Sheaf Cohomology for Fragment-Sequencing in Hierarchical Block Rectangular Matrices with Spectral Gaps in the Presence of Random Effects and White Noise: The Chain

2.1. Fragment-Sequencing

This section summarizes the issue posed in [1] after [2].

The contact map is the matrix whose entries correspond to the pairwise ‘contact frequencies’ between two vectors of bins. The definition of the contact-map matrix is taken from [23] p. 26 is taken as

It is worth noting that, in contrast, [1], $m_{ij}$ typically assigns a value of 1 when the distance is less than a given threshold. An alternative definition of a contact map is issued from [24] as

Accordingly, this definition is well-suited for defining pairwise sequencing.

The analysis from [1] is here reappraised.

Let $\overrightarrow{p}$ be the row vector of m consecutive bins, i.e., $\overrightarrow{p}=(p_1,p_2,\dots ,p_m)$ .

Let $\overrightarrow{q}$ be the row vector of n consecutive bins, i.e., $\overrightarrow{q}=(q_1,q_2,\dots ,q_n)$ .

Clearly, vectors p and q have different dimensions.

This is a characteristic of rectangular matrices, which are used for fragment sequencing within clusters.

The hierarchical structure of clusters is more apparent after the comparison with [1]. With all the above parameterizations in place, the scenario is prepared for the definition of contact networks.

The notions of weighted graphs and those of unweighted graphs can therefore be recalled.

Let $\left\{C\right\}$ be the set of clusters in $N_{\overrightarrow{p}}(V,\mathcal{E},w)$ ; the following definition holds

The Shavit-Walker-Lio’ (SWL) matrices $\widehat{B}$ ’s are defined as

• $\widehat{B(1)}$ an $m\times m$ matrix whose entries $b{(1)}_{ij}$ are as Equations (6a) and (6b)

  (6a) $$\begin{array}{cc}\hfill & b{(1)}_{ij}=1,i,j\in c,c\in C,c\subset V,\hfill \end{array}$$
  (6b) $$\begin{array}{cc}\hfill & b{(1)}_{ij}=0otherwise;\hfill \end{array}$$
  and

• $\widehat{B(s)}$ an $m\times m$ matrix whose entries $b{(s)}_{ij}$ are as Equations (7a) and (7b)

  (7a) $$\begin{array}{cc}\hfill & b{(s)}_{ij}=1,i,j\in c,c\in C_{s-1},c\subset V,\hfill \end{array}$$
  (7b) $$\begin{array}{cc}\hfill & b{(s)}_{ij}=0otherwise;\hfill \end{array}$$

If the structure is hierarchical, clusters may be present. For this reason, Hierarchical Block Matrices (HBMs) are sought.

The HBM matrices of N can be studied here.

2.2. Singular-Value Decomposition (SVD) of Ordered Block Matrices: Theorems

From [16], the underlying ‘signal’ matrix with ordered block structure $\widehat{D}$ is considered, i.e., $\widehat{D}\in {\mathbb{R}}^{m\times n}$ that is starting index of J as $m_1\times n$ to $m_M\times n$ .

A rectangular ‘observation’ matrix $\widehat{Y}$ is considered, under the hypothesis that there are M blocks of sizes $(m_1,m_2,\dots ,m_M)$ with $su{m}_{i=1}^{i=M}{m}_i=m$ and N blocks of sizes $(n_1,n_2,\dots ,n_N)$ with ${\sum }_{j=1}^{j=N}{n}_j=n$ .

Without loss of generality, $m\le n$ is assumed.

The observation matrix $\widehat{Y}$ can be decomposed according to the entries Equation (9)

The column vectors ${\eta }_{H(i)}$ and ${\nu }_{v(j)}$ are defined as continuous random variables with vanishing means and standard deviation ${\sigma }_{\eta }$ and ${\sigma }_{\nu }$ , respectively, i.e., they represent white noise.

The underlying constant signal $\widehat{D}$ is therefore isolated from the other components in Equation (9).

Let ${\widehat{J}}_{a\times b}$ be an $a\times b$ matrix with all entries equal to 1. The matrices $\widehat{D}$ , $\widehat{\eta }$ and $\widehat{\nu }$ in Equation (9) are written as

3.1. New Definition of the Chain from the Contact Map in the Presence of Random Effects and White Noises

The request of [23] that the distances between the contact maps be Euclidean is here accomplished.

The Matrix $\widehat{Y}$ from [15] is here rewritten using its singular-value decomposition, as shown in Equation (10)

The matrix $\widehat{\upsilon }$ and the matrix $\widehat{v}$ are reduced to vectors: the vector $\overrightarrow{u}$ is recovered in the general case; differently, in the present analysis, if $\widehat{Y}$ is compact, the metric is Euclidean when that of the vector $\overrightarrow{\upsilon }$ is Hilbert (while the definition of $\overrightarrow{u}$ is not in general Hilbert). In this way, the chain $\widehat{Q}$ is defined on a surface endowed with a Hilbert metric. The chain associated with the adjacency matrix corresponds to a topological Markov chain.

The Euclidean distance from [14] is used to define the ‘average pairwise distance’.

From [18], the Cameron-Martin metric is defined from $\mid \gamma \mid$ as Equation (11)

On the vector $\overrightarrow{v}$ of the singular-value decomposition Equation (10); the vectors $\overrightarrow{\upsilon }$ and the vector $\overrightarrow{v}$ are made to coincide, in the case of fragment sequencing. The role of $\varsigma$ is that of the variance, from which the covariance is taken, which allows one to specify marginal distributions; the choice of $\varsigma$ is specified in Theorem 2.

On the vector $\overrightarrow{v}$ of the singular-value decomposition Equation (10), the vectors $\overrightarrow{\upsilon }$ and the vector $\overrightarrow{v}$ are made to coincide in the case of fragment sequencing is studied. The role of $\varsigma$ is that of the variance, from which the covariance is taken, which allows one to specify marginal distributions; the choice of $\varsigma$ is specified in Theorem 2.

The hypothesis from [18] on the expectation value and on the variance are kept as Equations (12) and (13)

3.2. Clusters as Hidden Markov Models

As in [18], the obtained clusters are Hidden Markov Models of ‘multivariate Gaussian data’ when the covariance matrix is fixed. It is the purpose of the next Section to fix the covariance $\varsigma$ .

The mean vector is newly estimated from the log-emission function from [18] in terms of the Baum-Welch backwards probabilities and for the Baum-Welch forward ones and is required to vanish in the form of [15].

In the Baum-Welch algorithm, the likeli-hood function L is here written from [25] on the $\lambda$ collection of model parameters. The Baum-Welch backward probabilities $\alpha (t)$ and the Baum-Welch forward probabilities $\beta (t)$ are here considered for the observation of the probability space O as Equations (14a) and (14b)

The multivariate distribution mean is calculated as Equation (15)

The techniques for analytically obtaining a multivariate mean of zero are discussed in Section 4.1. In the following subsection, the paradigm developed here is shown to generate a Markov process. The covariance matrix is now fixed from [20], and the Topological Hidden Markov Models are newly built.

3.3. The new Topological Hidden Markov Models (of Clusters)

As studied from [20], a Gaussian distribution with vanishing mean is taken. The specification of the variance $\varsigma$ allows one to specify the marginal distributions.

Gaussian probability measures with ‘prescribed marginals’ are defined after the joint probability density $\mathcal{P}$ with marginals ${\mathcal{P}}_{c_1},{\mathcal{P}}_{c_2},\dots ,{\mathcal{P}}_{c_l}$ . The $c_l$ here used are those from the subset Equation (5).

A class of Gaussian measures with prescribed marginals is newly found after [20] as there exists the covariance matrix $\widehat{\varsigma }$ and $\widehat{\varsigma }$ is unique. The probability space $(states,observation,filter)$ is now constructed for the Topological Hidden Markov Models of clusters. The filter is constructed based on the probability function, and the measure of the probability space is subsequently defined.

From [20] p. 139, the random vector field $\overrightarrow{X}$ is considered, with a Gaussian distribution and with vanishing mean and positive-definite covariance $\widehat{\varsigma }$ . The density $p(\overrightarrow{X})$ is written as Equation (17)

Marginal densities $p_{\Gamma }(\overrightarrow{X})$ are defined for arbitrary subsets $\overrightarrow{\Gamma }$ of $\overrightarrow{X}$ . The following Proposition is drawn from Proposition 1 from [20].

The following new proposition is newly derived after Proposition 2 from the same source as

The particular cases of the pairwise sequence are now studied for comparison with [18]. The covariance obeys the following:

The generalized covariance here used is $\widehat{\varsigma }(\alpha ,\beta )$ . The generalized covariance matrix $\widehat{\varsigma }$ is defined as in Equation (18)

The following two theorems are recalled in [20] from [26].

The sheaf-cohomology techniques from [17] can be used to model the scenario with noise and random effects.

3.4. Topological Framework of Contact Probabilities

The role of contact probability is inscribed within a topological framework.

In the work of Lieberman-Aiden et al. [27], massive parallel sequencing is used to demonstrate the 3-dimensional features of the complete genome with respect to proximity-based ligation. Open and closed chromatin are shown to exhibit spatial segregation.

As mentioned before, the long-range interactions between chosen pairs of loci are discriminated using Chromosome Conformation Capture (CCC), where spatially constrained ligation plays a crucial role. Hi-C is a methodology based on massive unbiased sequencing, whereas CCC does not permit unbiased genome-wide analysis.

In the work of Kalhor et al. [28], the Tethered Conformation Capture (TCC) is described as a technique for genome-wide mapping of chromatin interactions. Ibidem, the TCC is outlined to enhance the signal-to-noise ratio, which highlights inter-chromosomal interactions; diverse combinations of interactions are hypothesized to be present in cells; 3-dimensional genome-wide structures are highlighted. As a result, the statistical analysis is limited, and chromosomal interactions are investigated only in the human genome.

As discussed, only a few structural aspects that rule the organization of chromatin are nowadays reported to be understood at the genome scale. Various factors limit the understanding of Chromosome Conformation Capture (CCC). Low signal-to-noise ratios in chromosome capture experiments reduce the ability to map low-frequency interactions, and individual structures are now hypothesized to vary across the cell population.

The capture of conformational data into 3-dimensional structural models is now, therefore, an open challenge. Accordingly, theoretical folding models [27] have been applied to genome-wide conformation capture data.

In the same source, the TCC is treated as a modified conformation capture method, in which a higher signal-to-noise ratio is calculated, allowing analysis of inter-chromosomal interactions. The resulting analysis technique is probabilistic and enables one to describe some features of the genome. As a methodology, massive parallel sequencing is performed, which links the initial contacts to the locations of the paired loci in the genome. The obtained contact maps accurately account for the observed patterns: the results are in accord with [27].

In the work of Misteli [29], a characterization of the genome is depicted. As previously discussed, the organization of the genomic sequence is described as being determined by spatial and temporal factors at three hierarchical scales: the functioning of nuclear properties, higher-order mechanisms associated with the chromosome fiber, and the spatial arrangement of genomes within the cell nucleus. Genome stability is understood to be influenced by these three factors, which also play a role in gene expression.

The three factors discussed above are pivotal for understanding large-scale DNA sequence mapping. Consequently, the cellular mechanisms that determine genome positioning and their effects on genome regulation must be comprehended to complete the sequencing process. The nature of transcription complexes is thus highly dynamic, with their behavior additionally modulated by compartmentalization.

In the work of Branco et al. [30], the compartmentalization processes are analyzed as responsible for gene expression after accounting for chromatin interactions within distal chromatin organization. More specifically, interactions between distal chromatin segments are reported to influence transcriptional regulation. The topology of chromosomes is introduced in [31], where the chromosome topology is also the capability to undergo the nuclear processes.

In the work of Haaf and Schmid [32], the topology of chromosomes is shown to follow several rules that dictate the number of attachment sites on each chromosome. The arrangement of DNA ‘families’ repeats is studied ibidem. The topological structures demonstrate patterns also in evolutionarily distant species. In turn, topological structures help define transcriptional processes. The compartmentalization of the processes regulating transcription is not yet fully understood.

In the work of Zhang et al. [33], the topology of chromosomes is explained as shaping the energy landscapes, which are described in the present paper within Markov Models and are now newly analyzed as apt to be arranged within the framework of the Markov State Model.

In [33], the energy landscape is also found to exert a backreaction on the three-dimensional genome organization. In the same work, the energy landscape is described using a maximum-entropy approach, which yields a least-biased effective energy landscape.

In the work of Boulos et al. [34], graph theory is applied in the description of the human genome as far as chromatin interaction (HiC) is concerned. The main replication regions are shown to be in correspondence with DNA loci of maximal network centrality; furthermore, these loci are demonstrated to constitute a set of ‘interconnected hubs’ both at the chromosome level and at the scales implied for different chromosomes. The genomic mechanisms of replication and of transcription can be framed within a graph-theoretical organization, which can be exploited to validate the polymer models of the nuclear organization. The DNA sequences are represented as networks, where critical positions are assigned based on a centrality hierarchy that distinguishes among degree centrality, betweenness centrality, and eigenvector centrality. The ranking accounts for the total weights of the incident edges; within this analysis, the degree centrality [35] is a local centrality measure, the betweenness centrality [35] accounts for to which extent a vertex is located between other vertices on the geodesics of the graph (it is here recalled that for these purposes the graph must be positioned on a manifold), and the eigenvector centrality [36] discriminated the vertices which are connected with ‘well-connected’ vertices. In this theoretical framework, the three-dimensional conformations are analyzed by representing genomic loci as vertices on a plane.

In the work of Sexton et al. [37], the contact map is constructed from the Drosophila species. More precisely, a high-resolution contact map is written from a modified genome-wide chromosome conformation capture approach. The data analysis is presented as demonstrating that the genome exhibits a linear partition into ‘well-demarcated’ domains, which overlap extensively with active epigenetic marks and repressive ones. Intra-chromosome and inter-chromosome contacts define contact density and clusters.

In the work of Hou et al. [38], the chromosome domains are proven to be defined after epigenetic marks.

In the work of Dixon et al. [39], the 3- dimensional organization of the human genome is summarized. More specifically, megabase-sized local chromatin interaction (“topological”) domains are identified. Moreover, their boundaries are also characterized. Topological domains are assigned a directionality index, which quantifies the degree and type of interaction bias of each genomic region. A Hidden Markov State Model is then used to identify the biased states, thereby pinpointing the locations of topological domains. As a result, genomic DNA is described as partitioned into spatial modules linked by chromatin segments, with topological boundary regions defined by regions of chromatin disorganization.

4.1. The Use of Dirichlet Forms for Obtaining a Vanishing Multivariate Distribution Mean Analytically

The use of Dirichlet forms is indicated in [16] and in [40].

The results from [16] are suited for constructing Topological Markov Models from Jukes-Cantor-inspired sequencing, i.e., as issued by [41].

On the other hand, the results from [40] are suited for the analytical solution of Equation (15) and for the implementation of the potential theory for the calculation of the rewards (which is recalled from [42] 2.2); the Cameron-Martin formula is recalled in Section 4.

With regard to the calculation of rewards, the presence of absorbing states within a fragment can be further examined.

From [43] and [44], the definition of vector fields for Dirichlet forms on Markov processes enables the analytical solution of Equation (15). In particular, [43] provides a definition of vector fields on mapping spaces for this purpose. The role of weights can be generalized for machine learning purposes, as in [45].

4.2. Generalized Constructions of the Chain of Fragment Comparison

The method of fragment comparison is described here, following [46].

From [46], the method is developed, in which a particular probability kernel is constructed with the suitable space of probability measures for the definition of the chain; the work [46] is to be implemented with the choice of the likelihood function as from [25] implementation of Equation (15). The results presented here are compatible with the most general construction [46] when a Gaussian distribution with vanishing mean is taken for the definition of the measure, i.e., the filter, of the probability space.

The comparison of the two fragments $(x^n)$ and $(y^n)$ here is accomplished on a chosen probability space $(\Omega ,\mathcal{F},Pr)$ with $Pr$ assumed on normed spaces $E_0$ and E, respectively.

Let A and B be fixed Borel subsets of $E_0$ and E, respectively.

$X^n$ is the state space $\sigma \{x_0,x_1,\dots ,x_n\}$ defined on a Borel subset with its $\sigma$ -algebra.

Analogously, $Y^n$ is $\sigma \{y_0,y_1,\dots ,y_n\}$ . The Hidden Markov Process $(x^n)$ is characterized by a transition kernel $\mathcal{T}(x,d{x}^{\prime })$ , and the observation process $(y^n)$ is characterized by the transition kernel ${\mathcal{T}}_1^{x^1}(y,d{y}^{\prime })$ as Equations (21a) and (21b)

The kernels $\mathcal{T}$ are probability measures for the fixed Borel subsets A and B; ${\mathcal{T}}_1$ is chosen as in Equation (22)

The Markov Process is one with Equation (23)

The evolution of the system is described after Equation (6) from [46]. It is noted here that the methods based on a Gaussian Markov distribution are applicable; that is, the transition kernels can induce a measure for the filter of the probability space.

The Cameron-Martin space is newly presented in [47]. The Ornstein-Uhlenbeck semigroup mapping is reappraised in [19] to implement the Ornstein-Uhlenbeck process described in [18]. An example of representing the Kantorovich–Rubinstein distance on a centered Gaussian measure over the Borel $\sigma$ -field is provided in [19] to compute distances between sequences. This example applies to both pairwise and fragment sequencing. In the same work, examples are provided for which the total variation of the norm is shown to be minorizable. Additionally, the existence of a mapping operator of norm 1, as stated in Lemma 2.1 of [48], is recalled; the lemma is restated as follows:

Lemma 1 allows one to define the probability space for the hierarchical block-matrix Markov shift. In contrast, the singular-value decomposition of experimental data matrices for complex non-Gaussian random variables is presented in [21], and the techniques developed in the present subsection are also applicable. Indeed, the derivation presented here, following [46], is independent of the Gaussian nature of the variables, allowing for the construction of a different chain.

4.3. Applications in Sequencing

The example from [18] can now be analyzed using the paradigms developed in the present paper to unveil the structures of Hidden Topological Markov Models. More specifically, it is demonstrated here that knowledge of the metric allows the definition of differential operators to be applied to matrices, replacing entries where necessary [17]. The metric defines the manifolds on which the graphs reside, from which the edges can be selected; the union of these edges determines the paths that describe the processes.

The question posed in [1] concerns how hierarchical block matrices can encode information about the topology of block-wise segmentation, particularly regarding the topology of neighboring regions. This investigation is motivated, among other reasons, by the tasks proposed in [22].

This section describes how the data can be represented on a topological manifold and specifies the corresponding metric. In the present case, a Hilbert metric will be determined, which defines the probability space of the process. Indeed, from the clarifications in [26], the one-dimensional segmentations are scrutinized, after which the numerical method is implemented, in which the likelihood with respect to the block boundaries is maximized. It is further noted that the likelihood can be maximized analytically.

The use of the Cameron–Martin distance enables the extraction of Markov property models from the set of “Brownian-motion-like” schemes to which the segmentation techniques may correspond.

Following [49], the problem is extended to a locally compact, connected, separable Hausdorff space equipped with a Radon measure. From [16], it is then possible to describe the time evolution of the eigenvalues of the relevant Markov-property models using the kernels on which the Radon measure is defined; the associated Dirichlet form implies the Bochner formula.

The use of the Hilbert space, which forms an $L^2$ structure, is proven in [17]; the employment of this space is proven to be needed straightforwardly in the case one takes into account the prescriptions from [50]. The Markov-property models studied here are those that determine the block decomposition of the topological shift.

It is noted here that the numerical model proposed in [50] can thus be solved analytically.

4.4. Applications in Machine Learning

In the work of Khan et al. [51], the methods of blockchain are addressed. Protocols for optimizing data availability are considered within the blockchain framework. The roles of blockchain and machine learning are compared, and the use of Hyperledger technology is analyzed; as a result, the integration of machine learning with blockchain distributed ledger technology is examined.

In the work of Khan et al. [52], blockchain-based platforms are applied, thereby addressing the challenges of data fluctuations. In the same work, the use of blockchain to minimize resource consumption is considered.

In the work of Khan et al. [53], the convergence of artificial-intelligence-enabled machine learning techniques, such as artificial neural networks, support vector machines, reinforcement learning, and deep learning, is analyzed. The use of adaptive control, convolutional neural networks, and recurrent neural networks in data processing is compared, and the integration of artificial intelligence with blockchain technology is proposed. In the same work, the application of artificial neural networks for assessing optimization parameters is also discussed.

In the work of Khan et al. [54], the challenge of data retention is addressed, and control systems are examined. The possibility of reshaping data analysis in the context of fog computing is also considered.

In the work of Khan et al. [55], the issue of autonomous decision-making in machine learning is examined. The aim of this study is to assess the balance between the use of artificial neural networks and Particle Swarm Optimization-enabled metaheuristic optimization methods. The hierarchy of automation is understood in terms of the artificial intelligence system, with a specific focus on cloud-native building blocks [56]. In the same work, the control plane functions are examined as being decoupled from the user planes.

In the work of Khan [57], the combination of gamification and general awareness training is explored. Generative artificial intelligence, in conjunction with gamification, is shown to replace traditional hierarchies. Moreover, generative AI and gamification-based learning and training are used to define a new metric for evaluating learners’ progress, with the goal of rewarding game-based learning.

In the work of Khan et al. [58], the focus is on proposing a lightweight middleware proof of elapsed time in blockchains. The concept of a permissioned chain is explained as enabling better single-entity control, and the key aspects of blockchain technology that ensure efficiency through a lightweight topology are summarized. In the same work, the use of multithreading to enhance system scalability is also discussed.

In the work of Khan et al. [59], Deepfake technology is investigated, and a critique of the assessment measures used to evaluate model performance is provided. The features of computational effectiveness and efficiency are described. In the same work, the use of cross-model ledger technology for evaluating Deepfakes across models within resilient systems is proposed as a topic for further investigation.

In the work of Khan et al. [60], blockchains and edge computing are proposed for authentication via a scalable, lightweight system based on Hyperledger Indy. Time latency is reduced through edge computing, and a hybrid cryptographic technique enables system integration. In the same work, the use of a permissioned blockchain is shown to facilitate compliance.