we call it the truncated Hop distance between two
discrete curves, is defined as a matrix composed of the differences of hop
distances of vertices between two discrete curves. We will show below by
example that treating a protein as a discrete curve and applying this simply
defined and computed parameter give us an important property in the formation
of amyloid fibril. We will show below by example that by treating a protein as
a discrete curve, this simply defined and computed parameter give us an
important property of the formation of amyloid fibril.

On the other hand, we introduced a version of discrete curvature and torsion
from [48] which is based on the osculating circle of a discrete curve. We will
introduce an example that this version of torsion could detect the BHB between
layers of amyloid fibril.

###  1.4. Topological Model

Topological Data Analysis (TDA) is a mathematical field focus on the ‘shapes’
of the data. It provides a set of mathematical and computational techniques to
analyze complex high-dimensional data, and identify key features of the data
which may be difficult to be extracted with traditional methods. Recently, TDA
has been witnessed applications in a variety of disciplines, such as
computational chemistry [70], nano material [73], environmental science [52,
53], natural language processing [23], image segmentation [14], medical
imaging [64, 46, 57], deep learning model [58, 12], complex network [26, 72],
economy [63], biomolecular analysis [17, 39], the winning of D3R Grand
Challenges [49, 50] and the evolutionary mechanism of SARS-CoV-2 [11], etc.
Compared with traditional dimensionality reduction methods, such as principal
component analysis (PCA), TDA retains more effective information while
reducing the dimensionality of data.

Up to now, various tools have been developed under the heading of TDA, such as
(i) Mapper. Mapper first divides the original data into several clusters based
on the preassigned filter functions. Then, two clusters are connected if they
have common data points. The output of mapper is a combinatorial graph. Monica
N. et al. analyzed the microarray gene expression data by adopting Mapper and
disease-specific genomic analysis transformation, and successfully identified
a unique subgroup of Estrogen Receptor-positive (ER+) breast cancers[51]. (ii)
Persistent topological laplacians (PDL). PDL captures information about data
based on the spectra of the laplacian operator. Specifically, the harmonic
spectra relates to the Betti number of data, while the non-harmonic spectra
provides extra geometry information [29, 38]. PDL-based approaches have been
applied to analyze the structure of protein-protein interaction networks [19].
(iii) Persistent homology (PH). PH is one of the most popularly used tool in
TDA, which creates a multiscale family of simplicial complexes and describes
the evolution of these complexes and their associated homology groups over
scales. We will focus on the PH in the following materials.

Persistent homology has been widely applied in data science, such as
crystalline material [33], chemical engineering[65], especially computational
biology[39]. For example, the element-specific PH combined with deep learning
winned the D3R Grand Challenges[49, 50], the element- and site-specific PH
adopted for prediction the evolutionary mechanism of SARS-CoV-2[11] and
hypergraph-based PH to predict protein-ligand binding affinity[40]. Theories
related to PH have been developed rapidly, including weighted PH [60, 3],
multiparameter PH [28, 61, 41], etc. Moreover, the open source packages, such
as Gudhi [42], Javaplex [1], Ripser[2], etc., have contributed to the wide
range of applications in different areas of PH. In this paper, we introduced
basic definitions of the PH. More details can be found in [77, 20]. Moreover,
We will give an example of the application showing that these tools give
better indicators to distinguish amyloid fibrils than the commonly used
indicator RMSD.

###  1.5. Arrangement of the Paper

The reminder of this paper is organized as follows. In Section 2, we defined
the truncated Hop distance to compare the structure of two discrete curves.
Moreover, we explained how to treat one layer of amyloid fibril as a discrete
curve. And we suggest that the effect in the formation of amyloid fibril is
relatively mild based on the result of the truncated Hop distance.
Preliminaries of the definitions of curvature and torsion on the discrete
curves are presented in Section 3. By computing the discrete curvature and
torsion on single layers of amyloid fibril, we suggest that the anomalies
reflect slight affects of the layer-layer hydrogen bonds. Finally in Section
4, we introduced the distances between persistent diagrams (PDs), and showed
that how these distances work in quantifying the difference of the structure
of amyloid fibrils.

##  2\. Discrete Curve and Hop Distance

A discrete curve is a polygonal curve in $\mathds{R}^{2}$ or $\mathds{R}^{3}$
given by its vertices via a map $\gamma:\mathds{Z}\to\mathds{R}^{3}$. In this
section, we will introduce a direct parameter that not only could distinguish
two different discrete curves, but also showed meaning in analyzing the
difference of the structure of protein with well-folded form and amyloid form.

###  2.1. Hop distance

Let $\gamma$ and $\gamma^{\prime}$ be two different discrete curves, and
$\gamma_{i}$ and $\gamma^{\prime}_{i}$ be the corresponding points of
vertices. Let $d(\gamma_{i},\gamma_{i+k})$ be the Euclidean distance between
the $i$-th and $(i+k)$-th vertices, also called the $k$-hop distance between
$\gamma_{i}$ and $\gamma_{i+k}$.

######  Definition 2.1.

The $N$-truncated hop distance, where $N$ is a fixed positive integer, between
the two discrete curve $\gamma$ and $\gamma^{\prime}$ is given by the $N\times
N$ matrix $D$ with

(2)
$D_{ij}=|d(\gamma_{i},\gamma_{j})-d(\gamma_{i}^{\prime},\gamma_{j}^{\prime})|.$

###  2.2. Applications to Analysis of the Structure of Amyloid fibrils

To explore structural or functional properties of proteins in practice, we
mostly consider the heavy atoms of residues, the backbone of peptide chain, or
the set of $C_{\alpha}\text{s}$ [5, 71] to simplify the structure of protein.
This simplication reduce the redundant and irrelevant information, but also
preserve the significant structural features of proteins, which can improve
the efficiency of research. In this case, we may treat a peptide chain as a
discrete curve to explore its geometry invariants. It could be helpful to
extract the structural features of protein, and further, to analyze the
functions of protein. In this example, we focused on the analysis of the
structure of point mutant V30M TTR with well-folded form and amyloid form, as
shown in Fig.2.

Figure 2. Convert peptide chains into discrete curves. (A) The top is the
cartoon representation of chain A of TTR V30M mutant (PDB entry: 4TL4), and
the bottom is its corresponding discrete curve. (B) The top is the cartoon
representation of chain A of one amyloid form of 4TL4 (PDB entry: 7OB4), and
the bottom is its corresponding discrete curve. Red balls represent the
$C_{\alpha}$ atoms.

It has been discovered that amyloid fibril serves as the key pathological
entity in different neurodegenerative diseases, such as Alzheimer’s disease
and Parkinson’s disease [54, 34]. However, the process by which amyloid fibril
is formed is so far obscure, hindering further research into the causes of
diseases. Here, we used the truncated hop distance to explore the strength of
effect in the formation of amyloid form, as shown in Fig.3.

Figure 3. The truncated hop distance between chain A of 4TL4 and 7OB4. The
unit is the Angstrom (Å).

It is clear that the values near the diagonal is less than 1.5Å, which
illustrates that there could not be stretching and compressing in the process
of the formation of amyloid fibril. We suggest that the effect is relatively
mild in the formation from chain A of 4TL4 to chain A of 7OB4.

##  3\. Discrete Curvatures and Discrete Torsions

The discrete torsion based on the cross ratio and osculating circle was first
introduced in [48], which is used in computer vision to estimate the
differences between the background smooth curve and the discrete sample curve.
In the paper [37], we introduced this important parameter to the analyzing the
structure of proteins. We found that compared with the traditional tools like
backbone dihedral angles, discrete curvature resp. discrete torsion is a order
two resp. order three parameter which is more sensitive to the weak
interactions. In this section, we will give a detailed introduction on these
parts.

###  3.1. Curvatures and Torsions for Smooth Curves

The curvature and the torsion are two important geometric invariants
parametrizing the property of a smooth space curve. We will first recall the
corresponding constructions in the smooth case. Let
$\gamma(s)\subset\mathds{R}^{3}$ be a smooth space curve with $s$ the arc
length parameter. Recall that the tangent field $T(s)$ is unit for all
$s\in\mathds{R}$. The principal normal field is given by
$N(s):=\frac{T^{\prime}(s)}{\parallel T^{\prime}(s)\parallel}$ and the
binormal field by $B(s):=N(s)\times T(s)$ while both of them are unit vector
fields. It is clear that the three unit vectors $T(s),\ N(s)$ and $B(s)$ are
pairwise normal, hence they form a frame at each point.

The curvature $\kappa$ is defined as $\kappa:=\parallel
T^{\prime}(s)\parallel$ parametrizing the bending degree of the curve.
Geometrically, there exists a space circle passed $\gamma(s)$ which has $N(s)$
as radial vector and $\frac{1}{\kappa}$ as radius. This circle is called the
osculating circle, which has contact of order at least 2 with $\gamma$ at
$\gamma(s)$. Therefore, they have the common tangent plane which is spanned by
$T(s)$ and $N(s)$ and we call it the osculating plane.

Since $B(s)$ is an unit field, its differential $B^{\prime}(s)$ should tangent
to $B(s)$. Moreover, it is clear that the following equation holds.

(3)
$0=\frac{d}{ds}<B(s),T(s)>=<B^{\prime}(s),T(s)>+<B(s),T^{\prime}(s)>=<B^{\prime}(s),T(s)>$

Hence, $B^{\prime}(s)$ should parallel to $N(s)$, the torsion of the curve
$\gamma$ at $\gamma(s)$ as the number $\tau$ satisfying the property that
$B^{\prime}(s)=-\tau N(s)$. From all these constructions, the torsion $\tau$
parametrized the speed of a curve moving away from its osculating plane.

###  3.2. Basics in quaternions

Let $\mathds{H}$ be the ring of quaternionics which could be treated as a skew
field whose elements can be identified with $\mathds{R}\times\mathds{R}^{3}$.
There exist an isomorphism of Euclidean 3-space $\mathds{R}^{3}$ with the
imaginary parts of $\mathds{H}$. In this paper, we will write $\mathds{H}$ in
the following way

(4) $\mathds{H}:=\\{[r,v]:r\in\mathds{R},v\in\mathds{R}^{3}\\}$

The first part $r=\operatorname{Re}q$ resp. the second part
$v=\operatorname{Im}q$ of a quaternion $q=[r,v]$ is called the real part of
$q$ resp. the imaginary part of $q$. The set of all pure imaginary quaternions
is denoted as $\operatorname{Im}\mathds{H}$, i.e.,
$\operatorname{Im}\mathds{H}:=\\{[0,v]:v\in\mathds{R}^{3}\\}$. Some operators
on $\mathds{H}$ under this form are listed in the Table 1.

Addition | $[r,v]+[s,w]=[r+s,v+w]$
---|---
Multiplication | $[r,v]\cdot[s,w]=[rs-<v,w>,rw+sv+v\times w]$
Conjugation | $\overline{q}=[r,-v]$
Norm | $|q|=\sqrt{q\cdot\bar{q}}=\sqrt{r^{2}+\parallel v\parallel^{2}}$
Inverse | $q^{-1}=\frac{\bar{q}}{|q|}$
Cross Ratio | $\operatorname{cr}(a,b,c,d):=(a-b)(b-c)^{-1}(c-d)(d-a)^{-1}$
Table 1. Some operators on $\mathds{H}$

In this paper, the polar representation of a quaternion is quite useful, for a
quaternion $q=[r,v]$ its polar representation is defined as
$q=|q|[\cos\phi,v\sin\phi]$ with $\parallel v\parallel=1$ and
$\phi\in[0,\pi]$. In this case, the square root of $q$ is defined as
$\sqrt{q}:=\sqrt{|q|}[\cos\frac{\phi}{2},v\sin\frac{\phi}{2}]$. Note that such
defined square root is not unique if $q\in\mathds{R}_{\leq 0}$, however, we
will not meet this case in our application.

Recall that a Möbius transformation is a homeomorphism of
$\mathds{R}^{n}\cup\\{\infty\\}$ which is a finite composition of inversions
in spheres and reflections in hyperplanes. In other words, any Möbius
transformation $\sigma$ can be written as the form

(5) $\sigma(x)=x+\frac{\alpha A(x-a)}{\parallel x-a\parallel^{\epsilon}},\
x\in\mathds{R}^{n}\cup\\{\infty\\}$

where $\alpha\in\mathds{R}$, $a,b\in\mathds{R}^{n}$, $A$ is an orthogonal
matrix and $\epsilon$ equals 0 or 2. It is clear to check that the cross-ratio
above is invariant under Möbius transformation i.e. the following equation
holds for all Möbius transformation $\sigma$

(6)
$\operatorname{cr}(\sigma(a),\sigma(b),\sigma(c),\sigma(d))=\operatorname{cr}(a,b,c,d)$

Moreover, we have the following Lemma for the properties of the cross ratio.

######  Lemma 3.1.

Let $a,b,c,d,d^{\prime}\in\mathds{H}$ be four quaternions, then we have

  1. (1)

If $d^{\prime}\in\mathds{H}$, $a\neq c$ and
$\operatorname{cr}(a,b,c,d)=\operatorname{cr}(a,b,c,d^{\prime})$, then we have
$d=d^{\prime}$.

  2. (2)

If $a,b,c,d\in\operatorname{Im}\mathds{H}\cong\mathds{R}^{3}$, then
$\operatorname{cr}(a,b,c,d)\in\operatorname{Re}\mathds{H}$ if and only if
$a,b,c,d$ are concyclic.

###### Proof.

  1. (1)

The first is clear to check.

  2. (2)

The second is proved in [4].

∎

######  Lemma 3.2.

Let $a,b,c,d\in\operatorname{Im}\mathds{H}$ be four points not lying on a
common circle. Then, the imaginary part of the cross-ratio is the normal of
the circumsphere (or plane) at $a$, i.e., for a proper circumsphere with
center $m$, we have
$\operatorname{Im}\operatorname{cr}(a,b,c,d)\mathrel{/\mskip-2.5mu/}(m-a)$.

######  Remark 3.3.

If the four points $a,b,c,d$ are coplaner, then $m$ is the point of $\infty$.
The vector $\operatorname{Im}\operatorname{cr}(a,b,c,d)$ is orthogonal to the
common plane of $a,b,c,d$.

###### Proof.

This is Lemma 10 of [48]. ∎

Let us define the ‘diagonal’ points $f(a,b,c,d)$ as

(7)
$f(a,b,c,d):=((b-a)(c-a)^{-1}\sqrt{\operatorname{cr}(c,a,b,d)}+1)^{-1}((b-a)(c-a)^{-1}\sqrt{\operatorname{cr}(c,a,b,d)}c+b)$

We have the following lemmas

######  Lemma 3.4.

The ‘diagonal’ point $f(a,b,c,d)$ fullfills the equation that

(8) $\operatorname{cr}(c,a,b,f(a,b,c,d))=-\sqrt{\operatorname{cr}(c,a,b,d)}$

Moreover if we assume
$a,b,c,d\in\operatorname{Im}(\mathds{H})\cong\mathds{R}^{3}$ then the
’diagonal’ point satisfies $f(a,b,c,d)\in\operatorname{Im}(\mathds{H})$. Even
more $f(a,b,c,d)$ lies on the circumsphere of $a,b,c,d$.

######  Remark 3.5.

If $a,b,c,d$ are four points in
$\mathds{R}^{3}\cong\operatorname{Im}\mathds{H}$ that form a parallelogram
with $ab$ parallel to $cd$, the inserting point $f(a,b,c,d)$ is the
intersection of the two diagonals $ac$ and $bd$. This is the reason that it is
called a ‘diagonal’ point.

######  Corollary 3.6.

The formula $f$ is invariant under Möbius transformation.

###### Proof.

The Lemma 3.4 is the Lemma 11 and Corollary 4 in [48]. Then from the Lemma 3.1
and Lemma 3.4 the formula $f$ is Möbius transformation invariant. ∎

Now we will introduce one of the most important Proposition and also the basis
of the further construction.

######  Proposition 3.7.

Let $a,b,c,d\in\operatorname{Im}\mathds{H}\cong\mathds{R}^{3}$ be four
pairwise different points and consider the four inserting points given by $f$
and cyclic permutations

(9) $\displaystyle A=f(d,a,b,c)$ $\displaystyle B=f(a,b,c,d)$ (10)
$\displaystyle C=f(b,c,d,a)$ $\displaystyle D=f(c,d,a,b)$

Then the four points $A,B,C,D$ are concyclic with

(11) $\operatorname{cr}(A,B,C,D)=-1$

###### Proof.

We only need to prove the Equation 11 holds. From Lemma 3.4 the four points
$A,B,C$ and $D$ lie on a common sphere i.e. the circumsphere of $a,b,c,d$. Let
us fix an unit vector $v_{0}\in\mathds{R}^{3}$. It is clear that the space
$\mathds{C}_{v_{0}}:=\\{[r,kv_{0}]:r,v\in\mathds{R}\\}$ is isomorphic to the
complex plane $\mathds{C}$. Let $\sigma$ is the Möbius transformation that
maps the circumsphere of $a,b,c,d$ to $\mathds{C}_{v_{0}}$.

From Lemma 3.1, that we have
$\operatorname{cr}(A,B,C,D)=\operatorname{cr}(\sigma(A),\sigma(B),\sigma(C),\sigma(D))$.
Hence without loss of generality, we may assume $a,b,c,d$ are both complex
numbers. In this case, the Equation 11 comes from a direct compute, the
details are in Theorem 1 of [48]. ∎

###  3.3. Curvatures and Torsions for Discrete Curves

Let us consider the discrete case. Recall that a discrete curveis a polygonal
curve in $\mathds{R}^{2}$ or $\mathds{R}^{3}$ given by its vertices via a map
$\gamma:\mathds{Z}\to\mathds{R}^{3}$. We will see the definition of curvature
and torsion for a discrete curve follows similar philosophy as their smooth
versions. Historically, the curvature for a discrete curve has a long history
and is widely used in computer vision and study of complex networks. However,
the torsion is pretty new, one is first introduced in [48]. We will introduce
them in this section.

Let $\gamma_{i}$ be a vertex of the discrete curve. From the Proposition 3.7,
for each four elements tuple
$(\gamma_{i-1},\gamma_{i},\gamma_{i+1},\gamma_{i+2})$ there exist four
inserting points i.e.

(12) $\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle
f(\gamma_{i+2},\gamma_{i-1},\gamma_{i},\gamma_{i+1}),\ B_{i}$ $\displaystyle=$
$\displaystyle f(\gamma_{i-1},\gamma_{i},\gamma_{i+1},\gamma_{i+2})$
$\displaystyle C_{i}$ $\displaystyle=$ $\displaystyle
f(\gamma_{i},\gamma_{i+1},\gamma_{i+2},\gamma_{i-1}),\ D_{i}$ $\displaystyle=$
$\displaystyle f(\gamma_{i+1},\gamma_{i+2},\gamma_{i-1},\gamma_{i})$

The Lemma 3.4 implies that the four points lie on the circumsphere of
$\gamma_{i-1},\gamma_{i},\gamma_{i+1}$ and $\gamma_{i+2}$ which we take it as
the discrete osculating sphere at $\gamma_{i}$. Note that
$\gamma_{i-1},\gamma_{i},\gamma_{i+1}$ and $\gamma_{i+2}$ are not concyclic in
general, but the Proposition 3.7 implies that the four points $A_{i}$,
$B_{i}$, $C_{i}$ and $D_{i}$ are concyclic.

######  Definition 3.8.

Let $\gamma$, $\gamma_{i}$,$A_{i}$,$B_{i}$,$C_{i}$ and $D_{i}$ as above, the
circle denoted by $k_{i}$ given by $A_{i}$,$B_{i}$,$C_{i}$ and $D_{i}$ is the
osculating circle of $\gamma$ at $\gamma_{i}$. The inverse of the radius of
$k_{i}$ the discrete curvature at $\gamma_{i}$ denoted by $\kappa_{i}$.

######  Definition 3.9.

The normal unit vector $N_{i}$ of $\gamma$ at $\gamma_{i}$ to be the unit
normal vector of $k_{i}$ at $B_{i}$. The plane passing through $B_{i}$ and
normal to $N_{i}$ as the osculating plane.

######  Definition 3.10.

The discrete torsion of $\gamma$ at $\gamma_{i}$ is defined as following

(13)
$\tau_{i}:=-\frac{9<\operatorname{Im}\operatorname{cr}(\gamma_{i-1},\gamma_{i},\gamma_{i+1},\gamma_{i+2}),N_{i}>}{2\kappa_{i}\parallel\gamma_{i}-\gamma_{i+1}\parallel^{2}}$

######  Remark 3.11.

The connection of discrete torsion and smooth version are as following: Let us
assume in this Remark that $\gamma(t)$ be a smooth curve and $t$ be any
parameter. In this case the torsion of $\gamma$ at $\gamma(t_{0})$ could be
computed as

(14) $\displaystyle\tau$ $\displaystyle=$
$\displaystyle-\frac{<\gamma^{\prime}\times\gamma^{\prime\prime},\gamma^{\prime\prime\prime}>}{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel^{2}}$
$\displaystyle=$
$\displaystyle-\frac{\det(\gamma^{\prime},\gamma^{\prime\prime},\gamma^{\prime\prime\prime})}{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel^{2}}$
$\displaystyle=$
$\displaystyle\frac{\det(\gamma^{\prime\prime\prime},\gamma^{\prime},\gamma^{\prime\prime})}{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel^{2}}$
$\displaystyle=$
$\displaystyle\frac{<\gamma^{\prime}\times\gamma^{\prime\prime\prime},\gamma^{\prime\prime}>}{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel^{2}}$

Recall that
$\kappa=\frac{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel}{\parallel\gamma^{\prime}\parallel^{3}}$
and

(15) $N=B\times
T=\frac{\gamma^{\prime\prime}<\gamma^{\prime},\gamma^{\prime}>-\gamma^{\prime}<\gamma^{\prime},\gamma^{\prime\prime}>}{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel\parallel\gamma^{\prime}\parallel}=\frac{1}{\parallel\gamma^{\prime}\parallel^{2}\kappa}\gamma^{\prime\prime}-\frac{<\gamma^{\prime},\gamma^{\prime\prime}>}{\parallel\gamma^{\prime}\times\gamma^{\prime\prime}\parallel\parallel\gamma^{\prime}\parallel}\gamma^{\prime}$

Hence, we have the following

(16)
$\tau=\frac{<\gamma^{\prime}\times\gamma^{\prime\prime\prime},N>}{\kappa\parallel\gamma^{\prime}\parallel^{4}}$

In the discrete case, the role of
$\gamma^{\prime}\times\gamma^{\prime\prime\prime}$ is replaced by
$\operatorname{Im}\operatorname{cr}(\gamma_{i-1},\gamma_{i},\gamma_{i+1},\gamma_{i+2})$.
We will show below that such defined discrete torsion satisfies our
expectations.

Now let us introduce without proof several properties of the definitions above
which reveal the fact that the discrete curvature, normal vector and torsion
defined above is the approximation of the smooth version.

######  Proposition 3.12.

The discrete torsion vanishes for planer discrete curve.

###### Proof.

If $\gamma$ is a discrete curve lying on the plane $P$, the vector
$\operatorname{Im}\operatorname{cr}(\gamma_{i-1},\gamma_{i},\gamma_{i+1},\gamma_{i+2})$
is orthogonal to $P$ from the Lemma 3.2. From the Definition 3.9,
$N_{i}\subset P$. Hence the discrete torsion vanish at all points. ∎

######  Theorem 3.13.

Let $s:\mathds{R}\to\mathds{R}^{3}$ be a smooth space curve, let
$\mu,\epsilon\in\mathds{R}$ be real numbers and let the discrete curve
$\gamma:\mathds{Z}\to\mathds{R}^{3}$ with $\gamma_{i}:=s(\mu+(2i-1)\epsilon)$
be the sample of $s$. Let $\kappa$, resp. $N$ and $\tau$ be the curvature
reps. unit normal vector and torsion of $s$ at $s(\mu)$. We have the following
properties

  1. (1)

The discrete curvature is a second approximation of $\kappa_{s}$, i.e.
$\kappa_{0}=\kappa_{s}+o(\epsilon^{2})$,

  2. (2)

The center $m_{0}$ of the discrete osculating circle $k_{0}$ of $\gamma$
converges to the center of the smooth osculating circle of $s$ at the same
rate, i.e. $m_{0}-s(\mu)=\frac{N(\mu)}{\kappa_{s}(u)}+o(\epsilon^{2})$,

  3. (3)

The discrete Frenet frame $(T_{0},N_{0},B_{0})$ is a second-order
approximation of the smooth Frenet frame $(T,N,B)$,

  4. (4)

The discrete torsion is a second-order approximation of the smooth torsion
i.e. $\tau_{0}=\tau+o(\epsilon^{2})$

###### Proof.

This is the Theorem 4, Lemma 14, Lemma 15 and Theorem 5 in [48]. The main
points is similar as the proof of the Proposition 3.7, we first map all points
to a complex plane by a Möbius transformation. Then we could check by direct
computation. ∎

###  3.4. Applications to Analysis of the Structure of Amyloid fibrils

We have shown a way of treating a peptide chain as a discrete curve and its
application in the Section 2. In this section, we considered another way as
following: Recall that from the Figure 1, the backbone consists of amino
nitrogens, $\alpha$-carbons, carbonyl carbons and carbonyl oxygens. For a
fixed peptide chain which is also a layer of the amyloid fibrils, let $N_{i}$
resp. $C_{\alpha_{i}}$ and $C_{i}$ be the coordinates of amino nitrogens resp.
$\alpha$-carbons and carbonyl carbons with $i\in\mathds{Z}^{\geq 1}$ being the
indices of the residues in this layer. This is a discrete curve whose vertices
points are $N_{i}$, $C_{\alpha_{i}}$ and $C_{i}$. By applying the definitions
in Section 3, we will compute the curvatures and torsions as below

(17) $\displaystyle C_{i-1},N_{i},C_{\alpha_{i}},C_{i}$ $\displaystyle\to$
$\displaystyle\hbox{Curvatures and Torsions at }N_{i}$ $\displaystyle
N_{i},C_{\alpha_{i}},C_{i},N_{i+1}$ $\displaystyle\to$
$\displaystyle\hbox{Curvatures and Torsions at }C_{\alpha_{i}}$ $\displaystyle
C_{\alpha_{i}},C_{i},N_{i+1},C_{i+1}$ $\displaystyle\to$
$\displaystyle\hbox{Curvatures and Torsions at }C_{i}$

The pseudo-code of computing curvature and torsion is shown as the Algorithm
1.

1

input : A series of shape $3l\times 3$ consists of coordinates of all amino
nitrogens, $\alpha$-carbons and carbonyl carbons

output : A series of shape $3l\times 2$ consists of curvatures and torsions at
each atom

2 for  _$i\leftarrow 1$ to $l-3$_ do

3

4 compute inserting points $A,B,C,D$

5 compute circumcircle center $O$ by solving equations
$\displaystyle(X-\frac{A+B}{2})\overrightarrow{BA}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle(X-\frac{A+C}{2})\overrightarrow{CA}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle(\overrightarrow{AB}\times\overrightarrow{AC})\overrightarrow{XA}$
$\displaystyle=$ $\displaystyle 0$

6 compute $\kappa=\frac{1}{\parallel\overrightarrow{AO}\parallel}$

7 compute
$N=\frac{\overrightarrow{AO}}{\parallel\overrightarrow{AO}\parallel}$

8 compute torsion $\tau$

9 end for

Algorithm 1 Algorithm of computing curvature and torsion

We conducted the computation to several amyloid fibrils. We found that the
torsions at carbonyl carbons are significantly larger than the one in other
places (we list one example 7OB4 in Figure 4). We showed that this anomalies
reflect slight affects of the layer-layer hydrogen bonds. Without loss of
generality, we may assume the nearby three layers of 7OB4 is marked as A,B and
C. We processed the correlation analysis between $|\tau_{i}|$, the absolute
value of torsion, and $|(\tilde{d}^{i}_{B,-})^{2}-(\tilde{d}^{i}_{B,+})^{2}|$,
where $\tilde{d}^{i,-}_{B}$ is the Euclidean distance of $O_{i}(B)$ and
$N_{i+1}(A)$ and $\tilde{d}^{i,+}_{B}$ is the Euclidean distance between
$O_{i}(B)$ and $N_{i+1}(C)$. We found that although the six atoms form a
peptide bond are coplaner, they have a tend to leave this plane under the
influence of the layer-to-layer hydro-bond.

Figure 4. Distributions of absolute values of Torsions for 7OB4. The $x$-axis represents the absolute value of torsion and the $y$-axis represents the number of atoms. We take purple bar for nitrogens, orange for alpha carbons and green for carbonyl carbons. The discrete torsion is dimensionless. Table 2. The correlation of $|\tau_{i}|$ and $|(\tilde{d}^{i}_{\cdot,-})^{2}-(\tilde{d}^{i}_{\cdot,+})^{2}|$ for 7OB4. S.D stands for the standard error | Slope | Intercept | Pearson |  $p$-value | S.D slope | S.D intercept
---|---|---|---|---|---|---
7OB4 | -0.187 | 17.765 | -0.194 | $2.325\times 10^{-9}$ | 0.032 | 1.094

##  4\. Topological Data Analysis

###  4.1. Persistent Homology in a Nutshell

We are going to introduce the definition of PH first in a general setting. Let
us take $\mathds{K}$ be a fixed ring. Let $X$ be a set and $K$ be a simplicial
complex constructed from $X$ which is a collection of non-empty finite subsets
of $X$, such that for every set $\sigma$ in $K$ and every subset
$\sigma^{\prime}\subset\sigma$, the set $\sigma^{\prime}$ also belongs to $K$.
We will call $\sigma\in K$ a simplex.

Suppose further that each simplex $\sigma$ is oriented, there exists a chain
complex

(18) $\cdots\to C_{n+1}(K)\xrightarrow{\partial_{n+1}}C_{n}(K)\to\cdots$

where $C_{n}(K)$ is the $\mathds{K}$-linear combination of simplices
$\sigma=[x_{0},\cdots,x_{n}]$ in $K$ of length $n+1$ and the boundary map
$\partial_{n+1}:C_{n+1}(K)\to C_{n}(K)$ is a $\mathds{K}$-linear map defined
in a natural way.

######  Definition 4.1.

The filtration of $K$ is a nested sequence of subcomplexes $\\{K_{t}\\}_{t\in
T}$ of $K$ with $T\subset\mathds{R}$ such that if $t\leq t^{\prime}$ then
$K_{t}\subset K_{t^{\prime}}$ and $\cup_{t\in T}K_{t}=K$. Note that the index
set $T$ could be finite or infinite.

Suppose $t\leq t^{\prime}$ be two different real numbers, the inclusion map
$K_{t}\hookrightarrow K_{t^{\prime}}$ is also a map of chain complexes namely
the following diagram commutes

(19)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n+1}(K_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n+1}^{t}}$$\textstyle{C_{n}(K_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n}^{t}}$$\textstyle{C_{n-1}(K_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n+1}(K_{t^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n+1}^{t^{\prime}}}$$\textstyle{C_{n}(K_{t^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n}^{t^{\prime}}}$$\textstyle{C_{n-1}(K_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

where $\partial_{n}^{t}$ is the restriction of $\partial_{n}$ to
$C_{n}(K_{t})$. Hence, it induces a map of homology group $H_{n}(K_{t})\to
H_{n}(K_{t^{\prime}})$.

######  Definition 4.2.

The $(t,t^{\prime})$-persistent $n$-th homology group is defined as the image
of the induced map $H_{n}(K_{t})\to H_{n}(K_{t^{\prime}})$. In other words, it
is the following group

(20) $H_{n}^{t\to t^{\prime}}=Z_{n}^{t}/(B_{n}^{t^{\prime}}\cap Z_{n}^{t}),$

where $Z_{n}^{t}$ is the $n$-th cycle group of $K_{t}$, i.e.,
$\ker\partial_{n}^{t}$, and $B_{n}^{t^{\prime}}$ is the $n$-th boundary group
of $K_{t^{\prime}}$, i.e., $\operatorname{Im}\partial_{n+1}^{t^{\prime}}$. The
rank of $H_{n}^{t\to t^{\prime}}$ is called the $(t,t^{\prime})$-persistent
$n$-th Betti number denoted as $\beta_{n}^{t\to t^{\prime}}$.

The value $t_{0}$ where for every $\delta>0$ the induced map
$H_{n}(K_{t_{0}-\delta})\to H_{n}(K_{t_{0}+\delta})$ is not an isomorphism is
called a homological critical value.

######  Remark 4.3.

In the real world applications, people always assume the coefficient ring
$\mathds{K}$ to be $\mathds{Z}/2\mathds{Z}$ in which case the orientation of
the simplex is not important, since in such circumstances the boundary of the
simplex is not related to the orientation. Hence we will usually omit the
choice of orientation in the TDA process.

######  Definition 4.4.

A persistence $\mathds{K}$ module $M$ is a family of $\mathds{K}$-modules
$M_{i}$, together with homomorphisms $\varphi_{i}:M_{i}\to M_{i+1}$. We say a
persistence module $M$ of finite type if each component module is a finitely
generated $\mathds{K}$-module, and if there exist positive integer $m_{0}$
such that the maps $\varphi$ are isomorphisms for $i\geq m_{0}$.

######  Remark 4.5.

There exists a functor from the category of persistence modules of finite type
over $\mathds{K}$ to the category of finitely generated non-negatively graded
modules over the polynomial ring $\mathds{K}[x]$, namely take the grade on $M$
as the natural one and $x$ acts on $M$ via the map $x\to\varphi$. From the
Artin-Rees theory [22], this construction turns out to be an equivalence of
categories.

From our requirements, from now on let us assume in additional that
$\mathds{K}$ is a field and the above simplicial complex $K$ is finite. This
means each group in chain complex 18 is a finite dimensional vector space over
$\mathds{K}$. In this case, the ‘filtration’ on the chain complex 18, which
will be called a persistent complex[77], is actually of finite length. In
other words, the total chain complex $C(K)$ and the homology module
$H(C(K),\mathds{K})$ are both persistent module of finite type. The total
chain complex $C(K)$ is not only a graded $\mathds{K}$-module but also a
$\mathds{K}[x]$-module with the variable $x$ of degree 1 acts as shift map via
the filtration. The total homology module $H(C(K),\mathds{K})$ retains this
$\mathds{K}[x]$-module structure. And from the structure theorem for principal
integral domains (PID), note that $\mathds{K}[x]$ is a PID for a field
$\mathds{K}$, we have the following decomposition

(21)
$H(C(K),\mathds{K})\cong\mathop{\oplus}\limits_{i}x^{t_{i}}(\mathds{K}[x])\oplus(\mathop{\oplus}\limits_{j}x^{r_{j}}(\mathds{K}[x]/(x^{s_{j}}\mathds{K}[x])))$

Here $t_{i}$, $r_{j}$ and $s_{j}$ are all integers represent index of
filtration. The decomposition theorem has the following meaning: the free
parts of Equation 21 are in bijective correspondence with those homology
generators birth at parameter $t_{i}$ and persist for all future parameter
values. The torsion parts correspond to those homology generators birth at
parameter $r_{j}$ and death at parameter $r_{j}+s_{j}$.

PD is one of the common visualization methods of the PH. PD is a multiset of
points in the extended two-dimensional plane $\overline{\mathds{R}}^{2}$ where
$\overline{\mathds{R}}=\mathds{R}\cup\infty$, whose coordinates are
respectively correspond to the birth and death time of the homology
generators, union the diagonal points [20]. Given two PDs $P_{1}$ and $P_{2}$,
there are two commonly used metrics to compare them, that is, the
_Wasserstein- $q$ distance_ $W_{q}$ and _Bottleneck distance_ $W_{\infty}$.
They are defined as following:

(22) $W_{q}(P_{1},P_{2})=(\inf\limits_{\eta:P_{1}\rightarrow
P_{2}}\sum\limits_{p\in P_{1}}||p-\eta(p)||_{\infty}^{q})^{\frac{1}{q}},$

(23) $W_{\infty}(P_{1},P_{2})=\inf\limits_{\eta:P_{1}\rightarrow
P_{2}}\sup\limits_{p\in P_{1}}||p-\eta(p)||_{\infty},$

where $\eta$ runs over all the bijections between the points in the two
diagrams. The _Wasserstein- $q$ distance_ is capable of capturing local
differences in the PDs, while the _Bottleneck distance_ is more concerned with
relatively global differences.

###  4.2. The Vietoris-Rips (VR) complex

For different types of data, there are various complexes to characterize the
data, such as (1) for point cloud data (PCD), Alpha complex, VR complex, Čech
complex and Witness complex are commonly used; (2) for imaging data, cubical
complex may be a good choice; (3) for graph or complex network data, path
complex may perform better. By treating a layer of amyloid fibril as a
discrete curve, there are several candidate complexes to characterize this
PCD. Here, we adopted the VR complex since it is precise and easy to compute.

Let $X$ be a finite point set with a index in $n$-dimensional Euclidean space
$(\mathds{R}^{n},d)$, where $d$ is the natural metric on the Euclidean space
and $\mathds{K}$ be the field $\mathds{Z}/2\mathds{Z}$.

######  Definition 4.6.

The Vietoris-Rips (VR) complex (denoted by $\operatorname{VR}(X)$) of $X$ with
parameter $\epsilon$ is the set of all simplices
$\sigma=\\{x_{0},\cdots,x_{k}\\}$, such that $d(x_{i},x_{j})\leq 2\epsilon$
for all $(i,j)$.

As we explained before, the orientation of the simplex could be omitted. And
there exist a natural filtration on the Vietoris-Rips (VR) complex given by
the parameter $\epsilon$. Since the number of $X$ is finite, this filtration
is always of finite length, i.e.,

(24)
$\emptyset=\operatorname{VR}(X)_{0}\subseteq{\operatorname{VR}(X)_{1}}\subseteq{\cdots}\subseteq{\operatorname{VR}(X)_{m}}$

We will modify our notations a bit as follows, let $i$ and $p$ be integers

(25) $H_{n}^{i,p}:=H_{n}^{i\to i+p}=Z_{n}^{i}/(B_{n}^{i+p}\cap Z_{n}^{i}),$

And we have similar modification for the notations of Betti number
$\beta_{k}^{i,p}:=\beta_{k}^{i\to i+p}$.

Figure 5 is the PD of the C-terminal fragment of chain A of 7OB4 using the VR
complex. The data set is given by taking the coordinates of
$C_{\alpha}\text{s}$ on the C-terminal fragment. Since each chain of 7OB4 is
almost planner, we only computed the persistence on $H_{0}$ and $H_{1}$. Red
points and blue points represent the 0-dimensional and 1-dimensional
homological generators respectively in the Fig.5.

Figure 5. The PD of the C-terminal fragment of 7OB4. Red points and blue
points represent the 0-dimensional and 1-dimensional homological generators,
respectively. The unit is the Angstrom (Å).

###  4.3. Applications to Analysis of the Structure of Amyloid fibrils

The pathological function of amyloid fibril is largely determined by its
structure. It is important to discriminate the structure of amyloid fibrils,
followed by exploring the functions which are closely related to those
structures. In biology, the RMSD is a common indicator to quantify the
difference between structures after aligning. Let
$\text{Prot}_{\text{1}}=\\{c_{1},\dots,c_{n}\\}$ and
$\text{Prot}_{\text{2}}=\\{c_{1}^{\prime},\dots,c_{n}^{\prime}\\}$ be the
coordinates of $C_{\alpha}\text{s}$ of two proteins, respectively. The RMSD
between $\text{Prot}_{\text{1}}$ and $\text{Prot}_{\text{2}}$ is defined as

(26)
$\text{RMSD}=\sqrt{\frac{\sum\limits_{i=1}^{n}(c_{i}-c_{i}^{\prime})^{2}}{n}}.$

It is clear that the RMSD only considers the difference of pairwise distance.
Here, we adopted the PH to quantify the difference between two amyloid fibril
structures from multiscales. The pseudo-code of comparing two amyloid fibrils
is shown as the Algorithm 2. The results are listed in Table3.

Amyloid fibrils | RMSD (Å) |  $W_{\infty}$(Å) |  $W_{2}$(Å) |  $W_{1}$(Å)
---|---|---|---|---
6SDZ-8ADE (N) | 0.393 | 0.053 | 0.206 | 0.991
6SDZ-8ADE (C) | 0.449 | 1.028 | 1.142 | 4.428
Table 3. Indicators to quantify the difference of the structure of two layers
of amyloid fibrils (PDB entries: 6SDZ and 8ADE). The letter in parentheses
represents the terminal fragment of chain A of each amyloid fibril.

We found that the RMSDs of two fragments are both less than 1Å. It indicates
that the structure of 6SDZ and 8ADE are considered as almost the same in the
RMSD context. However, the patient carries 6SDZ and the patient carries 8ADE
have different clinical manifestations [66], which implies that there should
be differences between the structure of these amyloid fibrils. We find that
the global difference between these amyloid fibrils can be captured by the
$W_{\infty}$ distance, while the local difference between these amyloid
fibrils is scaled up as the exponent order $q$ decreases. PH considers the
higher-order relations among these $C_{\alpha}\text{s}$, which could give us
more information to distinguish the structure of amyloid fibrils.

1

input : The coordinates of atoms of two fibril structures
$\text{Prot}_{\text{1}}$ and $\text{Prot}_{\text{2}}$ , the maximal filtration
value $l$, the exponent order $q$

output : Distances $\\{d_{\infty},d_{q}\\}$ to quantify the difference between
the two fibril structures

2 for  _$i\leftarrow 1$ to $2$_ do

3 $f_{i}\leftarrow$ the filtration of $P_{i}$ based on a suitable rule, that
is, $f_{i}=\\{v_{j}\\}_{j=1}^{n}$ where $v_{n}\leq l$

4 $\\{K^{i}_{j}\\}_{j=1}^{n}\leftarrow$ a nested sequence of simplicial
complexes based on $f_{i}$

5 $\text{PD}_{i}\leftarrow$ the persistent homology analysis of
$\\{K^{i}_{j}\\}_{j=1}^{n}$

6 end for

7

8$d_{\infty}\leftarrow W_{\infty}(\text{PD}_{1},\text{PD}_{2})$

$d_{q}\leftarrow W_{q}(\text{PD}_{1},\text{PD}_{2})$

Algorithm 2 Algorithm of using TDA to compare fibril structures

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