A path analytic model of health beliefs on the behavioral adoption of breast self-examination

Soo-Foon Moey; Norfariha Che Mohamed; Bee-Chiu Lim; Soo-Foon Moey; Norfariha Che Mohamed; Bee-Chiu Lim

doi:10.3934/publichealth.2021002

AIMS Public Health

2021, Volume 8, Issue 1: 15-31. doi: 10.3934/publichealth.2021002

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Research article

A path analytic model of health beliefs on the behavioral adoption of breast self-examination

1.
Department of Diagnostic Imaging and Radiotherapy, Kulliyyah of Allied Health Sciences, International Islamic University Malaysia (IIUM), Kuantan Campus, Pahang, Malaysia
2.
Clinical Research Centre, Hospital Tengku Ampuan Afzan (HTAA), Kuantan, Pahang, Malaysia

Received: 07 November 2020 Accepted: 17 December 2020 Published: 21 December 2020

Background

In Malaysia, breast cancer accounted for 34.1% of all female cancer cases with women presenting breast cancer at late stages. Breast cancer has a higher five-year survival rate if detected early. An increase of approximately 30% in the five-year survival rate is indicated if breast cancer is detected at stage III compared to stage IV. Thus, survival rate of breast cancer can be increased by creating awareness and encouraging breast cancer screening amongst women. Breast self-examination (BSE) is highly recommended for breast cancer screening due to its simplicity with no incurred cost. The Health Belief Model is used in this study to explain and predict the adoptive behavior of BSE amongst women in Kuantan, Pahang.

Materials and methods

This study employed a multi-stage sampling method using a simple proportion formula at 5% type 1 error, p < 0.05 and absolute error at 2% which resulted in a sample of 520 participants. The data for the study was obtained using a validated bilingual self-constructed questionnaire and the model constructed using Mplus software.

Results

Perceived severity, benefits and barriers were found to significantly influence the behavioral adoption of BSE. Married women aged from 45 to 55 years and knowledge were found to significantly moderate the relationship between perceived benefits and behavioral adoption of BSE. Further, self-efficacy was found as the core construct that mediates the relationship between married women aged 45 to 55 years and the behavioral adoption of BSE.

Conclusion

Self-efficacy is found in the study to influence the behavioral adoption of BSE. This is undeniable as self-efficacy can promote confidence in initiating and maintenance of behavioral change if the perceived change is beneficial at an acceptable cost.

Keywords:

Citation: Soo-Foon Moey, Norfariha Che Mohamed, Bee-Chiu Lim. A path analytic model of health beliefs on the behavioral adoption of breast self-examination[J]. AIMS Public Health, 2021, 8(1): 15-31. doi: 10.3934/publichealth.2021002

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Abstract

Background

Materials and methods

Results

Conclusion

Abbreviations

BSE:

Breast self-examination;

HBM:

Health belief model;

EFA:

Exploratory factor analysis;

SEM:

Structural equation modeling

1. Introduction

Ligands, are biochemical modifiers of macromolecular structure and can impact biological function. These can be co-factors (iron, cobalt, copper, zinc), co-enzymes such as Nicotinamide- and Flavin- Adenine Dinucleotides (NAD, FAD) and full-length molecules with short binding sites ^[1,2]. Ligands, unlike substrates/co-substrates are either reversibly altered or not at all. The biochemical role of ligands, in vivo, is complex and can influence both, enzyme-mediated substrate catalysis and non-enzymatic association and dissociation interactions. Whilst, the interaction with competitive inhibitors, co-factors or co-enzymes involves definitive and direct modifications to the active site residues, the effect of a ligand can be allosteric and indirect ^[3,4,5]. The latter involves both long-distance conformational changes and non-covalent interactions (hydrogen, Van der Waals, hydrophobic, electrostatic) ^{[6,7,8,9,10,11]}. Empirical data suggests that the binding affinity or the strength-of-association of a macromolecule for its ligand is a critical determinant of function ^{[7,8,9,10,11]}. For example, 2, 3-Bisphophoglycerate is a potent modifier of Hemoglobin function and does so by shifting the oxygen-dissociation curve to the right. In its absence Hemoglobin retains high affinity for molecular oxygen (left-shift of the oxygen dissociation curve), an undesirable effect on its role as a transporter ^[10,11]. Similarly, Ascorbic acid maintains iron in its reduced state in the gastrointestinal tract and as part of the active site of non-haem iron (Ⅱ)- and 2-oxoglutarate-dependent dioxygenases ^[12,13]. Deficiency of Ascorbic acid is implicated in tardy iron absorption in the ileum as well as a range of collagen disorders such as scurvy (Prolyl- and Lysyl-hydroxylases) ^[14,15]. Conversely, proteins which are modified such as those that may originate from missense or amino acid substitutions and are secondary to genomic variants such as single nucleotide polymorphisms (SNPs) and insertions-deletions (indels), will also result in several clinical outcomes ^[16,17]. Here, too, enzyme catalysis is directly affected if these are present at the active site or is impacted indirectly (folding, stability, complex formation) when present elsewhere ^[16,17,18].

There is a large volume of literature which describes macromolecules in terms of either residues (amino acids, nucleotides) interacting or an all atom-based interaction matrix. The proponent of the 2D approach is the Gaussian network model (GNM), while the Anisotropic network model (ANM) is representative of the 3D approach ^{[19,20,21,22]}. The fundamental premise of both these approaches is the elastic network model (ENM) ^[19]. Here, an atom or residue is modeled as an elastic mass and the interaction between a pair of atoms/residues is dependent on the selection of a pre-determined cut-off distance ^[22]. The force constant, although, not a parameter by definition, has also been studied and shows good correlation with B-factor data ^[23]. A major application of these studies is normal mode analysis (NMA), which has been used to glean valuable insights into the structural dynamics of the investigated macromolecule and into B-factor distribution ^[23,24]. Despite this success, there are significant limitations of this approach, including inadequate descriptors for the type of interactions computed by the Hessian matrix and data points that are dependent on a preselected cut-off distance (5–10 Ang, GNM; 10–15 Ang, ANM) ^[20,22,25]. Parameter-free versions of the ENM (pfENM), GNM (pfGNM) and ANM (pfANM), to resolve the latter, have been described and compared to establish B-factor distribution (isotropic, anisotropic) ^[25]. Additionally, many of these studies have focused on inferring biophysical characteristics such as cross-correlational fluctuations and mean square displacements. From a functional standpoint, however, it is not clear whether these data can be utilized to derive/study parameters such as the Michaelis-Menten constant (Km) or the association/dissociation (Ka/Kd). Since, these depend on the presence of an organic or inorganic modifier, i.e., ligand/substrate/co-factor/co-substrate, in addition to the modeled macromolecule, its exclusion is another major lacuna of these studies.

Despite the availability of clinical, empirical, analytical and computational data, a mathematically rigorous explanation for the heterogeneity in biochemical function, for a ligand-macromolecular complex, is missing. The work presented models a ligand and macromolecule as a homo- or hetero-dimer and subsumes a finite and equal number of atoms/residues per monomer. The pairwise interactions of the resulting square matrix will be chosen randomly from a standard uniform distribution. The resulting eigenvalues will be analyzed and modeled in accordance with known literature on biochemical reactions to generate biologically viable and usable dissociation constants. The theoretical results will be complemented by numerical studies where applicable. Additionally, and through various theorems, lemmas and corollaries, a schema to partition ligands into high- and low-affinity variants will also be discussed. The suitability of the transition-state dissociation constants as a model for ligand-macromolecular interactions will be inferentially assessed by analyzing the clinical outcomes of amino acid substitutions of selected enzyme homodimers. The relevance of the model to biochemical function will be discussed by examining the ligand-macromolecular complex for known ligands of non-haem iron (Ⅱ)- and 2-oxoglutarate-dependent dioxygenases (Fe2OG) and the major histocompatibility complex (Ⅰ) (MHC1).

The general outline of the manuscript includes an initial section where the system to be modeled, rationale for this study, and formal definitions are introduced (Section 2). The model is analyzed, formulated and presented as theorems, lemmas and corollaries (Section 3). This section also includes a numerical study to demonstrate and validate the theoretical assertions made. The biological relevance of these findings are discussed with an analysis of clinical outcomes of enzyme sequence variants and case studies of enzyme- and non-enzymatic complex formation (Section 4). A brief conclusion that summarizes the presented study, limitations and future directions is included at the end of the manuscript (Section 5). Details of all proofs are included after the conclusions (Section 6).

2. Model description and definitions

Consider the generic interaction between macromolecule $\left(c\right)$ and ligand $\left(\mu \right)$ ,

$\begin{array}{cc}c\mu \begin{array}{c}\stackrel{{r}_{f}}{\to }\\ \underset{{r}_{b}}{\leftarrow }\end{array}c+\mu & RXN\left(1\right)\end{array}$

We can represent this interaction/reaction, at a steady state, with the rate equations ^[26],

${R}_{d}\left(c\mu \right) = {r}_{f}.{\left[c\mu \right]}^{{L}_{c\mu }\ge 0}$

(1)

${{R}_{a}\left(c\mu \right) = {r}_{b}.\left[c\right]}^{{L}_{c}\ge 0}{\left[\mu \right]}^{{L}_{\mu }\ge 0}$

(2)

At steady state,

${R}_{d}\left(c\mu \right) = {R}_{a}\left(c\mu \right)$

(3)

$\Rightarrow \frac{{r}_{f}}{{r}_{b}} = \frac{{\left[c\right]}^{{L}_{c}\ge 0}.{\left[\mu \right]}^{{L}_{\mu }\ge 0}}{{\left[c\mu \right]}^{{L}_{c\mu }\ge 0}}$

(4)

$\frac{{r}_{f}}{{r}_{b}} = Kd\left(c\mu \right)$

(5)

Here,

$c: = Macromolecule$

$\mu : = Ligand$

$\left[.\right]: = Molar\;concentration\;of\;reactant\;in\;standard\;form\;\left(M\right)$

${R}_{a}\left(c\mu \right): = Rate\;of\;association\;of\;complex\;\left(M{s}^{-1}\right)$

${R}_{d}\left(c\mu \right): = Rate\;of\;dissociation\;of\;complex\;\left(M{s}^{-1}\right)$

${r}_{f}: = Rate\;constants\;of\;forward\;reaction\;\left({s}^{-1}\right)$

${r}_{b}: = Rate\;constants\;of\;reverse\;reaction\left({M}^{-1}{s}^{-1}\right)$

$L: = Stoichiometry\;of\;reactant\left(s\right)$

$Kd\left(c\mu \right): = Dissociation\;constant\;for\;ligand-macromolecular\;complex\left(M\right)$

It is clear that a ligand-macromolecular complex may exist in one of three distinct states. These include: a) perfect association, b) perfect dissociation and c) an intermediate- or transition-state; and can be represented in terms of the dissociation constant,

Case (1) $Perfect\;association$ Def. (1)

$\frac{1}{{r}_{b}}\to \infty ;{r}_{f}\to 0$

(6, 7)

$\Rightarrow Kd\left(c\mu \right)\approx 0$

(8)

Case (2) $Perfect\;disassociation$ Def. (2)

${r}_{f} > > > {r}_{b}$

(9)

$\Rightarrow Kd\left(c\mu \right)\ge 1$

(10)

Case (3) $Transient-state\;disassociation\;constant$ Def. (3)

${r}_{f}\lessgtr {r}_{b}$

(11)

$\Rightarrow Kd\left(c\mu \right)\in \mathbb{R}\cap \left(\mathrm{0, 1}\right)$

(12)

Consider an atom/residue-based representation (amino acids/nucleotides) of a generic set of monomeric macromolecules, ${\boldsymbol{\mathcal{C}}} = \{protein, DNA, RNA\}$ , with $z = \mathrm{1, 2}, \dots ., Z$ components each with $i$ -indexed ( $i = \mathrm{1, 2}, \dots ., I$ ) c-atoms/residues,

$\begin{array}{cc}{{\bf c}\equiv {{{\mathcal{C}}}}_{z}\in {\boldsymbol{\mathcal{C}}}|{\bf c} = \left[{c}_{1}{c}_{2}\dots .{c}_{i = I}\right]}^{T}, I\in \mathbb{N}& \end{array}$

(13)

The analogous model of a monomer ligand, ${\boldsymbol{\mathcal{L}}} = \{small\;molecule, peptide, oligonucleotide\}$ , with $j$ -indexed $(j = \mathrm{1, 2}, \dots , J)$ $\mu$ -atoms/residues is,

$\boldsymbol{\mu }\in {\boldsymbol{\mathcal{L}}}|{\bf{ \pmb{\mathsf{ μ}} }} = \left[{\mu }_{1}{\mu }_{2}\dots .{\mu }_{j = J}\right], J\in \mathbb{N}$

(14)

It is also assumed that the ligand-macromolecular complex is a homo- or hetero-dimer with an equal number of atoms/residues $\left(I = J\right)$ per monomer. The interaction matrix is,

$\begin{array}{rr}\left\langle{\boldsymbol{c}|\boldsymbol{\mu }}\right\rangle = {\left[{c}_{1}{c}_{2}..{c}_{i = I}\right]}^{T}\times \left[{\mu }_{1}{\mu }_{2}..{\mu }_{j = J}\right] = {{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }} = \left({{c}_{i}\mu }_{j}\right)\subset {\mathbb{R}}^{I\times J}& \end{array}$

(15)

The numerical values of this matrix are chosen randomly from the standard uniform distribution,

$\begin{array}{rr}{{\boldsymbol{\mathcal{C}}}}_{z\mu } = \left({{c}_{i}\mu }_{j}\right)\in {\mathcal{U}}_{\left[\mathrm{0, 1}\right]}& \end{array}$

(16)

The rationale for this choice is that each pairwise interaction is subsumed to be a function of an arbitrary number of non-bonded interactions (long- and short-range) and is therefore, unique. Clearly, this implies the existence of $\left\{I, J\right\}$ -linear independent vectors,

$\begin{array}{cc}rank\left({{\boldsymbol{\mathcal{C}}}}_{z\mu }\right) = \left\{I, J\right\}& \end{array}$

(17)

Since ${{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}$ is diagonalizable there exists a diagonal matrix, $\mathit{\boldsymbol{\mathcal{K}}{\boldsymbol{\mathcal{C}}}}_{z\mu }$ ,

$\mathit{\boldsymbol{\mathcal{K}}{\boldsymbol{\mathcal{C}}}}_{z\mu } = {\boldsymbol{X}}^{-1}{{\boldsymbol{\mathcal{C}}}}_{z\mu }\boldsymbol{X}$

(18)

${z}_{i = j} = z\in diag\left(\mathit{\boldsymbol{\mathcal{K}}{\boldsymbol{\mathcal{C}}}}_{z\mu }\right)\subset \mathbb{C}$

(19)

${{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}$ , is non-symmetric the computed eigenvalues of the modeled ligand-macromolecular interaction matrix can have positive and negative real parts,

$\left\{{\alpha }_{ij} = Re\left(z\right)\in \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}|i = j, z\in \mathbb{C}\right\}$

(20)

$\left\{{\alpha }_{ij} = Re\left(z\right)\in \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}|i = j, z\in \mathbb{C}\right\}$

(20.1)

A further subdivision can be made in accordance with established literature,

$\left\{{\alpha }_{ij} = Re\left(z\right)\in \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}|i = j, z\in \mathbb{C}\right\} {\omega }_{\infty } = Kd\left(c\mu \right)\ge 1$

(21)

$Perfect\;association\equiv Reverse \;\; {\omega }_{0} = Kd\left(c\mu \right) = 0$

(22)

This selection generates the set,

$\omega = {\alpha }_{ij}\in \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}\cap \left(\mathrm{0, 1}\right)$

(23)

$\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Def.\; {\mathit{(4)}}$

$\#\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right) = A$

(23.1)

We can combine these to get a preliminary definition of the transition-state disassociation constants. These are the strictly positive real part of all complex eigenvalues that characterize a ligand-macromolecular complex with an equal number of atoms/residues per monomer and belong to the open interval (0, 1),

$\left\{\omega = \mathcal{R}\mathrm{e}\left(\mathrm{z}\right)\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\subset \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}\cap \left(\mathrm{0, 1}\right)|z\in \mathbb{C}\right\}$

(

$Def. \; {\mathit{(5a)}}$ )

3. Results

3.1. Mathematical analysis of transition-state dissociation constants of the modeled ligand-macromolecular complex

Whilst, the states of perfect association and dissociation are key determinants of whether a reaction will occur or not, the transition-state dissociation constants may offer insights into the origins of threshold values, feedback mechanisms and other regulatory checkpoints. However, in order to ascribe biological relevance to these findings we must establish various bounds which can then be utilized to assess and thence assay the function of a ligand-macromolecular complex.

Theorem 1 (T1): The linear map between the transition-state dissociation constants and the eigenvalues that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecular complex is the injection,

$g:\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(24)

Theorem 2 (T2): The transition-state dissociation constants that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecular complex is a monotonic and non-increasing sequence,

$\begin{array}{cc}{\left\{{\omega }_{a}\right\}}_{a\le a+1}|& \omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\end{array}$

(25)

Theorem 3 (T3): The transition-state dissociation constants that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecule complex are monotonic, bounded and therefore, convergent,

$\begin{array}{cc}\underset{a\to \infty }{\mathrm{lim}}{\left\{{\omega }_{a}\right\}}_{a\le a+1} = \left\{\mathrm{0, 1}\right\}|& \omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\end{array}$

(26)

Corollary 1 (C1): The transition-state dissociation constants that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecule complex is a sequence with defined greatest-lower and least-upper -bounds,

$inf{\left\{{\omega }_{a}\right\}}_{a\le a+1} < {\left\{{\omega }_{a}\right\}}_{a\le a+1} < sup{\left\{{\omega }_{a}\right\}}_{a\le a+1}$

(27)

Corollary 2 (C2; without proof): The cardinality of the set of transition-state dissociation constants that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecule complex is finite,

$\begin{array}{c}\#\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right) = A < \#\boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right) = \left\{I, J\right\}\end{array}$

(28)

Using T1–T3 and C1, C2 we can refine our definition of the transition-state dissociation constants for the modeled ligand-macromolecular complex,

$\begin{array}{cc}{\left\{{\omega }_{a}\right\}}_{a\le a+1}|& \omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\end{array};a = \mathrm{1, 2}\dots A$

(

$Def. \; {\mathit{(5b)}}$ )

where,

$\omega = \mathcal{R}\mathrm{e}\left(\mathrm{z}\right)\in \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}\cap \left(\mathrm{0, 1}\right)|z\in \mathbb{C}$

3.2. Annotation of a ligand on the basis of the transition-state dissociation constants of the modeled ligand-macromolecular complex

It is clear from the above results that the eigenvalue-based transition-state dissociation constants are continuous and can potentially model the multiplicity of intermediate- or transient-states that a ligand-macromolecular complex may adopt. It should therefore, be possible to partition the transition-state dissociation constants into functionally distinct subsets and will be characteristic for a specific ligand-macromolecular complex.

Theorem 4 (T4): The transition-state dissociation constants that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecule complex is a proper subset of the complete set of the real part of all complex eigenvalues that comprise the interaction matrix,

$\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\subset \boldsymbol{K}\boldsymbol{d}\left({{\boldsymbol{\mathcal{C}}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(29)

Corollary 3 (C3): The distribution of the transition-state dissociation constants that characterize the interactions of the homo- or hetero-dimer form of the modeled ligand-macromolecule complex will result in a schema by which we can annotate the ligand as a high $\left({\mu }_{high}\right)$ - or low $\left({\mu }_{low}\right)$ -affinity variant,

$\mu = \left\{{\mu }_{high}, {\mu }_{low}\right\}$

(30)

3.3. Modeling the transition-state dissociation constants of higher-order ligand-macromolecular complexes

Biologically relevant macromolecular complexes are characterized by heterogeneity and high-order $\left(Z\ge 2\right)$ . This multimer-form of a macromolecule is formed around a primary molecule and its interactions. These may be protein-protein, DNA/RNA-protein or DNA-RNA-protein.

The multimer-form of ligand-macromolecular complex is easily modeled using the mathematical framework defined earlier. Here, the ligand-macromolecular complex is considered as a set of interacting monomers and the binding to a ligand occurs via a single unique monomer,

$\left\{{{{\mathcal{C}}}}_{z}\in {\boldsymbol{\mathcal{C}}}|{{{\mathcal{C}}}}_{1} = {{{\mathcal{C}}}}_{2}\dots = {{{\mathcal{C}}}}_{z = Z}|Z\ge 2\right\}$

(

$Def. \; {\mathit{(6a)}}$ )

where,

${{{\mathcal{C}}}}_{z}\equiv {{{\mathcal{C}}}}_{z\mu }\equiv \mu {{{\mathcal{C}}}}_{z}$

(

$Def. \; {\mathit{(6b)}}$ )

Here,

$\mu : = Ligand$

${{{\mathcal{C}}}}_{z}: = Unique\;monomer\;of\;a\;macromolecule\;that\;associates\;with\;a\;ligand$

$\mu {{{\mathcal{C}}}}_{z}: = Ligand-macromolecular\;complex$

On the basis of these definitions we can re-index the remaining monomers, i.e., after excluding the unique monomer that binds to the ligand,

$\left\{{{{\mathcal{C}}}}_{y}\in \widetilde {{\boldsymbol{\mathcal{C}}}} = {\boldsymbol{\mathcal{C}}}\backslash \left\{{{{\mathcal{C}}}}_{z}\right\}|y = \mathrm{1, 2}\dots Y;Y = Z-1\right\}$

(

$Def. \; {\mathit{(7)}}$ )

We now derive an expression for the multimer (higher-order)-form of a ligand-macromolecular complex.

Theorem 5 (T5): The multimer-form of a ligand-macromolecular complex comprising identical monomer units and with an arbitrary unit associating with a ligand is,

$\begin{array}{rr}{\prod }_{y = 1}^{y = Y}{{{\mathcal{C}}}}_{z}.{\mathcal{C}}_{y} = {\mathcal{C}}_{z}.{\mathcal{C}}_{Y}^{y}|y = \mathrm{1, 2}\dots Y;Y = Z-1;Z\ge 2;z = \mathrm{1, 2}\dots Z& \end{array}$

(31)

Rewriting, this result in terms of the definition of the multimer form of a ligand-macromolecular complex,

$\begin{array}{cc}{\mathcal{C}}_{z}.{\mathcal{C}}^{Y}\equiv \mu {\mathcal{C}}_{z}.{\mathcal{C}}^{Y}& \end{array}$

(

$Def. \; {\mathit{(8)}}$ )

Theorem 6 (T6): The linear map between the transition-state dissociation constants that characterize the interactions of the monomer- and multimer-forms of a ligand-macromolecular complex is a bijection,

$\begin{array}{cc}{h}^{-1}\circ h:\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\leftrightarrow u\in \boldsymbol{K}\boldsymbol{d}\left(\mu \boldsymbol{\mathcal{C}}_{z}\right)& \end{array}$

(32)

Theorem 7 (T7): The linear map between the transition-state dissociation constants and the eigenvalues that characterize the multimer-form of a ligand-macromolecular complex is a composition and an injection,

$\begin{array}{cc}g\circ {h}^{-1}\circ h:u\in \boldsymbol{K}\boldsymbol{d}\left(\mu \boldsymbol{\mathcal{C}}_{z}.{\mathcal{C}}^{Y}\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mathcal{C}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)& \end{array}$

(33)

3.4. Numerical studies to establish biological relevance of the transition-state dissociation constants

The aforementioned theoretical results establish the mathematical rigor behind the definition and development of the transition-state dissociation constants as a model for ligand-macromolecular interactions (T1–T7, C1–C3). These assertions are complemented and numerically validated in R-4.1.2. Here, the R-packages, "ConvergenceConcepts" and "pracma" are utilized to investigate and analyze the stochastic convergence of the eigenvalues generated by the interaction matrix of a ligand-macromolecular complex (Supplementary Text 1) ^[27]. The R-scripts to establish convergence along with data processing are developed in-house (Supplementary Text 2). The stepwise algorithm to compute and numerically validate the transition-state dissociation constants is presented (Figure 1).

Figure 1. Numerical studies of ligand-macromolecular interactions as eigenvalue-based transition-state dissociation constants.

DownLoad: Full-Size Img PowerPoint

Step 1: A ligand and macromolecule with an equal number of atoms/residues ( $n = 25$ ) is chosen. Whilst, the complex can be modeled as a perfect homodimer, imperfect forms such as alternatively spliced isoenzymes are prevalent and commonly observed. Alternatively, the ligand can be modeled as a different macromolecule altogether.

Step 2: Populate the square interaction matrix with values randomly chosen from a uniform distribution, ${\mathcal{U}}_{\left[\mathrm{0, 1}\right]}$ . These will represent one of three potential states for each interacting pair of atoms/residues of the modeled ligand-macromolecular complex (association, complete disassembly, transition-state).

Step 3: Compute the complex eigenvalues of this matrix and extract the real part of each.

Step 4: Form a sequence of the subset comprising those values that are strictly positive and belong to the open interval $\left(\mathrm{0, 1}\right)$ .

Step 5: Establish the stochastic convergence in distribution and/or probability of the terms of this sequence to the expected upper (tsup)- and lower (tinf)-bounds, i.e., 0 and 1.

Step 5.1: Construct a sequence of random numbers, X, whose elements are uniquely mapped to the eigenvalue-based transition-state dissociation constants and represent intermediate- or transition-states of the modeled ligand-macromolecular complex,

$\left\{{X}_{a}\in \boldsymbol{X}\cap \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\subset \mathbb{R}\cap \left(\mathrm{0, 1}\right)|a = \mathrm{1, 2}\dots A\right\}$

(

$Def. \; {\mathit{(9)}}$ )

Here,

$A = \#\left(\boldsymbol{X}\cap \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right)$

(34)

Step 5.2: Establish convergence of this set of random numbers. Here, weak convergence will suffice (distribution, probability),

$\underset{a\to \infty }{\mathrm{lim}}\left({X}_{a}\right)\to \left\{tinf, tsup\right\} = \left\{\mathrm{0, 1}\right\}$

(35)

The parameters to accomplish this numerically are,

$\begin{array}{r}nmax: = Number\;of\;values\;to\;analyse\\ M: = Number\;of\;paths\\ \varepsilon : = Threshold\;value\\ tinf: = Lower\;limit\;of\;interval\;to\;establish\;convergence\\ tsup: = Upper\;limit\;of\;interval\;to\;establish\;convergence\end{array}$

The values of these parameters for the numerically studied example are,

$nmax = A = 11$

(35.1)

$M = 500$

(35.2)

$\varepsilon = 0.01$

(35.3)

$tinf = 0$

(35.4)

$tsup = 1$

(35.5)

4. Discussion

The eigenvalue-based model of transition-state dissociation constants of a ligand-macromolecular complex asserts that there are several intermediate- or transition-states of a complex and that each of these has the potential to modify the biochemical process that the complex participates in.

4.1. Transition-state dissociation constants as a model of biochemical function for ligand-macromolecular complexes

Ligand-macromolecular complexes, in vivo, possess a finite and in most cases, an incomparable number of atoms/residues. The theoretical results establish definition(s), bounds and metrics to assess biochemical function for both, monomer (T1–T4, C1–C3)- and multimer (T5–T7)-forms. The numerical data suggests that the set of transition-state dissociation constants can be finite, converge and retain statistical relevance (Figure 1).

Proposition (P): The transition-state dissociation constants for the monomer ( $z = Z = 1$ )- and multimer ( $Z\ge 2$ )-forms of a ligand-macromolecular complex with a finite number of atoms/residues of each $\left(I, J\right)$ per monomer,

$\left\{u\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }\boldsymbol{\mathcal{C}}_{\boldsymbol{z}}.\boldsymbol{\mathcal{C}}^{\boldsymbol{Y}}\right)|{\mathcal{C}}_{z}\in \boldsymbol{\mathcal{C}}, \mu \in {\boldsymbol{\mathcal{L}}};z = \mathrm{1, 2}\dots Z;A = \left\{I, J\right\}\right\}$

(

$Def. \; {\mathit{(10)}}$ )

is the finite set,

$\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }\boldsymbol{\mathcal{C}}_{\boldsymbol{z}}.\boldsymbol{\mathcal{C}}^{\boldsymbol{Y}}\right)\subset \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mathcal{C}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

Here,

$I = \#{\mathcal{C}}_{z}|{\mathcal{C}}_{z}\in \boldsymbol{\mathcal{C}}$

(36)

$J = \#\mu |\mu \in {\boldsymbol{\mathcal{L}}}$

(37)

where,

$\boldsymbol{K}\boldsymbol{d}\left(.\right): = Set\;of\;constrained\;eigenvalue-based\;transition-state\;dissociation\;constants$

$I: = Finite\;number\;of\;atoms\;or\;residues\;of\;macromolecule$

$J: = Finite\;number\;of\;atoms\;or\;residues\;of\;ligand$

${\mathcal{C}}_{\mu z}.{\mathcal{C}}^{Y}: = Multimer\;form\;of\;ligand-macromolecular\;complex$

Enzyme-mediated catalysis, or lack thereof, results in metabolic enzyme disorders and may be inherited (inborn errors of metabolism) or acquired ^[28]. In order to assess the biomedical relevance of modeling ligand-macromolecule interactions as transition-state dissociation constants, the clinical outcomes of amino acid substitutions of selected enzyme homo- or hetero-dimers are examined (). These outcomes, i.e., benign, likely benign, pathologic, likely pathologic, conflicting, uncertain significance, are defined in accordance with the prevalent nomenclature of the ClinVar database ^[16]. Here, the data annotated as "uncertain significance" are those sequence variants with a high likelihood ( $\approx 90–95\%$ ) of being "benign" or "pathogenic" ^[17]. This means they are likely to classified as "true positive", and if ignored will result in a "false negative". On the other hand, an outcome designated as with a "conflicting interpretation" is likely to be due to unresolved contradictory findings in the presence or absence of confounding factors. If we assume perfect contradiction, i.e., 50%, and couple this with the previous result, we get a $\approx 70–73\%$ possibility that the variant of interest is a "true positive". This means, that here too, if missed a "false negative" will result. The metric of choice is the Recall (R) percentage,

$R = \frac{TP}{TP+FN}\times 100$

(38)

$R: = Recall$

$\begin{array}{l} TP: = Known\;positives(benign, likelybenign,\\ pathogenic, likely\;pathogenic) \end{array}$

(

$Def. \; {\mathit{(12)}}$ )

$\begin{array}{l} FN: = Likely\;positives(conflicting\\ \;data, uncertain\;significance) \end{array}$

(

$Def. \; {\mathit{(13)}}$ )

Table 1. Analysis of clinical outcomes of amino acid substitutions of selected enzyme homo- and hetero-dimer forms present in ClinVar.

	Enzyme	EC	CV	SNP	M	Co	US	B	LB	P	LP	FN	TP	R (%)
1	Glucokinase	2.7.1.2	778	619	370	49	157	4	4	65	113	206	186	47.45
2	Pyruvate kinase	2.7.1.40	1239	629	209	14	135	7	7	30	20	149	64	30.05
3	Cathepsin A	3.4.16.x	1776	1270	501	20	382	26	23	26	34	402	109	21.33
4	Pyruvate dehydrogenase	1.2.4.1	2317	1591	584	25	391	35	46	62	43	416	186	30.9
5	Phosphofructokinase 1	2.7.1.11	165	32	10	2	4	2	1	1	1	6	5	45.45
6	Phosphofructokinase 2	2.7.1.105	206	76	23	1	17	1	1	2	1	18	5	21.74
7	Cystathione beta-synthase	4.2.1.22	930	744	255	21	172	3	4	46	32	193	85	30.58
8	DNA topoisomerase Ⅱ	5.6.2.2	466	319	152	0	122	11	14	3	1	122	29	19.21
9	Guanylate cyclase 1	4.6.1.2	1572	1117	576	29	417	11	10	34	59	446	114	20.36
10	Phenylalanine hydroxylase	1.14.16.1	1314	1102	631	5	182	0	3	161	254	187	418	69.09
Note: EC: Enzyme commission number; CV: Number of clinical variants; SNP: Single nucleotide polymorphisms; M: Missense mutations; Co: Conflicting data; US: Uncertain significance; B: Benign; LB: Likely; P: Pathogenic; LP: Likely pathogenic; FN: False negative (Co + US); TP: True positive (B + LB + P + LP); R: Recall ( $\frac{TP}{TP+FN}\times 100$ ).

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It is clear from these data that amino acid substitutions (nature, type), either alone or in combination, comprise distinct transition-states and are significant contributors to the biochemical function of each enzyme dimer ( $Recall\approx 19–70\%$ ). Since each of these states will result in a distinct $Kd$ , it is easily inferred that the transition-state dissociation constants, for a complex, may be more representative of biochemical function (T1–T4, C1–C3).

4.2. Ligand-macromolecule biochemistry as a function of its transition-state dissociation constants

The discussion, vide supra, presents and highlights the biomedical relevance of modeling ligand-macromolecular interactions as transition-state dissociation constants. The results for monomer- and multimer-forms of ligand-macromolecular complexes are mathematically rigorous and have been validated, in silico. The results are now examined in context of biochemical function (enzyme, non-enzyme) for selected cases.

Case 1: Oxygen sensitive variants of non-haem iron (II)- and 2-oxoglutarate-dependent dioxygenases

The non-haem iron (Ⅱ)- and 2-oxoglutarate-dependent dioxygenases $\left(EC1.14.x.y\right)$ , comprise a large superfamily of enzymes, are present in all kingdoms of life and is chemically diverse (variable reaction chemistry, multiple substrates) ^[12,13,29]. Clinically relevant members include Phytanoyl-CoA dioxygenase (PHYT), Lysine Hydroxylases, and the Proline 4-Hydroxylases (P4H) amongst several others ^{[12,13,14,15,29]}. These enzymes have important roles in Phytanic acid metabolism and collagen maturation, with sub-optimal activities contributing to diseases such as Refsum and the Ehlers-Danlos (ED)-syndrome ^[14,15,30]. Here, too, the outcomes (clinical, non-clinical) associated with substitution mutations for PHYT suggest that modeling ligand-macromolecular interactions as transition-state dissociation constants may be a better index of biochemical function (T1–T4, C1–C3, P) ^[31].

P4Hs, are classified as being either hypoxia-sensitive $\left(H-P4H\equiv HP4H;EC\mathrm{1.14.11.29}\right)$ or collagen transforming $\left(C-P4H\equiv CP4H;EC\mathrm{1.14.11.2}\right)$ ^[28]. These reactions may be written,

$\begin{array}{rr}HP4H-HIF+{O}_{2}+2OG+Proline\begin{array}{c}\stackrel{{r}_{f}}{\to }\\ \underset{{r}_{b}}{\leftarrow }\end{array}Hydroxyproline+SA+{CO}_{2}& RXN\left(2\right)\\ CP4H+{O}_{2}+2OG+Proline\begin{array}{c}\stackrel{{r}_{f}}{\to }\\ \underset{{r}_{b}}{\leftarrow }\end{array}Hydroxyproline+SA+{CO}_{2}& RXN\left(3\right)\end{array}$

$\begin{array}{rrr}{r}_{f}, {r}_{b}& : = & Rate\;constants\;for\;forward\;and\;backward\;reactions\;at\;steady\;state\\ HP4H& : = & Hypoxia\;inducible\;factor-dependent\;Proline\;4-Hydroxylase\\ HIF\equiv {\mu }_{high}& : = & Hypoxia-inducible\;factor(high-affinity\;modifier)\\ 2OG& : = & 2-oxoglutarate\\ SA& : = & Succinic\;acid\\ C{O}_{2}& : = & Carbon\;dioxide\end{array}$

The amino acid identity between $HP4H$ and $CP4H$ notwithstanding, there are significant differences between the molecular biology that they exhibit. This implies that despite the similarity of co-factor $\left(iron\left(II\right)\right)$ , substrate $\left(L-Proline\right)$ and co-substrate $\left(2-oxoglutarate\right)$ , the binding affinities for molecular dioxygen vary considerably ^[7,32],

${Km}_{HP4H} = 0.1-0.76\;mM$

(39)

${Km}_{CP4H} = 0.03-1.5\;mM$

(40)

The turnover numbers for the cognate substrate, too, differ significantly ^[7],

${Kcat}_{HP4H} = 0.015-0.733{s}^{-1}$

(41)

${Kcat}_{CP4H} = 0.0188-0.02{s}^{-1}$

(42)

Clearly, a plausible explanation for these disparate empirical observations is the binding of the hypoxia-inducible factors (HIF) to P4H. The hypoxia-inducible factors (HIFs), are a family $(n = 3)$ of transcription factors which sense hypoxia and trigger the upregulation of hypoxia-dependent genes ^[32,33,34]. Here, although hypoxia-inducible factor, is a full length protein, the actual binding site is the C-terminal end of HP4H ^[7,35].

$Kd\approx 0.000016-0.023\;mM$

(43)

Some of these observations may be inferred from the partitioning of the transition-state dissociation constants into distinct subsets (Table 2):

Table 2. Ligand-macromolecular biochemistry as a function of its transition-state dissociation constants.

	Case 1	Case 2
Ligand $\left(\boldsymbol{\mu }\in {\boldsymbol{\mathcal{L}}}\right)$	Hypoxia-inducible factor	Peptide
Macromolecule $\left(\boldsymbol{c}\equiv {\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}\in \boldsymbol{\mathcal{C}}\right)$	HP4H, CP4H	$M1\beta$
Primary complex $\left\langle {} \right.\boldsymbol{c}\|\boldsymbol{\mu }\left. {} \right\rangle$ High-affinity variant $:=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}\right)$ Low-affinity variant $:=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}\right)$	$\left\langle{\boldsymbol{H}\boldsymbol{P}{\bf{4}}\boldsymbol{H}\|\boldsymbol{\mu }}\right\rangle$ , $\left\langle{\boldsymbol{C}\boldsymbol{P}{\bf{4}}\boldsymbol{H}\|\boldsymbol{\mu }}\right\rangle$ $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{H}\boldsymbol{P}{\bf{4}}\boldsymbol{H}\|{\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}\right)=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}\boldsymbol{H}\boldsymbol{P}{\bf{4}}\boldsymbol{H}\right)$ $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{C}\boldsymbol{P}{\bf{4}}\boldsymbol{H}\|{\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}\right)=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}\boldsymbol{C}\boldsymbol{P}{\bf{4}}\boldsymbol{H}\right)$	$\left\langle{\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\|\boldsymbol{\mu }}\right\rangle$ $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\|{\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}\right)=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\right)$ $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\|{\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}\right)=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\right)$
Higher-order complex $\left\langle {} \right.\boldsymbol{c}\|\boldsymbol{\mu }\left. {} \right\rangle .{{\mathcal{C}}}^{y}$ High-affinity variant $:=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{{\mathcal{C}}}^{y}\right)$ Low-affinity variant $:=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{{\mathcal{C}}}^{y}\right)$	--- --- ---	$\left\langle{\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\|\boldsymbol{\mu }}\right\rangle.PLC$ $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\|{\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}.\boldsymbol{P}\boldsymbol{L}\boldsymbol{C}\right)=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }.\boldsymbol{P}\boldsymbol{L}\boldsymbol{C}\right)$ $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }\|{\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}.\boldsymbol{P}\boldsymbol{L}\boldsymbol{C}\right)=\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }.\boldsymbol{P}\boldsymbol{L}\boldsymbol{C}\right)$
Functional relevance	${R}_{HP4H}\left(t\right)\gg {R}_{CP4H}\left(t\right)$	$M1\beta \|{\mu }_{high}.PLC\to \alpha M1\beta$ (Anterograde) ${M1\beta \|\mu }_{low}.PLC\to rM1\beta$ (Retrograde)
Note: ${\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}$ : All pairwise-atom/residue based square interaction matrix of ligand and macromolecule, $\left\langle {} \right.\boldsymbol{c}\|\boldsymbol{\mu }\left. {} \right\rangle$ ; $\mathit{\boldsymbol{\mathcal{K}}\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}$ : Diagonal matrix of the interactions of a ligand and macromolecule; ${z}_{i=j}=z$ : Set of eigenvalues of $\mathit{\boldsymbol{\mathcal{K}}\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}$ where $z\in diag\left(\mathit{\boldsymbol{\mathcal{K}}\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{C}$ ; $\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$ : Set of eigenvalue-based transition-state dissociation constants for monomer- and multimer-forms of ligand-macromolecular complexes, $\left\{\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)={\alpha }_{i=j}=Re\left(z\right)\in \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\cap \left(\mathrm{0, 1}\right)\right\}$ ; ${\mu }_{high}, {\mu }_{low}$ : High- and low-affinity variants of an arbitrary ligand, $\mu =\{{\mu }_{high}, {\mu }_{low}\}\in {\boldsymbol{\mathcal{L}}}$ ; $\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{{\mathcal{C}}}^{y}$ : Higher-order complex of ligand and macromolecule; $\boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}$ : Subset of transition-state dissociation constants of ligand-macromolecular interaction; $\boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}$ :Subset of transition-state dissociation constants of ligand-macromolecular interaction; $R\left(t\right)$ : Rate of reaction; HP4H: Hypoxia stimulated- and Collagen Proline 4-Hydroxylase; CP4H: $M1\beta$ : Heterodimer of major histocompatibility complex I (MHC1) with beta-2 microglobulin; PLC: Peptide loading complex.

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In particular, binding of HIF restricts the range of the binding affinity of HP4H for molecular oxygen significantly,

$\frac{\Delta {Km}_{HP4H}}{\Delta {Km}_{CP4H}}\times 100\approx 45\%$

(44)

Here, the set of conformers of HP4H once bound to HIF ensures that the catalytic activity of HP4H for HIF is significantly reduced in the presence of hypoxia. This will extend the half-life of HIF and facilitate transcription of HIF-responsive genes ^[36,37]. In contrast, CP4H exhibits no such differential activity. Furthermore, there is a significant variation in the catalytic activity (turnover number) of these enzymes for their cognate substrate (L-Proline),

$\frac{max\left({Kcat}_{HP4H}\right)-min\left({Kcat}_{HP4H}\right)}{max\left({Kcat}_{CP4H}\right)-min\left({Kcat}_{CP4H}\right)} = \frac{\Delta {Kcat}_{HP4H}}{\Delta {Kcat}_{CP4H}}\approx 600$

(45)

These data suggest that a ligand when bound to a macromolecule can affect the rate at which the resulting complex assembles or disassembles and thereby influence biochemical function. Hence, partitioning the transition-state dissociation constants of the ligand-macromolecular complex into distinct subsets may offer valuable insights into the in vivo function of enzymes in physiological and pathological states (T4, C2, C3).

Case 2: Generic model of MHC1-mediated high-affinity peptide export

The peptide loading complex (PLC) is a higher-order $(Z > 2;Tapasin, ERp57, MHC1)$ complex that assembles at the endoplasmic reticulum (ER)-membrane and functions to transport cytosolic peptides into the ER-lumen en route to the plasma membrane ^[38,39]. Whilst, regulation of this process, by Tapasin is well studied, the role of peptides and the possible mechanism(s) of action is unclear ^{[40,41,42,43]}. A low-affinity peptide-driven (LAPD)-model of the MHC1-mediated export of high-affinity peptides to the plasma membrane of nucleated cells has been proposed and investigated in silico ^[44]. A major proponent of this study was simulating the differential disassembly of PLC in response to peptides with varying affinities (high, low) for the MHC1- ${\beta }_{2}$ -microglobulin ^[44]. In fact, data from the simulations suggested that low-affinity peptides may not only actively participate in the transport of high-affinity peptide export, but could also regulate the same ^[44]. Another interesting observation discussed was the role of low-affinity peptides in priming the MHC1-export apparatus, such that irrespective of the nature of the cellular insult (acute, chronic), export of high-affinity peptides was rapid, continuous and efficient ^[44].

Utilizing the partition schema for the transition-state dissociation constants from the current analysis (Table 2), we can model and rewrite the differential disassembly of the PLC,

$\begin{array}{rr}\left\langle{\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }|{\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}}\right\rangle.PLC\begin{array}{c}\stackrel{{r}_{f}}{\to }\\ \underset{{r}_{b}}{\leftarrow }\end{array}Tapasin-ERp57+aM1\beta & RXN\left(4\right)\\ \left\langle{\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }|{\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}}\right\rangle.PLC\begin{array}{c}\stackrel{{r}_{f}}{\to }\\ \underset{{r}_{b}}{\leftarrow }\end{array}Tapasin-ERp57-{\mu }_{low}+rM1\beta & RXN\left(5\right)\end{array}$

$\begin{array}{l}{r}_{f}, {r}_{b} : = Rate\;constants\;for\;forward\;and\;backward\;reactions\;at\;steady\;state\\ M1\beta : = Heterodimer\;of\;MHC1\;with\;beta-2\;microglobulin\\ {\mu }_{low} : = Low-affinity\;peptide\\ {\mu }_{high} : = High-affinity\;peptide\\ \left\langle{\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }|{\boldsymbol{\mu }}_{\boldsymbol{l}\boldsymbol{o}\boldsymbol{w}}}\right\rangle : = Set\;of\;peptides\;with\;low\;affinity\;for\;MHC1\;and\;in\;complex\;with\;MHC1\\ \left\langle{\boldsymbol{M}{\bf{1}}\boldsymbol{\beta }|{\boldsymbol{\mu }}_{\boldsymbol{h}\boldsymbol{i}\boldsymbol{g}\boldsymbol{h}}}\right\rangle : = Set\;of\;peptides\;with\;high\;affinity\;for\;MHC1\;and\;incomplex\;with\;MHC1\\ eM1\beta : = Net\;exportable\;complex\;of\;high-affinity\;peptide\;with\;MHC1\\ aM1\beta : = Anterograde-derived\;exportable\;complex\;of\;high-affinity\;peptide\;with\;MHC1\\ rM1\beta : = Retrograde-derived\;exportable\;complex\;of\;high-affinityp\;eptide\;with\;MHC1\\ PLC : = Peptide\;loading\;complex\\ ERp57 : = Endoplasmic\;reticulum\;protein\;disulfide\;isomerase\end{array}$

The appropriate dissociation constants are,

(46)

(47)

Rewriting these equations in terms of the peptide-bound MHC1,

${Kd}_{RXN4} = \frac{\zeta }{{\left[M1\beta |{\mu }_{high}\right]}^{{L}_{M1\beta |{\mu }_{high}}\ge 0}}$

(48)

where,

$\zeta = \frac{\left[Tapasin-ERp57\right]\left[aM1\beta \right]}{{\left[PLC\right]}^{{L}_{PLC}\ge 0}}$

(48.1)

${Kd}_{RXN5} = \frac{\zeta }{{\left[M1\beta |{\mu }_{low}\right]}^{{L}_{M1\beta |{\mu }_{low}}\ge 0}}$

(49)

where,

$\zeta = \frac{\left[Tapasin-ERp57-{\mu }_{low}\right]\left[rM1\beta \right]}{{\left[PLC\right]}^{{L}_{PLC}\ge 0}}$

(49.1)

Clearly,

${Kd}_{RXN4}\propto \frac{1}{{\left[M1\beta |{\mu }_{high}\right]}^{{L}_{M1\beta |{\mu }_{high}}\ge 0}}$

(50)

${Kd}_{RXN5}\propto \frac{1}{{\left[M1\beta |{\mu }_{low}\right]}^{{L}_{M1\beta |{\mu }_{low}}\ge 0}}$

(51)

These results suggest that,

${Kd}_{RXN4}\simeq 1.0\left(Perfect\;disassociation\right)and\left[aM1\beta \right]\to \infty$

(52)

${Kd}_{RXN5}\simeq 0.0\left(Perfect\;association\right)and\left[rM1\beta \right]\to 0$

(53)

and is in accordance with existing empirical and simulation data,

$\left[{\mu }_{high}\right] < < < \left[{\mu }_{low}\right], \left[aM1\beta \right] > > > \left[rM1\beta \right]$

(54)

Here, the partitioning of transition-state dissociation constants into low- and high-affinity peptides for the MHC1 can provide valuable insights into the underlying molecular biology of MHC1-mediated high-affinity peptide transport under physiological and pathological conditions (T5–T7, P) ^[42,43,44].

5. Conclusions

The work presented models ligand-macromolecular interactions as eigenvalue-based transition-state disassociation constants. The interaction matrix is an all-atom/residue pairwise comparison between the ligand and macromolecule and comprises numerical values drawn randomly from a standard uniform distribution. The transition-state dissociation constants are the strictly positive real part of all complex eigenvalues of this ligand-macromolecular interaction matrix, belong to the open interval (0, 1) and form a sequence whose terms are finite, monotonic, non-increasing and convergent. The findings are rigorous, numerically robust and can be extended to higher-order complexes. The study, additionally, suggests a schema by which a ligand may be partitioned into high- and low-affinity variants. This study, although theoretical offers a plausible explanation into the underlying biochemistry (enzyme-mediated substrate catalysis, assembly/disassembly and inhibitor kinetics) of ligand-macromolecular complexes. Future investigations may include assigning weights to each interaction, investigating origins of co-operativity in enzyme catalysis and inhibitory kinetics amongst others.

6. Proofs

This section provides formal proofs for the included theorems, corollaries and proposition.

Proof (T1):

From Defs. (4) and (5),

${\text{For every}}\; \omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\exists g\left(\omega \right)\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mathcal{C}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)|{g}^{-1}\circ g\left(\omega \right) = \omega$

(55)

Let,

${\omega }_{x}, {\omega }_{y}\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)|g\left({\omega }_{x}\right), g\left({\omega }_{y}\right)\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mathcal{C}}_{\boldsymbol{z}\boldsymbol{\mu }}\right);x\ne y;\left\{x, y\right\}\le A$

If,

$g\left({\omega }_{x}\right) = g\left({\omega }_{y}\right)$

(56)

then,

${g}^{-1}\circ g\left({\omega }_{x}\right) = {g}^{-1}\circ g\left({\omega }_{y}\right)$

(57)

$\Rightarrow {\omega }_{x} = {\omega }_{y}$

(58)

If,

$t = 0|t\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mathcal{C}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(59)

From (55),

${g}^{-1}\circ g\left(t\right) = 0\notin \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

(60)

Similarly, For,

$t\ge 1|t\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mathcal{C}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(61)

From (55),

${g}^{-1}\circ g\left(\omega \right)\ge 1\notin \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

(62)

From (58), (60) and (62),

Proof (T2): (By induction)

For $a = 1$ ,

${\omega }_{a} = \mathrm{max}\left(\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right) < 1 \ \ \ \ \ \ \ \ \ \ {\rm{(By\ Def.\ (5)), }}$

(63)

Assume $a = A-1$ ,

${\omega }_{a}\ge {\omega }_{A-1}$

(64)

Then $\exists a = A$ ,

${\omega }_{a}\ge {\omega }_{A-1}\ge {\omega }_{A}$

(65)

For $a = 1$ ,

${\omega }_{a} = \mathrm{min}\left(\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right) > 0 \ \ \ \ \ \ \ \ \ \ ({\rm{By}}\ Def.\ (5)),$

(66)

${\omega }_{a}\le {\omega }_{A-1}$

(67)

${\Rightarrow 0 < \left\{{\omega }_{a}\right\}}_{a\le a+1}\sim \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right) < 1 \forall \omega$

Proof (T3):

From (T2),

${\left\{{\omega }_{a}\right\}}_{a\le a+1} < 1$

Choose $\varepsilon \in {\mathbb{R}}_{+}, \varepsilon \to 0$ ,

${\varepsilon }^{a > A} < \left|{\omega }_{a > A}-1\right| < \varepsilon$

For every $a > A$ ,

$\begin{array}{l} \left|\underset{a\to \infty }{\mathrm{lim}}{\omega }_{a > A}-1\right| < \varepsilon \\ \underset{a\to \infty }{\mathrm{lim}}|{\omega }_{a > A}-1| < \varepsilon\\ \ \ \ {\underset{a\to \infty }{\mathrm{lim}}{\omega }_{a > A}} = 1 \end{array}$

(69)

Similarly,

${\left\{{\omega }_{a}\right\}}_{a\le a+1} > 0$

Choose $\varepsilon \in {\mathbb{R}}_{+}, \varepsilon \to 0$ ,

$\frac{1}{\varepsilon } > \left|{\omega }_{a > A}-0\right| > \varepsilon$

For every $a > A$ ,

$\begin{array}{c} \left|\underset{a\to \infty }{\mathrm{lim}}{\omega }_{a > A}-0\right| > \varepsilon \\ \underset{a\to \infty }{\mathrm{lim}}|{\omega }_{a > A}-0| > \varepsilon \\ \underset{a\to \infty }{\mathrm{lim}}{\omega }_{a > A} = 0 \end{array}$

(70)

From (69) and (70),

$\underset{a\to \infty }{\mathrm{lim}}{\left\{{\omega }_{a}\right\}}_{a\le a+1} = \left\{\mathrm{0, 1}\right\}$

Proof (C1):

Assume,

$sup{\left\{{\omega }_{a}\right\}}_{a\le a+1} = max{\left\{{\omega }_{a}\right\}}_{a\le a+1} = {\omega }_{a = 1}$

Choose $\varepsilon \in {\mathbb{R}}_{+}, \varepsilon \to 0$

then for any $a = \mathrm{1, 2}..A$ , we can find,

${\varepsilon }^{A}.{\omega }_{a = 1}⋘{\omega }_{a = 1}$

(71)

${\varepsilon }^{A}.{\omega }_{a = 1}\le {\omega }_{a = A}$

(72)

${\omega }_{a = 1}\le {\omega }_{a = A}.\left(\frac{1}{{\varepsilon }^{A}}\right)$

(73)

Let $\delta \in {\mathbb{R}}_{+}, \delta \to 0$ ,

${\omega }_{a = 1}-\delta < {\omega }_{a = A}.\left(\frac{1}{{\varepsilon }^{A}}\right)$

(74)

Assume,

$inf{\left\{{\omega }_{a}\right\}}_{a\le a+1} = min{\left\{{\omega }_{a}\right\}}_{a\le a+1} = {\omega }_{a = A}$

Choose $\varepsilon \in {\mathbb{R}}_{+}, \varepsilon \to 0$

then for any $a = \mathrm{1, 2}..A$ , we can find,

${\varepsilon }^{A}.{\omega }_{a = A} < {\omega }_{a = A}$

(75)

${\omega }_{a = A}⋘{\omega }_{a = A}.\left(\frac{1}{{\varepsilon }^{A}}\right)$

(76)

${\omega }_{a = 1}\le {\omega }_{a = 1}.\left(\frac{1}{{\varepsilon }^{A}}\right)$

(77)

Let $\delta \in {\mathbb{R}}_{+}, \delta \to 0$ ,

${\omega }_{a = 1} < {\omega }_{a = 1}.\left(\frac{1}{{\varepsilon }^{A}}\right)+\delta$

(78)

From (74) and (78),

Proof (T4):

From (T2 and T3), (C1 and C2)

Case (1)

${\left\{{\omega }_{a}\right\}}_{a > A} = \boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}\subset \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

then,

${\left\{{\omega }_{a}\right\}}_{a\le A}\in \boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\sim \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

(79)

Case (2)

${\left\{{\omega }_{a}\right\}}_{a > A} = \boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\subset \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

then,

${\left\{{\omega }_{a}\right\}}_{a\le A}\in \boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}\sim \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

(80)

From (79) and (80),

$\boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\cap \boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p} = \left\{{\varnothing }\right\}$

(81)

Since,

$\boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\supset \left(\boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\cup \boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}\right)\cup \left(\boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\cap \boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}\right)$

From (79)–(81),

$\begin{array}{l} \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\supset \boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\cup \boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}\\ \ \ \ \ \ \Rightarrow \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\subset \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right) \end{array}$

(82)

Proof (C3):

If,

$\exists \mu \in {\boldsymbol{\mathcal{L}}}|\#\boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p} > > > \#\boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f}\forall a$

Then,

$\mu \equiv {\mu }_{low}$

(83)

Similarly, if,

$\exists \mu \in {\boldsymbol{\mathcal{L}}}|\#\boldsymbol{K}\boldsymbol{i}\boldsymbol{n}\boldsymbol{f} > > > \#\boldsymbol{K}\boldsymbol{s}\boldsymbol{u}\boldsymbol{p}\forall a$

Then,

$\mu \equiv {\mu }_{high}$

(84)

From (83) and (84),

Proof (T5) (By induction)

Assume $z = 1;Z\ge 2$ , $y = 1;Z = 2$ ,

${\mathcal{C}}_{1}.{\prod }_{y = 1}^{y = Y = Z-1}{\mathcal{C}}_{y} = {\mathcal{C}}_{1}.{\prod }_{y = 1}^{y = 1}{\mathcal{C}}_{y}$

(85)

$= {\mathcal{C}}_{1}.{\mathcal{C}}_{Y}$

(85.1)

$= {\mathcal{C}}_{1}.{\mathcal{C}}^{Y}$

(85.2)

Assume truth for $y = Y\gg 1$ ,

${\mathcal{C}}_{1}.{\prod }_{y = 1}^{y = Y = Z-1}{\mathcal{C}}_{y} = {\mathcal{C}}_{z}.\left({\mathcal{C}}_{1}{\dots .{\mathcal{C}}}_{Y}\right)$

(86)

$= {\mathcal{C}}_{1}.{\mathcal{C}}^{Y}$

(86.1)

For $y = Y+1$ ,

${\mathcal{C}}_{1}.{\prod }_{y = 1}^{y = Y+1}{\mathcal{C}}_{y} = {\mathcal{C}}_{1}.{\prod }_{y = 1}^{y = Y+1}{\mathcal{C}}_{y}$

(87)

$= {\mathcal{C}}_{1}.\left({\prod }_{y = 1}^{y = Y}{\mathcal{C}}_{y}\right).\left({\prod }_{y = 1}^{y = 1}{\mathcal{C}}_{y}\right)$

(87.1)

$= {\mathcal{C}}_{1}.{\mathcal{C}}_{Y}^{Y}.{\mathcal{C}}_{1}^{1}$

(87.2)

$= {\mathcal{C}}_{1}.{\mathcal{C}}^{Y+1}$

(87.3)

From (85)–(87),

Proof (T6):

From Defs. (6–8),

$\mu {\mathcal{C}}_{z}.{\mathcal{C}}^{Y}\equiv \mu {\mathcal{C}}_{z}\equiv ⟨\boldsymbol{c}|\boldsymbol{\mu }⟩$

(88)

$\Rightarrow \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\sim \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

(89)

$\Rightarrow \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\cap \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

(90)

From Def. (5),

${\omega }_{a\in [1, A]}\in \left(\boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\cap \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right)$

(91)

$\left({\omega }_{a\in [1, A]}\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\right)\cap \left({\omega }_{a\in [1, A]}\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right)$

(92)

From (T2–T4), (C1–C3),

$\forall a\left({\omega }_{a = \mathrm{1, 2}\dots A}\subseteq \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\right)\cap \left({\omega }_{a = \mathrm{1, 2}\dots A}\subseteq \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right)$

(93)

$\Rightarrow \left({\left\{{\omega }_{a}\right\}}_{a = \mathrm{1, 2}\dots A}\sim \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\right)\cap \left({\left\{{\omega }_{a}\right\}}_{a = \mathrm{1, 2}\dots A}\sim \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\right)$

(94)

Conversely, let,

$u\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right);\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

From (92) and (94),

$\forall \omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\exists u\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)|u = h\left(\omega \right)$

(95)

$\forall u\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\exists \omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)|\omega = {h}^{-1}\left(u\right) = {h}^{-1}\left(h\left(\omega \right)\right)$

(96)

From (95) and (96),

Proof (T7):

From (T6),

${h}^{-1}\circ h:\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)$

From (T1), Def. (4)

$g:\omega \in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{c}\boldsymbol{\mu }\right)\cup \mathbb{R} = \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

Rewriting,

$g:\omega \mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(97)

$g\left({h}^{-1}\left(u\right)\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(98)

$g\circ {h}^{-1}\left(h\left(\omega u\right)\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(99)

$g\circ {h}^{-1}\circ h\left(\omega \right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(100)

Proof (P):

From (T1–T7) and Defs. (4–8, 10),

For $z = Z = 1$ ,

${\mu {\mathcal{C}}}_{z}.{\mathcal{C}}^{Y} = \mu {\mathcal{C}}_{z}$

(101)

$g\circ {h}^{-1}\left(u\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}\right)\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}$

(102)

$g\circ {h}^{-1}\circ h\left(\omega \right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(102.1)

$\Rightarrow \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}\right)\subset \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(103)

For $Z\ge 2$ ,

$g\circ {h}^{-1}\left(u\in \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)\subset \mathbb{R}$

(104)

$g\circ {h}^{-1}\circ h\left(\omega \right)\mapsto \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(104.1)

$\Rightarrow \boldsymbol{K}\boldsymbol{d}\left(\boldsymbol{\mu }{\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}}.{\boldsymbol{\mathcal{C}}}^{\boldsymbol{Y}}\right)\subset \boldsymbol{K}\boldsymbol{d}\left({\boldsymbol{\mathcal{C}}}_{\boldsymbol{z}\boldsymbol{\mu }}\right)$

(105)

From (103) and (105),

Acknowledgments

This work is funded by an early career intramural grant (Code No. A-766) awarded to SK by the All India Institute of Medical Sciences (AIIMS, New Delhi, INDIA).

Conflict of interest

The author declare there is no conflict of interest.

Acknowledgments

The authors would like to acknowledge the help rendered by the polyclinics involved in the study. We also wish to thank all participants for voluntarily giving their time in completing the questionnaire. This study is funded by the Fundamental Research Grant (FRGS/1/2017/SKK05/UIAM/02/1), Ministry of Higher Education (MOHE), Malaysia.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	World Cancer Research Fund International, American Institute for Cancer Research Worldwide cancer data: Global cancer statistics for the most common cancers (2020) .Available from: https://www.wcrf.org/dietandcancer/cancer-trends/worldwide-cancer-data.
[2]	Ministry of Health MalaysiaMalaysia National Cancer Registry Report 2012–2016. Putrajaya, Malaysia: National Cancer Institute.
[3]	Ferlay J, Soerjomataram I, Ervik M, et al. (2015) GLOBOCAN 2012 v1. 0, Cancer Incidence and Mortality Worldwide: IARC CancerBase No. 11. International Agency for Research on Cancer, Lyon, France. 2013. Globocan Iarc Fr .
[4]	Abdullah NA, Wan Mahiyuddin W, Muhammad NA, et al. (2013) Survival rate of breast cancer patients in Malaysia: A population-based study. Asian Pac J Cancer Prev 14: 4591-4594. doi: 10.7314/APJCP.2013.14.8.4591
[5]	Ibrahim NI, Dahlui M, Aina E, et al. (2012) Who are the breast cancer survivors in Malaysia? Asian Pac J Cancer Prev 13: 2213-2218. doi: 10.7314/APJCP.2012.13.5.2213
[6]	Bhoo Pathy N, Yip CH, Taib NA, et al. (2011) Breast cancer in a multi-ethnic Asian setting: Results from the Singapore-Malaysia hospital-based breast cancer registry. Breast J 20: 75-80. doi: 10.1016/j.breast.2011.01.015
[7]	Kanaga KC, Nithiya J, Noor Shatirah MFV (2011) Awareness of breast cancer and screening procedures among Malaysian women. Asian Pac J Cancer Prev 12: 1965-1967.
[8]	Akhtari-Zavare M, Aliyan-Fini F, Ghanbari-Baghestan A, et al. (2018) Development and validation of breast cancer knowledge and beliefs questionnaire for Malaysian student population. Pertanika J Soc Sci Hum 26: 2541-2554.
[9]	Yip CH, Pathy NB, Teo SH (2014) A review of breast cancer research in Malaysia. Med J Malaysia 69: 7-22.
[10]	Dahlui M, Eng DHG, Taib NA, et al. (2012) Predictors of breast cancer screening uptake: a pre intervention community survey in Malaysia. Asian Pac J Cancer Prev 13: 3443-3449. doi: 10.7314/APJCP.2012.13.7.3443
[11]	Al-Naggar RA, Bobryshev YV, Al-Jashamy K (2012) Practice of breast self-examination among women in Malaysia. Asian Pac J Cancer Prev 13: 3829-3833. doi: 10.7314/APJCP.2012.13.8.3829
[12]	Akhtari-Zavare M, Juni MH, Said SM, et al. (2013) Beliefs and behavior of Malaysia undergraduate female students in a public university toward breast self-examination practice. Asian Pac J Cancer Prev 14: 57-61. doi: 10.7314/APJCP.2013.14.1.57
[13]	Ghahremani L, Mousavi Z, Kaveh MH (2016) Self-care education programs based on a trans-theoretical model in women referring to health centers: breast self-examination behavior in Iran. Asian Pac J Cancer Prev 17: 5133-5138. doi: 10.7314/APJCP.2016.17.3.1157
[14]	Mohamed NC, Moey SF, Lim BC (2019) Validity and reliability of Health Belief Model questionnaire for promoting breast self-examination and screening mammogram for early cancer detection. Asian Pac J Cancer Prev 20: 2865-2873. doi: 10.31557/APJCP.2019.20.9.2865
[15]	(2002) Institute of MedicineSpeaking of health: assessing health communication strategies for diverse populations. Washington D.C.: The National Academies Press 28-75.
[16]	Murphy CC, Vernon SW, Diamond PM, et al. (2014) Competitive testing of health behavior theories: how do benefits, barriers, subjective norm, and intention influence mammography behavior? Ann Behav Med 47: 120-129. doi: 10.1007/s12160-013-9528-0
[17]	Department of Statistics Malaysia The Source of Malaysia's Official Statistics (2020) .Available from: https://www.dosm.gov.my/v1/index.php?r=column/ctheme&menu_id=L0pheU43NWJwRWVSZklWdzQ4TlhUUT09&bul_id=MDMxdHZjWTk1SjFzTzNkRXYzcVZjdz09.
[18]	Huang HT, Kuo YM, Wang SR, et al. (2016) Structural factors affecting health examination behavioral intention. Int J Environ Res Public Health 13: 395-409. doi: 10.3390/ijerph13040395
[19]	Hair JF, Black WC, Babin BJ, et al. (2009) Multivariate data analysis. Exploratory factor analysis Upper Saddle River, NJ: Pearson Prentice-Hall, 90-150.
[20]	Hu LT, Bentler PM (1999) Cutoff criteria for fit Indexes in covariance structure analysis: Conventional criteria versus new alternatives. Struct Equ Modeling 6: 1-55. doi: 10.1080/10705519909540118
[21]	Kline RB (2011) Principles and practice of structural equation modeling New York: The Guilford Press.
[22]	Schermelleh-Engel K, Moosbrugger H, Muller H (2003) Evaluating the fit of structural equation models: Tests of significance and descriptive goodness-of-fit measures. Methods Psychological Res 8: 23-74.
[23]	Bryne BM (2010) Structural equation modeling with AMOS basic concepts, applications, and programming New York: Taylor and Francis Group, LLC.
[24]	Wang J, Wang X (2012) Structural equation modeling: applications using Mplus United Kingdom: John Wiley & Sons Ltd. doi: 10.1002/9781118356258
[25]	Whittaker TA (2012) Using the modification index and standardized expected parameter change for model modification. J Exp Educ 80: 26-44. doi: 10.1080/00220973.2010.531299
[26]	Tarka P (2018) An overview of structural equation modeling: Its beginnings, historical development, usefulness and controversies in the social sciences. Qual Quant 52: 313-354. doi: 10.1007/s11135-017-0469-8
[27]	Avci IA (2008) Factors associated with breast self-examination practices and beliefs in female workers at a Muslim community. Eur J Oncol Nurs 12: 127-133. doi: 10.1016/j.ejon.2007.11.006
[28]	Noroozi A, Jomand T, Tahmasebi R (2011) Determinants of breast self-examination performance among Iranian women: An application of the Health Belief Model. J Canc Educ 26: 365-374. doi: 10.1007/s13187-010-0158-y
[29]	Maddux JE, Rogers RW (1983) Protection motivation and self-efficacy: a revised theory of fear appeals and attitude change. J Exp Social Psychol 19: 469-479. doi: 10.1016/0022-1031(83)90023-9
[30]	Beatson R, McLennan J (2011) What applied social psychology theories might contribute to community bushfire safety research after Victoria's ‘Black Saturday’? Aust Psychol 46: 171-182. doi: 10.1111/j.1742-9544.2011.00041.x
[31]	Westcott R, Ronan K, Bambrick H, et al. (2017) Expanding protection motivation theory: investigating an application to animal owners and emergency responders in bushfire emergencies. BMC Psychol 5: 1-14. doi: 10.1186/s40359-017-0182-3
[32]	Paton D, Johnston D (2001) Disasters and communities: vulnerability, resilience and preparedness. Disaster Prev Manag 10: 270-277. doi: 10.1108/EUM0000000005930
[33]	Paton D (2013) Disaster resilient communities: developing and testing an all-hazards theory. J Integr Disaster Risk Manag 3: 1-17. doi: 10.5595/idrim.2013.0050
[34]	Heidari Z, Mahmoudzadeh-Sagheb HR, Sakhavar N (2008) Breast cancer screening knowledge and practice among women in southeast of Iran. Acta Med Iran 46: 321-328.
[35]	Tavafian SS, Hasani L, Aghamolaei T, et al. (2009) Prediction of breast self-examination in a sample of Iranian women: an application of the Health Belief Model. BMC Womens Health 9: 37-43. doi: 10.1186/1472-6874-9-37
[36]	Kessler TA (2012) Increasing mammography and cervical cancer knowledge and screening behaviors with an educational program. Oncol Nurs Forum 39: 61-68. doi: 10.1188/12.ONF.61-68
[37]	Lotfi B, Hashemi SZ, Ansari-Moghadam A (2011) Investigation of the impact of HBM-based training on BSE in women referred to health centers in Zahedan in 2010–2011. J Health Scope 1: 39-43. doi: 10.5812/jhs.4698
[38]	Khiyali Z, Aliyan F, Kashfi SH, et al. (2017) Educational intervention on breast self-examination behavior in women referred to health centers: application of Health Belief Model. Asian Pac J Cancer Prev 18: 2833-2838.
[39]	Demirkiran F, Ozgun H, Eskin M, et al. (2011) Cognition of breast cancer among gestational age Turkish women: a cross-sectional study. Asian Pac J Cancer Prev 12: 277-282.
[40]	Elobaid YE, Aw TC, Grivna M, et al. (2014) Breast cancer screening awareness, knowledge, and practice among Arab women in the United Arab Emirates: a cross-sectional survey. PloS One 9: e105783. doi: 10.1371/journal.pone.0105783
[41]	Moey SF, Mutalib AMA, Mohamed NC, et al. (2020) The relationship of socio-demographic characteristics and knowledge of breast cancer on stage of behavioral adoption of breast self-examination. AIMS Public Health 7: 620-633. doi: 10.3934/publichealth.2020049
[42]	Akhtari-Zavare M, Juni MH, Ismail IZ, et al. (2015) Barriers to breast self-examination practice among Malaysian female students: a cross-sectional study. Springerplus 4: 692-697. doi: 10.1186/s40064-015-1491-8
[43]	Liu YL, Wang DW, Yang ZC, et al. (2019) Marital status is an independent prognostic factor in inflammatory breast cancer patients: an analysis of the surveillance, epidemiology, and end results database. Breast Cancer Res Treat 178: 379-88. doi: 10.1007/s10549-019-05385-8
[44]	Kirag N, Klzllkaya M (2019) Application of the Champion Health Belief Model to determine beliefs and behaviors of Turkish women academicians regarding breast cancer screening: a cross sectional descriptive study. BMC Womens Health 19: 1-10. doi: 10.1186/s12905-019-0828-9
[45]	Dewi TK, Massar K, Ruiter RAC, et al. (2019) Determinants of breast self-examination practice among women in Surabaya, Indonesia: An application of the health belief model. BMC Public Health 19: 1-8. doi: 10.1186/s12889-018-6343-3
[46]	Didarloo A, Nabilou B, Khalkhali HR (2017) Psychosocial predictors of breast self-examination behavior among female students: an application of the health belief model using logistic regression. BMC Public Health 17: 1-8. doi: 10.1186/s12889-017-4880-9
[47]	Glanz K, Rimer BK, Viswanath K (2008) Health behavior and health education: theory, research, and practice USA, San Francisco: Jossey-Bass Inc Pub.
[48]	Alkhasawneh IM (2007) Knowledge and practice of breast cancer screening among Jordanian nurses. Oncol Nurs Forum 34: 1211-1217. doi: 10.1188/07.ONF.1211-1217
[49]	Abolfotouh MA, Banimustafa AA, Mahfouz AA, et al. (2015) Using the health belief model to predict breast self-examination among Saudi women. BMC Public Health 15: 1-12. doi: 10.1186/s12889-015-2510-y
[50]	Minhat H, Mustafa J, Mohd Zain N (2014) The practice of breast self-examination (BSE) among women living in an urban setting in Malaysia. Int J Public Health Clin Sci 1: 91-99.
[51]	Sama CB, Dzekem B, Kehbila J, et al. (2017) Awareness of breast cancer and breast self-examination among female undergraduate students in a higher teachers training college in Cameroon. Pan Afr Med J 28: 91.
[52]	Gomez SL, Quach T, Horn-Ross PL, et al. (2010) Hidden breast cancer disparities in Asian women: disaggregating incidence rates by ethnicity and migrant status. Am J Public Health 100: S125-S131. doi: 10.2105/AJPH.2009.163931
[53]	Rendall MS, Weden MM, Favreault MM, et al. (2011) The protective effect of marriage for survival: a review and update. Demography 48: 481-506. doi: 10.1007/s13524-011-0032-5
[54]	Elmore L, Deshpande A, Daly M, et al. (2015) Postmastectomy radiation therapy in T3 node-negative breast cancer. J Surg Res 199: 90-96. doi: 10.1016/j.jss.2015.04.012
[55]	Dewi TK, Zein RA (2017) Predicting intention perform breast self-examination: application of the Theory of Reasoned Action. Asian Pac J Cancer Prev 18: 2945-2952.
[56]	Aljohani S, Saib I, Noorelahi M (2017) Women performance of breast cancer screening (Breast Self-Examination, Clinical Breast Exam and Mammography). Adv Breast Cancer Res 6: 16-27. doi: 10.4236/abcr.2017.61002
[57]	Yoshany N, Morowatisharifabad MA, Mihanpour H, et al. (2017) The effect of husbands' education regarding menopausal health on marital satisfaction of their wives. J Menopausal Med 23: 15-24. doi: 10.6118/jmm.2017.23.1.15
[58]	Bandura A (1997) Self-efficacy: The exercise of control New York: W. H. Freeman.
[59]	Barbeau K, Boileau K, Sarr F, et al. (2019) Path analysis in Mplus: A tutorial using a conceptual model of psychological and behavioral antecedents of bulimic symptoms in young adults. Quant Meth Psych 15: 38-53. doi: 10.20982/tqmp.15.1.p038

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A path analytic model of health beliefs on the behavioral adoption of breast self-examination

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Abstract

1. Introduction

2. Model description and definitions

3. Results

3.1. Mathematical analysis of transition-state dissociation constants of the modeled ligand-macromolecular complex

3.2. Annotation of a ligand on the basis of the transition-state dissociation constants of the modeled ligand-macromolecular complex

3.3. Modeling the transition-state dissociation constants of higher-order ligand-macromolecular complexes

3.4. Numerical studies to establish biological relevance of the transition-state dissociation constants

4. Discussion

4.1. Transition-state dissociation constants as a model of biochemical function for ligand-macromolecular complexes

4.2. Ligand-macromolecule biochemistry as a function of its transition-state dissociation constants

5. Conclusions

6. Proofs

Acknowledgments

Conflict of interest

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Abstract

1. Introduction

2. Model description and definitions

3. Results

3.1. Mathematical analysis of transition-state dissociation constants of the modeled ligand-macromolecular complex

3.2. Annotation of a ligand on the basis of the transition-state dissociation constants of the modeled ligand-macromolecular complex

3.3. Modeling the transition-state dissociation constants of higher-order ligand-macromolecular complexes

3.4. Numerical studies to establish biological relevance of the transition-state dissociation constants

4. Discussion

4.1. Transition-state dissociation constants as a model of biochemical function for ligand-macromolecular complexes

4.2. Ligand-macromolecule biochemistry as a function of its transition-state dissociation constants

5. Conclusions

6. Proofs

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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