Thermodynamics utilizing DNA methylation processes

Abstract

A framework consistent with the thermodynamic principle and used for deciphering a DNA methylation process utilizes probability density functions of DNA methylation information divergence to summarize statistical biophysics forming the basis of a spontaneous methylation background, and is related to the channel capacity of a molecular machine conforming to the Shannon's capacity theorem (Shannon's capacity theorem). The contribution of a molecular machine (enzyme) logic operation to a Gibbs entropy (Gibbs entropy) (S) and a Helmholtz free energy (Helmholtz free energy) (F) is intrinsic. Application in the biomedicine and biopharmaceutical industries can be achieved by estimating S on a methylation group data set. As a thermodynamic state variable, a single methylation group entropy is determined entirely from the current state of the system, biologically translated into a correspondence between an estimated entropy value and an observable phenotypic state. Analysis of entropy fluctuations in experimental data sets shows that there is a limit in the magnitude of whole genome methylation changes during the response of organisms to environmental changes, thereby allowing early diagnosis and prediction of epigenetic state changes.

Description

[0001] Statement Regarding Federally Sponsored Research or Development

[0002] This invention was made with government support under Award No. GM134056, awarded by the National Institutes of Health. The government has certain rights in this invention.

[0003] Cross - Reference to Related Applications

[0004] This application claims priority to U.S. Provisional Patent Application Serial No. 63 / 380,180, filed on October 19, 2022. U.S. Provisional Patent Application Serial No. 63 / 380,180 is hereby incorporated by reference in its entirety, including but not limited to the specification, claims, and abstract, as well as any figures, tables, appendices, or schematics therein. Technical Field

[0005] The present disclosure relates to, but is not limited to, DNA methylation, information thermodynamics, and epigenetics. More specifically, but not limited thereto, the present disclosure recognizes that the thermodynamics of the DNA methylation process has industrial applications. Background Art

[0006] The background description provided herein provides context for the present disclosure. Aspects of the work of the currently named inventors and descriptions of aspects that may not otherwise qualify as prior art at the time of filing are neither expressly nor impliedly admitted as prior art.

[0007] Some aspects of the subject matter of the present disclosure relate to some aspects of the subject matter of PCT / US2023 / 064913, filed on March 24, 2023, with the same applicant and the same inventors' names. PCT / US2023 / 064913 claims priority to U.S. Provisional Patent Application Serial No. 63 / 323,690, filed on March 25, 2022. U.S. Provisional Patent Application Serial No. 63 / 323,690 is hereby incorporated by reference in its entirety, including but not limited to the specification, claims, and abstract, as well as any figures, tables, appendices, or schematics therein.

[0008] DNA methylation is an epigenetic mechanism that plays an important role in various biological processes, including transcriptional and post-transcriptional regulation, genomic imprinting, aging, and stress responses to environmental changes and diseases.

[0009] Cytosine DNA methylation is one of the best-characterized epigenetic modifications to date. It plays important roles in various biological processes, including X-chromosome inactivation, genomic imprinting, transposon silencing, transcriptional regulation, and the aging process. Additionally, DNA methylation plays an important role in maintaining DNA stability, which means that the most frequent methylation changes contribute to maintaining the thermodynamic stability of DNA molecules.

[0010] The inventors have previously explained in detail how these methylation changes incorporate background activity, which must be distinguished from targeted differentially methylated positions (DMPs) introduced by methylation regulatory mechanisms. See Sanchez et al., “Discrimination of DNA Methylation Signal from Background Variation for Clinical Diagnostics,” Int. J. Mol. Sci. 20, 5343 (2019), which is hereby incorporated by reference in its entirety.

[0011] When evaluating samples from a single species under various experimental conditions, it is not difficult to find evidence of differential methylation activity in the control population through data analysis and simulation. See id. These DMPs are hypothesized to result from fluctuations inherent in any random process, i.e., properties summarized by the fluctuation theorem. Regardless of how constant the environment is, the probability of finding statistically significant methylation changes in the control population is greater than zero, which means that the randomness of the methylation process is derived from the inherent randomness of the biochemical system. Spontaneous natural methylation variations (“noise”) are expected in multicellular organisms, while methylation regulatory mechanisms (“signals”) direct the organism to adapt to fluctuations in the microenvironment and macroenvironment as well as developmental processes.

[0012] The present inventors have developed a model for the probability distribution of methylation variations (noise plus signal), expressed as the information divergence of methylation levels, which is derived for a constrained scenario based on statistical physics. See Sanchez et al., "Information thermodynamics of cytosine DNA methylation", PLoS One 11, e0150427 (2016), which is hereby incorporated by reference in its entirety. Background Methylation variations can be described by a generalized gamma probability distribution or a member of the generalized gamma distribution family. However, such modeling (see ibid.) can only be used as a transfer function in which the model parameters are still undetermined, which is useful for practical applications of modeling the output of the system for each possible input, but is not useful for understanding the thermodynamics of the methylation process.

[0013] The formal derivation of the generalized gamma distribution model for the cytosine DNA methylation process takes into account the continuous action of the second law of thermodynamics on biological processes and the subsequent application of Jaynes' Maximum Entropy Principle after Jaynes' attempt to characterize the information-theoretic interpretation of the second law. The statistical physics assumptions are set on the channel capacity of molecular machines (usually enzymes or macromolecular structures integrated by DNA and enzymes), which is closely related to Shannon's channel capacity. It is assumed that the energy scale of biomolecular machines is comparable to the thermal energy at the environmental temperature and is sensitive to thermal fluctuations. This kind of modeling can provide a physical interpretation for parameters that have not been done before.

[0014] Accordingly, there is a need in the art for practical applications based on thermodynamics and informatics that utilize the probability distribution (a member of the generalized gamma probability distribution family) of the information divergence of methylation levels that accurately describes the methylation process. For example, the entropy and Helmholtz free energy of a single methylation system can help to evaluate the biological significance of the theory for analyzing methylation variations in animals and plants. Summary of the Invention

[0015] DNA methylation plays an important epigenetic role in the development and environmental responsiveness of multicellular organisms. Appropriate assessment of the thermodynamics of methylation variation in natural populations helps to understand the system dynamics and distinguish methylation regulatory signals from background "noise". Modeling based on established physical principles can be an essential step in systematizing scientific methods and enhancing scientific insight and model prediction accuracy, depending on the application. Resolving the accuracy and reliability of the thermodynamic impact model of DNA methylation in cell populations, especially in clinical diagnosis and prognosis.

[0016] The present disclosure demonstrates the application of the principle of maximum entropy and the constraints derived from the channel capacity of molecular machines, which describe the methylation process not only by the probability distribution f(E) of the dissipated energy E, but also by the probability of maintaining the integrity of the DNA methylation message under environmental fluctuations (e.g., diseases, drug treatments, lifestyle, climate change, etc.). Formally, according to Shannon's theory, the derived probability distribution f(E) can be reinterpreted as the probability P y (x) such that if the message recovered at the receiving point is y, then the information divergence between y and the original message x generated by the source is χ.

[0017] The following objectives, features, advantages, aspects, and / or embodiments are not exhaustive and do not limit the entire disclosure. No single embodiment is required to provide each and every objective, feature, or advantage. Any of the objectives, features, advantages, aspects, and / or embodiments disclosed herein can be integrated with one another, in whole or in part.

[0018] The main objectives, features, and / or advantages of the present disclosure are to improve or overcome the deficiencies in the art.

[0019] While experimental science can only provide evidence to support research hypotheses and demonstration is only possible within a mathematical theoretical framework, additional objectives, features, and / or advantages of the present disclosure are to provide support for the following epigenetic reprogramming, respectively: (i) epigenetic reprogramming induced by the msh1 mutation in the plant Arabidopsis thaliana and (ii) epigenetic reprogramming induced by different cancer types in humans. Deviations from the rules shown in the present disclosure occur only in the dysfunctional states represented by the Arabidopsis methyltransferase mutant met1 and human cancer cells. In patients with different types of cancer, the research results show that severe information loss occurs during the transition from differentiated (healthy) tissue to cancer cells.

[0020] After reading the following brief and detailed descriptions of the accompanying drawings, these and / or other objectives, features, advantages, aspects, and / or embodiments will become apparent to those skilled in the art. The present disclosure encompasses (a) combinations of the disclosed aspects and / or embodiments and / or (b) reasonable modifications not shown or described. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Detailed descriptions and illustrations of several embodiments in which the present disclosure may be practiced are provided, where like reference numerals represent like components throughout the several views. The drawings are presented for illustrative purposes and are not to be scaled unless otherwise indicated.

[0022] Figures 1A - 1B A graphical summary of the information thermodynamics of the methylation process and its application to methylation analysis is shown.

[0023] Figure 1A is a flow chart in accordance with the thermodynamic entropy. The flow chart shows that applying Jaynes' maximum entropy principle (MEP) makes the Boltzmann distribution the most likely for the methylation system. Criteria derived from the channel capacity of the molecular machine and additional maximum likelihood estimates enable the theoretical derivation of the generalized gamma distribution model to be the best description of the genome-wide methylation changes observable in a single dataset, expressed in terms of the information divergence of the methylation changes: χ:E = χk B Tθ -1 . The state of the methylation system is described by the generalized gamma probability density function, and an analytical expression for the entropy of the methylation system is derived therefrom.

[0024] Figure 1B An illustration of methylation group variation based on empirical data analysis is shown. Analysis of the experimental dataset shows that the best-fit model is a member of the generalized gamma distribution model or the generalized gamma probability distribution family. Molecular thermal noise generated by biochemical processes induces methylation changes to stabilize the DNA molecule, and this background variation conforms to the laws of statistical physics ( Figure 1B left side). The methylation regulatory signal is a response to developmental and / or environmental conditions and can be discerned as starting from the threshold of the receiver (denoted here as χ ɑ＝0.05 ) and determined by each individual probability density function. In cases where the probability is greater than zero (highlighted in the colored area below the right part of the curve), there is a risk of false positive classification even if a subset of cytosine sites is not present in the data sample, because methylation fluctuations also exist in the background variation of the control sample. Signal detection and machine learning are implemented to estimate the optimal cut-off value χ 截止 ≥χ ɑ＝0.05To distinguish treatment or disease-induced methylation signals from naturally occurring background variations to reduce the risk of false positives, thereby allowing the classification of legitimate differentially methylated positions (DMPs). Thus, the probability density function of χ is the null hypothesis for identifying DMPs. As shown, the information contribution of DMPs to the system entropy, which is a thermodynamic state variable of the system, is very small.

[0025] Figures 2A - 2F An evaluation of the entropy fluctuations in the experimental dataset is shown.

[0026] Figure 2A A regression analysis for the dataset from six-generation memory lines and met1 mutants of Arabidopsis thaliana (in the subplot) is shown Relative to the expected value (average) v = <χ> of the J-information divergence χ.

[0027] Figure 2B A regression analysis for the human dataset from patients with different types of cancer and tissue controls is shown Relative to v. The regression analyses of both small plots support the linear model Only dysfunctional conditions, such as the Arabidopsis thaliana met1 mutant, human breast cancer, human metastasis (in red), or undifferentiated stem cells (hESC, in magenta), do not conform to the regression analysis of statistical significance Relative to v such that the considered regression analysis Relative to e -v .

[0028] Figure 2C A regression analysis for the dataset from six-generation memory lines and met1 mutants of Arabidopsis thaliana (in the subplot) is shown Relative to e -ν .

[0029] Figure 2D A regression analysis for the dataset of the human dataset from patients with different types of cancer and tissue controls is shown Relative to e -v . Figure 2C and 2D The regression analyses in or equivalently Only dysfunctional conditions, such as the Arabidopsis thaliana met1 mutant, human breast cancer, human metastasis (in red), or undifferentiated stem cells (hESC, in magenta), do not conform

[0030] If the regression model is valid, then the method e is introduced into the model -ν≈1 - v, the new model derived must also be valid within the experimental error range.

[0031] Figure 2E Shows the regression analysis for the dataset from six - generation memory lines and met1 mutants of Arabidopsis thaliana (in the sub - figure). Relative to ν.

[0032] Figure 2F Shows the regression analysis for the human dataset from patients with different types of cancer and tissue controls. Relative to ν. The regression analysis of both small figures supports Only dysfunctional cases, such as the Arabidopsis thaliana met1 mutant, human breast cancer, human metastasis (in red) or undifferentiated stem cells (hESC, in magenta), do not conform.

[0033] Figure 3A Shows the block diagram with the sum of the Boltzmann factors in the human dataset. The figure shows that breast cancer, metastasis, and stem cells do not conform.

[0034] Figure 3B Shows the bar graph with the estimated mean of the sum of the Boltzmann factors for the entire group of Arabidopsis thaliana and human samples. The number of individuals on each chromosome is represented in white on each bar. The statistical summary of five Arabidopsis thaliana chromosomes and ten human somatic chromosomes is shown at the top. The error bars correspond to the estimated standard deviation for each chromosome. The results show that there is no significant statistical difference in the sum of the estimated Boltzmann factors for the Arabidopsis thaliana and human datasets, thus supporting

[0035] Figures 4A - 4B Shows the estimated chromosomal Gibbs entropy for the following groups: TD and ASD. Figure 4A Related to males. Figure 4B Related to females. According to The best - probability generalized gamma model for each chromosome is used to estimate the Gibbs entropy. The unit of the entropy value in the figure is: joule × kelvin × mol -1 .

[0036] Figures 5A - 5B Shows the analysis of entropy fluctuations in placental tissues from TD and children with ASD. Figures 5A - 5BShows the analysis results of entropy fluctuations in autism in male and female children. The analysis of outliers from TD indicates that the feedback control of the methylation regulatory mechanism may fail in these individuals. In other words, the analysis of entropy fluctuations reveals the existence of unknown clinical conditions in the development of so-called "healthy" individuals.

[0037] Figure 5A Relates to male children. The range of entropy fluctuations in the TD samples is highlighted by the horizontal shaded band.

[0038] Figure 5B Relates to female children. For visual comparison, Figure 5B the horizontal shaded band in Figure 5A is set to cover the same range as in

[0039] One of ordinary skill in the art does not need to view the combinations of the nearly infinite different features described in the following detailed description in isolation in one or more figures to facilitate understanding of the present disclosure. Detailed Description

[0040] The present disclosure is not limited to what is described herein. Mechanical, electrical, chemical, procedural, and / or other changes can be made without departing from the spirit and scope of the present disclosure. Unless otherwise stated, the features shown or described are not essential for allowing the basic operation of the present disclosure.

[0041] A graphical summary of the theoretical approach in the current work is shown in FIG. 1. As shown, the basic knowledge of the natural variability of methylation signals in the control group and the treatment group comes from their probability distributions. The DNA methylation kinetics as a biological system follows the principles of thermodynamics. In biochemical terms, DNA methylation changes are biochemical reactions achieved by two classes of enzymes: methyltransferases and demethylases. These enzymes as molecular machines achieve methylation changes through several steps of logical operations. According to Landauer's principle, the minimum energy dissipation ε = k B Tln2 per bit of information for each machine operation; at human body temperature, 310.15 K: ε = 1.784 Jmol -1 . Therefore, according to Landauer's principle, any methylation change involves an associated amount of energy dissipation E ≥ k B T ln 2 per bit of information for each machine operation, where k B represents the Boltzmann constant and T represents the absolute temperature.

[0042] Statistical Physics Modeling of Methylation Background Processes

[0043] The most probable distribution of the methylation states on DNA molecules driven by spontaneous / stochastic fluctuations can be obtained by maximizing the thermodynamic entropy under general system constraints: i) ∑ i π i = 1 and ii) ∑ i π i E i = <e>, where π i is the (discrete) probability of observing dissipation of the energy value E i , and <e>is the mathematical expectation of E. Under these assumptions, Jaynes' maximum entropy principle (MEP) makes the Boltzmann distribution the most probable distribution of the system. Assume the energy E dissipated to reach the state i of the system i is actually a continuum with a certain density of states with methylation changes and the dissipated energy E, the probability of observing genome-wide energy dissipation between 0 and E can be estimated as:

[0044]

[0045] where represents the partition function of the system, and β = k B T is a scaling constant. That is, the number of methylation changes per unit energy at E(A(E,β,…)dE) is the number of methylation changes of the energy dissipated per bit of information in the infinitesimal range E to E + dE. In the probability of observing genome-wide energy dissipation between 0 and E, by definition, the expression under the integral together with the partition function is the probability density function expressed as follows:

[0046]

[0047] Note that for A(E,β,…) = 1, the last equation simplifies to the classical expression of the Boltzmann distribution. The probability density function is a general probability model of the methylation background process, which conforms to the exponential decay law. According to the probability density function, for any specific case of f(E|β,…), it is expected that the probability of observing methylation changes will decrease as the amount of energy dissipated per bit of information processed by the molecular machine (enzyme) increases. In the following sections, the information-thermodynamic constraints on the molecular methylation mechanism allow for maximum likelihood estimation of specific cases of the function f(E|β,…).

[0048] Channel Capacity of Methylation Mechanisms

[0049] The basic constraints for deriving the probability density function of DNA methylation changes involve the physics of information in molecular machine operations. The machine capacity is closely related to Shannon's channel capacity, that is, the maximum amount of information that a molecular machine can gain per operation. According to Schneider, the machine capacity is constrained as follows: where P y is the energy dissipated by the molecular machine, N y is the energy of thermal noise, and d 空间 represents the number of independent moving parts of the molecular machine involved in the operation.

[0050] According to Shannon, the received signal has an average energy E y = P y +N y and E 0 = N y , to indicate that with probability = 1 and d 空间 = v - 1 arrives at the energy dissipated at, which means that:

[0051]

[0052] Derivation of the Probability Density Function of Methylation Background

[0053] Search for a probability density function (PDF) from the family of generalized gamma (GG) distributions. The following assumptions must hold:

[0054] Let N i be the number of times the amount of energy E is dissipated in the interval [E i-1 , E i ).

[0055] Consider the number k of such events: N1, N2,..., N k , where

[0056] Let p i be the probability that the amount of energy E is dissipated in the interval [E i-1 , E i ), where

[0057] The number of ways in which a set {N i | i = 0,..., k} can be realized in a sequence of length N is given by the multinomial coefficient: And the probability of each sequence of events is: Therefore, N distinguishable methylation events produce N1 outcomes with energy dissipated in the interval [E0, E1), N2 outcomes with energy dissipated in the interval [E1, E2), and N k-1 , E k ) and the probability of the N k outcomes is given by the multinomial distribution:

[0058]

[0059] Therefore, the most probable distribution of the methylation state in the system (DNA molecule) is determined by a set of values {N i} and {p i} and the constant N, which in the current case is the number of cytosine sites in the DNA molecule. The continuous action of the second law of thermodynamics is expected to maximize the Boltzmann entropy within each cell, which in turn gives rise to the most probable density of the methylation state. Therefore, for a fixed set of probability values {p i}, N i value Maximize the probability P. Specifically, for p i and N i the following requirements are imposed, which are repeated here for better understanding:

[0060] 1) The probability p i is proportional to a specified power of the energy E i :

[0061]

[0062] where E 0 represents the energy dissipated with probability 1.

[0063] 2) For each choice of α, the following sum is a positive constant:

[0064]

[0065] where E > 0; N i is assumed to be a large number.

[0066] Solve this problem by writing the Lagrangian which consists of the logarithm of P and a linear combination of the relevant constraints:

[0067]

[0068] Using Stirling's formula to approximate the factorial for large N and setting the derivative yields the equation:

[0069]

[0070] After setting the equation can be used to obtain the expressions:

[0071]

[0072] and

[0073]

[0074] The last equation allows updating the discrete probability distribution π i under the assumptions 1 (the probability p i is proportional to a specified power of the energy E) and 2 (for each choice of α, the following sum is a positive constant): i :

[0075]

[0076] That is to say, can be interpreted as a discrete probability distribution associated with an energy dissipation event under system constraints 1 and 2. Alternatively, the above equation can be written as:

[0077]

[0078] After setting λ = β α and δ = v / α, the said equation can be rewritten as:

[0079]

[0080] After setting k → ∞ in the said equation, the size of the interval ΔE → 0 and the sum in the bracketed coefficient can be replaced by a definite integral, and the said definite integral gives:

[0081]

[0082] And thus:

[0083]

[0084] Therefore, the integration from 0 to E on the right side of the above equation satisfies the equation for determining the probability of observing genome-wide energy dissipation between 0 and E. Assuming that the energy E i dissipated to reach the state i of the system is actually a continuum, the probability density function expressed as f(E|β,…) becomes the generalized gamma probability density function f(E|β,δ).

[0085] Probability Density Function of Methylation Background Variation

[0086] The probability of observing genome-wide energy dissipation between 0 and E and the probability density function quantitatively summarize the statistical physics underlying the methylation changes not induced by the methylation mechanism. Applying the principles of thermodynamics to chromatin dynamics often maximizes the Boltzmann entropy, thereby generating the most probable methylation density state. Seeking to maximize the probability: P = (N1,…,N k ,N,p1,…,p k ), that is, N distinguishable methylation events occur with probabilities p1,…,p k to produce in the intervals [E1,E2),…,[E k-1 ,E k ) the results.

[0087] Two basic assumptions are imposed on p i , N i and E i :

[0088] 1) The probability p i is proportional to a specified power of the energy E i :

[0089]

[0090] 2) For each choice of α, the following sum is a positive constant:

[0091]

[0092] where E > 0; N i is assumed to be a large number.

[0093] The first hypothesis is derived from the current interpretation of the channel capacity of the molecular machine, as given by the channel capacity of the methylation mechanism. The second hypothesis implies that the parameter α carries information about the molecular machine, since v = αδ. On a thermodynamic basis, applying the Lienhard and Meyer approach to the specific scenario of DNA methylation provides the maximum likelihood estimate of the function f(E|β,…) in the section: Derivation of the Probability Density Function of Methylation Background , ibid. The above hypotheses yield the generalized gamma probability density function:

[0094]

[0095] where α > 0, β > 0, δ > 0, and E > 0. Consistent with the probability density function, the analytical expression for the partition function is derived from the generalized gamma probability density function:

[0096]

[0097] Therefore, the density A(E,β,…) can be expressed as:

[0098]

[0099] The information - theoretic divergence χ(p,q) of the methylation levels p and q will follow a distribution (generalized gamma, gamma, or Weibull distribution model) derived from the probability of observing the genome - wide energy dissipation between 0 and E, provided that the probability is proportional to the energy E. In this case, the dissipated energy E is the information per bit associated with the corresponding methylation change. For the information - theoretic divergence measure of the methylation level χ(p,q), the same analytical steps are followed to derive the generalized gamma probability density function (see section: Derivation of the Probability Density Function of Methylation Background , ibid), resulting in the probability density function for the information divergence χ(p,q):

[0100]

[0101] Hypothesis (where χ is in bits), the dissipated energy can be estimated as:

[0102]

[0103] For a molecular machine operating under ideal conditions, according to Landauer's principle, the minimum dissipated energy is E = χk B Tln 2, or θ = 1 / ln 2 under ideal conditions. A more general distribution including the position parameter μ is given by:

[0104]

[0105] The mean value is:

[0106]

[0107] And the variance is:

[0108]

[0109] Specifically, χ(p,q) can be expressed in terms of the Hellinger divergence given by Sanchez et al., "Discrimination of DNA Methylation Signals from Background Variations for Clinical Diagnosis", or in terms of the J-divergence. After performing goodness-of-fit tests on processed bisulfite sequence datasets from different species, it was found that the most common members of the generalized gamma distribution family are the Weibull distribution (δ = 1) and the gamma distribution (α = 1), which are obtained from the generalized gamma probability density function as specific cases. See Sanchez et al., "Integrative Network Analysis of Differentially Methylated and Expressed Genes for Biomarker Identification in Leukemia", Sci. Rep. 10, 2123 (2020), which is incorporated herein by reference in its entirety.

[0110] Association with Shannon Communication Theory

[0111] As the present inventors have pointed out in one or more previous reports, the genome-wide patterning of cytosine DNA methylation occurs at specific landmarks, which statistically implies the existence of some methylation language / code. See, e.g., Sanchez et al., "Genome-wide discriminatory information patterns of cytosine DNA methylation", Int. J. Mol. Sci. 17, 938 (2016), which is hereby incorporated by reference in its entirety. According to Shannon's communication theory, a communication system can be described by the conditional probability (density) P x (y) such that if the message x is generated by the source, then the message recovered at the receiving point will be y. Shannon defined the rate R1 of generating information for a given quality v1 = ∫∫ρ(x,y)P(x,y)dx dy of the replica as a function of fixed ν1 and variable P x (y) of where ρ(x,y) is a distance function. Shannon showed that the function ρ(x,y) behaves as the "distance" between x and y, measuring the improbability of receiving y given that x was transmitted, based on a fidelity criterion.

[0112] In Shannon's analysis, the conditional probability P y (x) that minimizes the rate R is given by the expression P y (x) = B(x)e -λρ(x,y) where B(x) is chosen to satisfy ∫B(x)e -λρ(x,y) dx = 1. In the function B(x), the transmitted message x can be represented at each cytosine site by the methylation level observed in the treatment or patient group. The methylation level is estimated as: where and nC i are the number of methylated and unmethylated cytosines observed at site i, respectively. The received message y can be specified as a reference methylation level, which can be the center of the group control or estimated from an independent subset of control samples from the control population. Next, the function ρ(x,y) can be represented by the symmetric information divergence χ(x,y) between the methylation levels x and y. For a fixed reference y, the equation χ(x,y) = χ(x) makes it possible to choose χ(x) to be the following:

[0113]

[0114] where dχ = χ′(x)dx and λ = 1 / θ. The conditional probability P y (x) can be reinterpreted after changing the variable as:

[0115]

[0116] This equation shows the following probability, that is, if the message recovered at the receiving point is y, then the information divergence between y and the original message x generated by the source is X. Applying Shannon's reasoning leads to the following conclusion. If the organism's methylation system conforms to a communication system, then the methylation information transfer with the best coding is described by P y (X|α,δ,θ). This means that if the methylation signal is the message obtained in the framework of a communication system, then the probability can correctly describe the methylation information transfer, and thus all knowledge derived from it is not a mathematical forgery.

[0117] Gibbs Entropy of the System

[0118] The Gibbs entropy of the system generated by methylation changes is defined by the integral:

[0119]

[0120] The said equation yields the known analytical expression (see Materials and Methods for Calculation Presented in Table 1 , as follows):

[0121]

[0122] where represents the digamma function. After considering the generalized gamma probability density function, the said function can be written as:

[0123]

[0124] where is the classical entropy term, and is the contribution from the moving parts of the molecular machine. Therefore, the entropy of a single methylation system is divided into a classical term and the contribution from the moving parts of the molecular machine:

[0125] S = S 经典 + S 机器 .

[0126] Based on the probability density function of information divergence, a rough estimate of the Gibbs entropy for different tissues / cells can be based on the information divergence χ i after expressing the energy E i with χ i :

[0127]

[0128] where the term It is a function of the model parameter related to the number of independent moving parts of the molecular machine (v = αδ).

[0129] Since log2x = lnx / ln 2, the Gibbs entropy for different tissues / cells can be written as:

[0130]

[0131] The term within the parentheses from the Gibbs entropy S (under constant temperature conditions) corresponds to the Shannon entropy H, in which case the Shannon entropy depends only on the distribution parameters estimated from the experimental data of each individual. Therefore, the Shannon entropy H can be written as:

[0132]

[0133] and

[0134] S = k B ln2H.

[0135] According to Schneider, for the decrease in methylation group entropy:

[0136] ΔS = S 之后 - S 之前 ,

[0137] The uncertainty of the change in genome-wide methylation will also correspondingly decrease:

[0138] ΔH = H 之后 - H 之前 .

[0139] After this uncertainty reduction, the methylation group gain information I m :

[0140] I m = -ΔH

[0141] Expressed in joules per kelvin as:

[0142] I m ≡ -k B ln2ΔH.

[0143] Therefore, although the information-theoretic entropy and the thermodynamic entropy are independent functions, they produce the same result up to the product of the Boltzmann constant and ln 2.

[0144] If I m > 0, then the methylation system has experienced information gain, otherwise if Im < 0, then the system has experienced information loss.

[0145] Thermodynamic Potential of Methylation Changes

[0146] Assuming that there is a balance between the methylation and demethylation processes along each DNA molecule, the total mass (number of molecules N) and volume (V) of the DNA molecule remain constant. This assumption holds in most experimental datasets because for a large enough genomic region, the sum of methylation level differences approaches zero. Under these conditions, and assuming a constant temperature (T), the methylation changes in the DNA molecule and its surrounding microenvironment can be considered a closed system that only undergoes mass transfer and not energy transfer. In statistical physics, these systems are called NVT systems, and the thermodynamic variables N, V, and T remain constant. The Helmholtz free energy (F) represents the driving force for an NVT system and is the thermodynamic potential that measures the "useful" work that can be obtained from a closed system at constant temperature and constant volume.

[0147] The Helmholtz free energy can be estimated from its definition: F = U - TS. Assuming that the internal energy U of the system is not changed by the operation of the molecular machine, it becomes: ΔF = -TΔS or

[0148]

[0149] The same result can be derived from the definition of the Gibbs free energy: G = H - TS. Given that the system pressure is not changed by the operation of the molecular machine (ΔH = 0): ΔG = -TΔS, the change in the Helmholtz free energy roughly estimates how much Helmholtz free energy will be involved in methylation. A rough estimate χ based on the information divergence can be used in this method:

[0150]

[0151] where β = k B T. From the entropy of a single methylation system, the Helmholtz free energy can be divided into a classical term and the contribution of the moving parts of the molecular machine:

[0152] ΔF = -βS 经典 -βS 机器 = ΔF 经典 +ΔF 机器 ,

[0153] where according to the analytical expression for the partition function: For S used in the Weibull distribution and the gamma distribution G and the specific cases of F(β) are obtained with parameter values δ = 1 and α = 1 respectively. Substituting the Gibbs entropy for different tissues / cells in the change in the Helmholtz free energy gives:

[0154] ΔF = -βln2H.

[0155] At constant temperature, ΔF decreases as the Shannon entropy in the system increases. The variation in the Helmholtz free energy between two system states (before and after) can be expressed as:

[0156] ΔΔF = ΔF 之后 -ΔF 之前

[0157] Alternatively, consider the methylome gain information:

[0158] ΔΔF = TI m

[0159] where the information loss (I m <0) will be associated with the free energy loss ΔΔF < 0.

[0160] Biological Applications

[0161] Entropy is a thermodynamic state variable of a system, which means that the value of the entropy is completely determined by the current state of the system, rather than by how the system reaches that state. Thus, of interest is to understand whether the entropy of the methylation system measured relative to a reference state is consistent with the observable phenotypic changes. Functions for entropy and Helmholtz free energy estimation, given respectively by a crude estimate of the Gibbs entropy and the change in the Helmholtz free energy for different tissues / cells, are included in the MethylIT platform (R software package). Entropy was estimated in a single-seed descent population of six consecutive generations of Arabidopsis Col-0 isolates (accession) (wild-type control, WT), the methyltransferase mutant met1, and the heritable epigenetic memory (mm) and full-sib non-memory (nm) lines derived from msh1 grown under two different growth conditions.

[0162] In Arabidopsis thaliana, seedlings grown in vitro and in pots under short and long day lengths may experience a huge impact on plant growth. Therefore, methylation group results were compared between wild-type Col-0 2-week-old seedlings grown on nutrient medium under a 24-hour day length and mature plants grown on potting medium under a 12-hour day length. CG methylation in plants is maintained by METHYLTRANSFERSE1 (MET1), and mutations that disrupt its activity induce genome-wide CG hypomethylation. Data from this mutant to be tested were used for the observable information loss in met1 plants grown under the same conditions relative to the wild type (34). In terms of the msh1 memory state, heritable epigenetic stress memory occurs after RNAi inhibition of the MutS HOMOLOG (MUTS) gene in Arabidopsis thaliana, such that approximately 20% of the RNAi transgenic null offspring have a heritable memory phenotype of delayed maturation and persistent stress response (mm), and the phenotypes of the remaining part remain unchanged and are called "non-memory" (nm). The six-generation pedigree of msh1 memory has been previously described, and both the first-generation memory (mm1) and non-memory (nm1) full-sibling types showed evidence of genome-wide cytosine methylation remodeling relative to the wild type. Here, the analysis included samples from the six-generation msh1 memory pedigree, and it was predicted that the impact of these variants on entropy variation would be greater than that of the met1 mutant. The results of the first generation (mm1, nm1) and the third generation (mm3) shown in Table 1 confirmed these predicted results.

[0163] Table 1: Estimation of Gibbs entropy for msh1 epigenetic memory (mm1, mm3), non-memory (nm), met1 mutant, and corresponding Col-0 control (WT) in Arabidopsis thaliana.

[0164]

[0165] Entropy values were estimated using Gibbs entropy S and J-divergence. After replacing the Boltzmann constant with the gas constant, the values were given in J×K -1 ×mol -1 . Information loss (I m ) was given by I m = -k B ln 2ΔH. Helmholtz free energy (ΔΔF) values were estimated using ΔΔF = TI m and J-divergence. The values were given in kJ×mol -1 . The symbols ‘**’ and ‘***’ represent highly statistically significant differences at p-value < 0.01 and p-value < 0.001, respectively, between the mutant state or memory state. The symbol represents the Wilcoxon paired test, otherwise the linear mixed model was used for testing.

[0166] The effect of the msh1 mutant strain on genome-wide methylation changes in epigenetic memory and non-memory progeny is reflected in a discrete increase in entropy and thus information loss: ΔS = S 对照 - S 突变体 < 0, i.e., I m < 0. This observation further demonstrates the epigenetic effect of generating a memory state. A significantly greater information loss was observed in the met1 mutant, which is consistent with the significant effect of genome-wide CG demethylation.

[0167] The results show that entropy is also a highly sensitive measure of the state of an organism. There are significant differences in the entropy values of the Col-0 wild-type controls WT3 and WT met1 . Although these wild-type control groups are derived from the same Arabidopsis isolate, there are differences in ontogeny among the control groups. WT met1 plants were grown in half-strength Gamborg's B5 medium under continuous light for two weeks, while WT3 plants were grown to maturity on a standard peat mixture in pots, maintaining a twelve-hour (12 h) day length and sampled at the bolting stage. These differences in plant stage and growth conditions are responsible for the observed significant entropy differences.

[0168] In human studies, the Gibbs entropy for different cancer cells and corresponding healthy tissue / cell controls is presented in Table 2.

[0169] Table 2: Gibbs entropy estimated in human cancer cells and corresponding normal tissues.

[0170]

[0171] Energy values were estimated using S = S 经典 + S 机器 and J-divergence. Values are given in J × K -1 × mol -1 . Human embryonic stem cell (HESC) values are provided as a reference for undifferentiated tissue.

[0172] The results show that the Gibbs entropy of all cancer cells increases relative to the corresponding normal tissues. Since the information divergence is calculated relative to the same reference individual, the observed entropy values indicate that breast cancer metastatic cells experience the most severe information loss (assuming that experimental errors are not large enough to affect the estimated values). The relationship between Gibbs entropy and Helmholtz free energy predicts the results shown in Table 3.

[0173] Table 3: Helmholtz free energy estimated in cancer cells and corresponding normal tissues.

[0174]

[0175] The energy values are given in kJ×mol -1 and the human embryonic stem cell (HESC) values are provided as a reference for undifferentiated tissue.

[0176] After methylation reprogramming that transforms differentiated healthy cells into a cancer cell state, the information potential of cancer cells seems to drop sharply relative to healthy tissue cells. These data reflect an important means of assessing the state of a biological system that has not been previously documented.

[0177] A large value ν = <χ> of the information divergence χ indicates a large amount of (genome-wide) energy dissipated by the methylation mechanism. Then, the physical work accomplished by the methylation mechanism necessarily results in a reduction of genome-wide methylation uncertainty, which must be reflected in the (dimensionless) entropy value. The regression analysis implemented on Arabidopsis thaliana and human datasets relative to ν supports this conjecture ( Figures 2A - 2B ). The final result enables an empirical estimate of the entropy fluctuations through regression analysis relative to e -ν ( Figure 2C -D), from which the equation is obtained:

[0178]

[0179] The regression analysis through Arabidopsis thaliana and human datasets relative to e -ν verifies the last equation ( Figures 2C - 2D ). The equation can be written as a quotient:

[0180]

[0181] This is another way of expressing the fluctuation theorem in the context of DNA methylation. The model parameter η is an experimentally measurable quantity that characterizes the effect of feedback control.

[0182] As Figure 2C and Figure 2D shown, larger values of the model parameter η are associated with a better feedback control effect, which is reflected in a better fit of the model to the dataset.

[0183] The model validity implies the validity of the model within experimental error, which is derived by introducing e -ν ≈ 1 - v into the initial model.

[0184] The results from regression analysis in Arabidopsis thaliana and human datasets relative to v support this expression ( Figures 2E - 2F )。This coincidence is found in the Arabidopsis mutant met1( Figure 2E dots with striped patterns in the sub - figure), breast cancer (dots with right - handed striped patterns), and stem cells (dots with left - handed striped patterns, Figure 2F ) except for the extreme cases found in

[0185] When considering the average value of the sum of the Boltzmann factor and e -ν the result shows that the average sum seems to be constant ( Figure 3A -B). Specifically, no statistical difference is found between the overall average values of the values from Arabidopsis( Figure 3A ) and humans ( Figure 3B ), which gives rise to a hypothesis:

[0186]

[0187] where λ = 1, or has a value close to 1. Thus, it can be written and has feedback control: This gives rise to the equation previously derived along an independent reasoning path:

[0188]

[0189] Data and the R script for construction Figure 3B are given in the following example.

[0190] Similarly, only extreme conditions do not conform to the model. From a biological perspective, the derivation of the quotient implies that the amplitude of genome - wide methylation changes resulting from the response to environmental changes is limited. Figure 2A -F and 3A - 3B show that only the extreme cases found in Arabidopsis met1 and cancer cells are unrestricted.

[0191] The results of the estimation of the Gibbs entropy for each chromosome from the control group and patients with autism are shown in the box plots from Figures 4A - 4B . For both groups of males and females, there are significant statistical differences in each chromosome between the TD group and the ASD group. However, the box plots also show the existence of atypical individuals, which in turn indicates the existence of structured groups where ASD individuals experience different severities of the disorder.

[0192] Since the entropy variation ΔS = S ASD -S TD An estimate of the average information gain or loss (I = -ΔS) is provided, and the block diagram also shows a significant statistical information loss (I < 0) (on average) in the ASD group (higher entropy value) compared to the TD group (lower entropy value). Fundamentally, the information loss experienced by ASD tissue cells translates into a loss of methylation regulatory signals that are typically visible in healthy individuals.

[0193] The resolution power of genes estimated from the identified important enriched gene networks (as described in the authors' previous works) together with the entropy estimate can provide an assessment of the disease severity at a specific stage. For example, among two patients carrying similar affected (basic) enriched gene networks, the patient with lower Gibbs entropy experiences less information loss and is thus expected to have a lower disease severity in such individuals.

[0194] Notably, depending on the individual and the disease, information loss is associated with either a negative or positive impact on the individual's health status. The information loss observed in patients with cancer or autism indicates a (general) negative impact on the patient's condition. However, some children with ASD may carry out methylation reprogramming of gene networks associated with mathematical intelligence performance, enabling the patient to have good mathematical skills, as shown by the network analysis of the ASD group (data not shown).

[0195] Figures 5A - 5B Shows the analysis of random fluctuations in TD and ASD children. As Figures 5A - 5B shown, it must be expected (depending on the tissue) that the feedback control from the methylation regulatory system should control the range of entropy fluctuations induced by external forces within one. As Figures 5A - 5B shown, there is a statistically significant difference between the entropy fluctuations from the TD group and the ASD group. However, the outliers in the TD group indicate the potential impact of clinical symptoms that were not detected in the evaluation tests for ASD many years after the birth of the children. In other words, since the entropy fluctuation analysis is not limited to any specific disease, the outliers observed in the TD ("healthy") group would be signals of developing unknown clinical conditions in individuals carrying extreme entropy fluctuations.

[0196] Discussion

[0197] Therefore, a theoretical premise for explaining the behavior of DNA methylation variation is proposed. The results shown describe the information thermodynamics of cytosine methylation, far beyond the simple application of the probability density function of information divergence used as the null hypothesis in methylation analysis. The results confirm that, as given by the generalized gamma probability density function, members of the generalized gamma probability distribution family quantitatively summarize the statistical physics underlying spontaneous methylation variation driven by random fluctuations. The parameters from the generalized gamma probability density function carry information about the channel capacity of molecular machines related to the Shannon capacity theorem.

[0198] According to Shannon's communication theory, the probability density function of information divergence can be interpreted as a conditional probability density distribution. The conditional probability interpretation of methylation given by the conditional probability P y (x) assumes that the message remains unchanged in the control population and that, under environmental changes or disease conditions, changes in the message occur in some subpopulations, as reflected in treatment or patient datasets.

[0199] Consistent with Shannon's theory, in the case of optimal coding, the conditional probability density P y (χ) indicates that if the message recovered at the receiving point is y, then P y (χ) will decrease exponentially with the information divergence χ(x,y) between y and the message x generated by the source. If DNA methylation conforms to a communication system, then the optimal coding of methylation messages is described by the probability density function for information divergence as a logical application of Shannon's results.

[0200] Methylation changes supporting DNA thermal stability are expected to occur with the highest frequency and have relatively small divergence values ( Figure 1B the left side of the probability density function in α＝0.05 ). The observed data from the control population show information divergence values χ(x,Y): χ(x,y) < χ 截止 ≤ χ Figure 1B smaller, thus representing background "noise" in the system. As 截止 shown, most of these values χ(x,y) > 0 also conform to the inequality χ(x,Y) ≤ χ 截止 , where χ Figure 1B is some estimated value used to minimize the false positive rate when evaluating differentially methylated positions (

[0201] the right side of the curve), representing treatment-related variation. In practice, machine learning methods can be used for such estimation.Methylation messages are encoded within the mechanical properties of DNA molecules. For example, the flexibility or rigidity of the DNA double helix is necessary for regulating nucleosome folding and the binding of transcription factors (TFs) to DNA sequence motifs. Depending on the DNA sequence context, adding or removing a methyl group to a cytosine nucleotide is expected to alter these local physical properties.

[0202] The Gibbs entropy and Helmholtz free energy, as given by S = S 经典 + S 机器 and ΔF = -βS 经典 -βS 机器 = ΔF 经典 + ΔF 机器 show that there are substantial differences between the classical statistical mechanics and statistical biophysics of the methylation process. The entropy contributions from molecular machines (enzymes) via conformational changes are included in the function φ(α, δ). Applying these equations to experimental datasets can provide important biological insights. The results shown in Table 1 indicate that, as thermodynamic state variables, the states of the methylation systems estimated by the entropy driven by S = S 经典 + S 机器 are consistent with the phenotypic observations. Epigenetic memory lines in Arabidopsis thaliana have an increasing effect on the information loss observed in nm1 to mm3. In contrast, the met1 mutants that undergo genome-wide loss of CG methylation provide a reference for extreme methylation changes and information loss (Table 1).

[0203] The results presented in Table 2 show that the Gibbs entropy estimated from the methylome data of patients with different cancer types is close to or even higher than the value of the Gibbs entropy in embryonic stem cells. DNA methylation remodeling seems to cause a related increase in the system entropy, making cancer cells closer to undifferentiated somatic cells. The transformation of normal cells into cancer cells results in an increase in entropy and thus information loss S = S 正常细胞 - S 癌细胞 . Biological evidence also shows that when cancer stem cells (subpopulations within the tumor mass) are derived from cancer cells, information from the original tissue is lost. The results from Tables 1 and 2 are consistent with these known effects, and the observations from this study indicate that the transformation of healthy cells into cancer cells disrupts the natural methylation programming and generates a genome-wide disordered state.

[0204] Experimental verification of each equation in this section in methylome datasets from human and Arabidopsis thaliana chromosomes is important for understanding the DNA methylation process. These equations show that the fluctuations of methylation satisfy specific constraints expressed in the quantitative relationship between the amplitude of the entropy fluctuations and the expected value of the J - information divergence. The mean v and variance σ 2 of the J - information divergence are given by the conditional probability density Π y The integration of the parameter values of (χ) is determined, which is consistent with Shannon's theory. Thus, the fluctuation constraints revealed by the equations in this section involve maintaining the fidelity of the methylated message at the receiving point with optimal coding. Such a result presumably allows sufficient variation in the methylation signal to ensure the organism's adaptation to environmental changes.

[0205] Samples reflecting extreme scenarios in Arabidopsis thaliana (met1) and humans (breast cancer, metastasis, and stem cells) do not conform to the model shown by the equations in this section (Figures 2 and 3). The met1 mutation causes almost complete loss of CG genomic methylation and massive ectopic hypermethylation of CHG and CHH genes and transposable elements. Similarly, methylation reprogramming in cancer cells results in significant information loss, as indicated by the results shown in Table 2. However, the situation in embryonic stem cells seems to be very different from that in met1 and cancer cells, because DNA methylation may not be necessarily required in this cellular context. Even when CG methylation is completely lost by jointly disrupting the three DNA methyltransferases Dnmt1, Dnmt3a, and Dnmt3b, the change in cell phenotype is minimal. As pluripotent undifferentiated cells, stem cells may not need to maintain a specific epigenomic state before differentiating into specialized tissues.

[0206] Although the main purpose of this study is to lay a theoretical foundation for understanding the behavior of DNA methylation, the research results suggest that the practical significance of entropy estimation may prove to be of great importance for early diagnosis and assessment of changes in the epigenetic state. These data in plant and human diseases indicate that it may be feasible to detect transitions in the epigenomic state, such as those occurring during a plant's response to environmental changes or the transformation of human precancerous cells, based on the physical-informational chromosomal data types provided.

[0207] Examples

[0208] Biological Experiment Datasets

[0209] The Arabidopsis thaliana methylome dataset (reported in Table 1) was derived from whole-genome bisulfite sequencing of samples from msh1 memory (generations 1 - 6) and non-memory (generation 1) sibling plants (5 plants / generation) and isogenic Col-0 wild-type controls (5 plants). The dataset was downloaded from the Gene Expression Omnibus (GEO) series GSE129303a and GSE118874.

[0210] The methylation datasets for met1 mutants and corresponding wild types (3 samples each) were obtained from the GEO series GSE122394. Fastq files from the Arabidopsis thaliana methylation met1 mutant and corresponding wild type datasets were downloaded from the European Nucleotide Archive (ENA, https: / / www.ebi.ac.uk / ena / browser / home). The raw read counts of methylated and unmethylated cytosines for met1 for additional methylation analysis were obtained as follows: The raw sequencing reads were quality controlled with FastQC (version 0.11.5), trimmed with TrimGalore! (version 0.4.1) and Cutadapt (version 1.15), and then aligned to the TAIR10 reference genome using Bismark (version 0.19.0) and bowtie2 (version 2.3.3.1). The deduplicate_bismark function in Bismark with default parameters was used to remove duplicate reads, and reads with coverage greater than 500 were removed to control for PCR bias. Methylated Cs (COV files) were obtained from the Bismark methylation extractor with default parameters.

[0211] The human cancer and healthy tissue control datasets (Table 2) were downloaded from the GEO series GSE52271. Blood B cell CD19 (GSM1279518) was used as a reference for calculating the information divergence J-divergence (JD). The Bi-seq datasets of primary human pluripotent cells have GEO accession numbers: GSM2041690, GSM2041691, and GSM2041692. More detailed descriptions of these datasets are given below.

[0212] Materials and Methods for Calculating the Results Presented in Table 1

[0213] 1. Hellinger information divergence and J information divergence of methylation levels

[0214] The methylation level p of a single i-cytosine site j ij corresponding to the probability vector p ij =(p ij , 1 - p ij ). Then, the information divergence between methylation levels. The methylation levels p 1 j and p 2 j at site j from individuals 1 and 2 are the vectors p 1 j=(p 1j , 1 - p 1j ) and p 2 j=(p 2j , 1 - p 2j ) the divergence between. If a vector of coverage is supplied, then the information divergence is estimated according to the following formula:

[0215]

[0216] This formula corresponds to the Hellinger divergence, as given by the inventors of the present disclosure in the first formula of Theorem 1 in the following reference: Kundariya, H. et al., "MSH1-induced heritable enhanced growth vigor through grafting is associated with the RdDM pathway in plants." Nat Commun 11, 5343 (2020), which is hereby incorporated by reference in its entirety. Otherwise:

[0217]

[0218] It is worth noting that several filtering conditions are provided to select cytosine positions with biological significance, which avoids experimental errors in downstream analysis. By filtering the reads, poor-quality data can be removed, which is at the edge of experimental errors generated by BS-seq sequencing. The user can check whether the cytosine positions used in the analysis have biological significance. For example, the counts of cytosine positions in the'ref' sample are mC1 = 10 and uC1 = 20, and mC2 = 1 and uC2 = 0 in the 'indv' sample will respectively produce methylation levels p1 = 0.333 and p2 = 1, and TV = p2 - p1 = 0.667, which clearly indicates a hypermethylated site. However, there are not enough reads to support p2 = 1. The Bayesian estimation of TV will disclose that this site is actually hypomethylated. Therefore, the best practice is to remove such sites. Remove this specific case under the default settings: minimum coverage = 4, minimum methylated = 4, and minimum unmethylated = 0 (see the uniqueGRfilterByCov function instance, which is called by "estimateDivergence" described in the subsequent section: as follows).

[0219] 2. J-divergence

[0220] Given the methylation levels from two individuals at a given cytosine site, this function calculates the J-information divergence (JD) between the methylation levels. The motivation for introducing JD in Methyl-IT can be seen in the items listed below:

[0221] 1. The divergence is the symmetric form of the Kullback - Leibler divergence: D KL (P‖Q) Kullback and Leibler themselves actually defined the divergence as:

[0222]

[0223] The divergence is symmetric and non - negative, where the probability distributions P and Q are defined on the same probability space.

[0224] 2. Generally, JD is highly correlated with the Hellinger divergence, which is the main divergence currently used by Methyl - IT (see the following example of the function estimateDivergence):

[0225]

[0226] The estimateDivergence function prepares data for estimating the information divergence and acts as a wrapper for calling the function that calculates the selected information divergence of the methylation level. In downstream analysis, the probability distribution of the given information divergence is used in Methyl - IT as the null hypothesis of the noise distribution, which allows distinguishing the methylation regulatory signal from the background noise in an additional signal detection step. For the current version, two information divergences for calculating the methylation level are calculated by default: 1) the Hellinger divergence (H) and 2) the total variation distance (TVD). In the context of methylation analysis, TVD corresponds to the absolute difference in methylation levels. Here, although the reported variable is the total variation (TV), the variable actually used in downstream analysis is TVD. Once the differentially methylated positions (DMPs) are identified in downstream analysis, TV becomes the standard metric for judging whether a cytosine position is hypermethylated or hypomethylated. Currently, an option for calculating the J - information divergence (JD) is provided. The motivation for introducing this divergence is given in the auxiliary of the function estimateJDiv.

[0227] 3. By construction, the measurement unit of JD is given in bits of information, which lays the foundation for additional information thermodynamics analysis.

[0228] The methylation level p at a given cytosine site i from individual j ij gives rise to the probability vector p ij =(p ij , 1 - p ij ). Then, the J - information divergence between the methylation level p ij (used as a reference individual) and q i =(q i , 1 - q i ) is given by the expression:

[0229]

[0230] The statistic obtained using the asymptotic chi-squared (χ 2 ) distribution is based on the statistic used for D KL . That is:

[0231]

[0232] where n1 and n2 are the total counts (coverage in the case of methylation) used to calculate the probabilities p ij and q i . To prevent zero counts, a basic Bayesian correction is added.

[0233] 3. Methylome data - human datasets

[0234] All methylome datasets were taken from the Gene Expression Omnibus (GEO) database.

[0235] b) The Bi-seq datasets from primary human pluripotent cells have GEO accession numbers: GSM2041690, GSM2041691, and GSM2041692.

[0236] c) Human cancer types and corresponding normal tissues with GEO accession numbers: GSE52271 and GSE56763.

[0237] 1. White matter of brain (GSM1279516)

[0238] 2. Glioma of brain (GSM1279532)

[0239] 3. Normal breast (GSM1279517)

[0240] 4. Breast cancer (GSM1279514)

[0241] 5. Breast metastasis (GSM1279513)

[0242] 6. Normal colon (GSM1279519)

[0243] 7. Colon cancer (GSM1279521)

[0244] 8. Colon metastasis (GSM1279520)

[0245] 9. Normal lung (GSM1279527)

[0246] 10. Lung cancer (GSM1279524)

[0247] 11. Normal blood B cells CD19 (GSM1279518)

[0248] Blood B cell CD19 samples were used as references for calculating information divergences: Hellinger divergence (HD) and J-divergence (JD).

[0249] Public raw datasets of methylation analysis by high-throughput sequencing from patients with autism spectrum disorder (ASD) and controls (typical development, TD) (accession number GSE178203) were downloaded from the Gene Expression Omnibus (GEO) and re-analyzed using MethylIT. The datasets consisted of placental tissues from 42 male individuals (20 TD and 22 ASD) and 23 female individuals (10 TD and 12 ASD). The raw data was initially reported in the published study from the reference.

[0250] Specifically, placental tissues were from patients with autism spectrum disorder (ASD) and controls (typical development, TD), with GEO accession number GSE178203.

[0251] a) TD male children

[0252] 1. GSM5381715

[0253] 2. GSM5381716

[0254] 3. GSM5381720

[0255] 4. GSM5381726

[0256] 5. GSM5381728

[0257] 6. GSM5381730

[0258] 7. GSM5381733

[0259] 8. GSM5381738

[0260] 9. GSM5381741

[0261] 10. GSM5381745

[0262] 11. GSM5381750

[0263] 12. GSM5381751

[0264] 13. GSM5381753

[0265] 14. GSM5381754

[0266] 15. GSM5381755

[0267] 16. GSM5381759

[0268] 17. GSM5381760

[0269] 18. GSM5381762

[0270] 19. GSM5381772

[0271] 20. GSM5381774

[0272] b) ASD male children

[0273] 1. GSM5381713

[0274] 2. GSM5381718

[0275] 3. GSM5381722

[0276] 4. GSM5381724

[0277] 5. GSM5381732

[0278] 6. GSM5381737

[0279] 7. GSM5381739

[0280] 8. GSM5381740

[0281] 9. GSM5381743

[0282] 10. GSM5381744

[0283] 11. GSM5381746

[0284] 12. GSM5381747

[0285] 13. GSM5381748

[0286] 14. GSM5381749

[0287] 15. GSM5381752

[0288] 16. GSM5381756

[0289] 17. GSM5381757

[0290] 18. GSM5381758

[0291] 19. GSM5381761

[0292] 20. GSM5381764

[0293] 21.GSM5381767

[0294] 22.GSM5381768 c) Female TD children

[0295] 1.GSM5381710

[0296] 2.GSM5381711

[0297] 3.GSM5381714

[0298] 4.GSM5381719

[0299] 5.GSM5381723

[0300] 6.GSM5381727

[0301] 7.GSM5381731

[0302] 8.GSM5381734

[0303] 9.GSM5381765

[0304] 10.GSM5381766 d) ASD female children 1.GSM5381712

[0305] 2.GSM5381717

[0306] 3.GSM5381721

[0307] 4.GSM5381725

[0308] 5.GSM5381729

[0309] 6.GSM5381735

[0310] 7.GSM5381736

[0311] 8.GSM5381742

[0312] 9.GSM5381763

[0313] 10.GSM5381769

[0314] 11.GSM5381770

[0315] 12.GSM5381771

[0316] 13.GSM5381773

[0317] 4. Methylation group data - Arabidopsis dataset

[0318] The Arabidopsis thaliana methylation dataset of msh1 memory and non-memory (normal appearance) sibling plants was derived from msh1 mutants. Basically, as described in reference (35) (main text), transgenic positive plants were self-pollinated, and the transgenes segregated in the offspring. Among the transgenic null plants, 20% of the plants showed delayed flowering, smaller size, and lighter green color, which is called the memory phenotype. The memory plants were self-pollinated for six generations, and each generation of plants was subjected to bisulfite sequencing. The datasets of generations 2-6 are available under the GEO accession numbers GSE129303 and GSE118874.

[0319] The RData files with J-information divergence and the best-fit models and corresponding goodness-of-fit used in all calculations reported in the main text were calculated using the J-divergence function outlined above.

[0320] 5. Methylation analysis

[0321] Methylation analysis was implemented using aspects of the MethylT R software package (0.3.2.2), which the inventors have described in technical literature incorporated by reference above, including but not limited to U.S. Provisional Patent Application Serial No. 63 / 323,690, Sanchez et al., "Differentiation of DNA Methylation Signals from Background Variations for Clinical Diagnosis", International Journal of Molecular Sciences 20, 5343 (2019), Sanchez et al., "Information Thermodynamics of Cytosine DNA Methylation", PLoS ONE 11, e0150427 (2016), and Sanchez et al., "Genome-Wide Resolved Information Patterns of Cytosine DNA Methylation", International Journal of Molecular Sciences 17, 938 (2016). The new R script with methylation analysis is available in the subsection starting with " R Script ".

[0322] Table 4: Best-fit PDF model parameters of the GG distribution for estimating Gibbs entropy and Helmholtz free energy.

[0323]

[0324]

[0325] Computational Tools and Statistical Analysis

[0326] The best non - linear fitting models of J - divergence, members of the generalized gamma distribution (probability density function of information divergence and a more generalized distribution including the location parameter μ), and the estimation of Gibbs entropy and Helmholtz free energy are realized using the MethylIT functions gibb_entropy and helmholtz_free_energy respectively. The estimation of the Boltzmann factor shown in Figure 2 is realized using the MethylIT function boltzman_factor. All R scripts for the results in Tables 1 - 3 with the following titles can be found in the above section: R Script for Analyzing Cancer Datasets , R Script for Analyzing Arabidopsis Datasets and R Script for Analyzing Entropy Fluctuations .

[0327] The group comparisons shown in Table 1 were realized in the lme4 software package (version 1.1 - 27.1), which applied a linear mixed model with a random effect of chromosome using the formula: entropy = group+(1|chromosome).

[0328] Table 5 applies the implementation of the numerical estimation of entropy that has been completed in an independent R software package, which is used to study generalized probability distributions. MethylIT - detected data (included in the MethylIT software package) is included in Table 5 and includes data related to the following: single control samples: C1, C2, C3, and treated samples: T1, T2, T3.

[0329] Entropy C1 C2 C3 T1 T2 T3 Theoretical Equations 3.941 3.918 3.929 4.457 4.469 4.515 Full - Digital Estimation 4.061 4.043 4.047 4.571 4.580 4.630

[0330] The difference between the results of the theoretical equation and the results of the all - digital estimation lies within the bounds of experimental error. That is, if someone applies some arbitrary theoretical information divergence to the methylation level and calculates the all - digital estimation of Gibbs entropy or Boltzmann entropy, then such a person / company will obtain results that mimic their entropy results up to the limit of the constant value.

[0331] R Script for Analyzing Cancer Datasets

[0332] This data set was downloaded from the Gene Expression Omnibus (GEO) to a local folder and read into R.

[0333] library(MethylIT)

[0334] 1. Read the data set

[0335]

[0336] 2. Hellinger divergence

[0337] Hellinger divergence estimation

[0338]

[0339]

[0340] To reduce the computational workload, only the J-divergence from chromosomes 7, 9, 17, and 22 was considered.

[0341]

[0342] 3. Nonlinear regression for JD. Only G gamma

[0343] The last file can be downloaded from GitLab using the following script:

[0344]

[0345]

[0346] The results from the nonlinear fitting can be downloaded from GitLab

[0347]

[0348]

[0349]

[0350] 4. Gibbs entropy

[0351]

[0352] 5. Helmholtz free energy

[0353]

[0354]

[0355] The energy unit is: J×mol -1 .

[0356] R Script for Analyzing Arabidopsis Datasets

[0357] The R script examples given here are limited to the 3rd generation but can be extended to all generations.

[0358] library(MethylIT)

[0359] library(MethylIT)

[0360] library(ggplot2)

[0361] library(ggpmisc)

[0362] library(dplyr)

[0363] For simplicity, only the wild-type third-generation control and the third generation of the memory line are analyzed here. The same R script applies to all sample groups.

[0364] If the read count dataset is available at the GEO database, the MethylIT function getGEOSuppFiles can be used to download the read count dataset from GEO. Users can usually download it manually by themselves and then read it into R using the function readCounts2GRangesList.

[0365] 1. Download files from GEO

[0366] Wild-type control samples:

[0367] wt3_files <- getGEOSuppFiles(GEO = c("GSM3704281","GSM3704282",

[0368] "GSM3704283","GSM3704284","GSM3704285"),

[0369] verbose = FALSE)

[0370] Third generation of the memory line:

[0371] mm3_files <- getGEOSuppFiles(GEO = c("GSM3704257","GSM3704258",

[0372] "GSM3704259","GSM3704260","GSM3704261"),

[0373] verbose = FALSE)

[0374] 2. Read the dataset

[0375] Read wild-type control samples:

[0376]

[0377] Read the third-generation dataset of the memory line:

[0378]

[0379] 3. Reference samples

[0380]

[0381] 4. J-divergence

[0382]

[0383] 5. Estimate the best - fit probability distribution model

[0384] Depending on the computing power, this step requires a rather long computing time. The RData file with the information divergence values can be downloaded from GitLab using the following script:

[0385]

[0386]

[0387] The best - fit probability distribution model can be found using the function gofReport:

[0388]

[0389] It can also be used for the memory strain dataset.

[0390] 6. Thermodynamic state variables

[0391] The Gibbs entropy can be estimated using the function gibb_entropy.

[0392] Wild - type entropy and Helmholtz free energy:

[0393]

[0394] The Gibbs entropy can be estimated using the function helmholtz_free_energy:

[0395]

[0396] Memory strain entropy and Helmholtz free energy:

[0397]

[0398] Gibbs entropy:

[0399]

[0400] R Script for Analyzing Entropy Fluctuations

[0401] Load the libraries and auxiliary functions required for the analysis.

[0402]

[0403]

[0404]

[0405] 1. Arabidopsis dataset

[0406] The Arabidopsis methylation datasets of msh1 memory and non-memory (normal appearance) sibling plants were derived from msh1 mutants. Basically, as described in reference (1) (main text), transgenic positive plants were self-pollinated, and the transgenes segregated in the offspring. Among the transgenic null plants, 20% of the plants showed delayed flowering, smaller size, and lighter green color, called the memory phenotype. Memory plants were self-pollinated for six generations, and each generation of plants was subjected to bisulfite sequencing.

[0407] The datasets of generations 2-6 are available through GEO accession numbers GSE129303 and GSE118874. The RData files with J-information divergence and the best-fitting model and corresponding goodness of fit used in all calculations reported in the main text are available at PSU GitLab: https: / / git.psu.edu / genomath / datasets.

[0408]

[0409]

[0410] 2. Plotting linear regression

[0411]

[0412]

[0413]

[0414] Regression analysis

[0415]

[0416]

[0417] 3. Cancer datasets

[0418]

[0419]

[0420]

[0421] 4. Plotting linear regression

[0422]

[0423] 5. Bar charts

[0424] Recombinant datasets:

[0425]

[0426]

[0427] Bar chart:

[0428]

[0429] Interference between breast cancer and stem cells:

[0430]

[0431] R Script for Generating Group Differences in Entropy Based on Arabidopsis Memory Lines

[0432] Group differences between 'control' and the mash1 memory line are presented here. Of interest is to understand whether the Gibbs entropy (gent) of methylation variation measured relative to some reference state conforms to the observable phenotypic changes. The Gibbs entropy was estimated for the Arabidopsis Col-0 ecotype (wild-type control, WT), the methyltransferase mutant met1(1), and the first and third generations of heritable epigenetic memory states (nm1, mm1, and mm3) derived as epigenetically modified offspring from the parental line following suppression of MSH1 expression.

[0433] library(MethylIT)

[0434] #> Warning: Replacing previous input 'lifecycle::last_warnings' with

[0435] #> 'rlang::last_warnings' when loading 'tibble'

[0436] #> Warning: Replacing previous input 'lifecycle::last_warnings' with

[0437] #> 'rlang::last_warnings' when loading 'pillar'

[0438] library(lmerTest)

[0439] library(lme4)

[0440] library(reshape2)

[0441] Next are helper functions to format the datasets into 'data.frames'

[0442]

[0443]

[0444] The RData file with the non - linear fitting model can be downloaded from GitLab using the following script:

[0445]

[0446]

[0447]

[0448]

[0449] 1. Linear models and generalized linear models for the first generation of the memory strain

[0450]

[0451]

[0452] 2. Linear models and generalized linear models for the non - memory strain

[0453]

[0454]

[0455] The linear mixed model cannot be fitted. Alternatively, if the linear mixed model fails, then if the difference x - y follows a normal distribution, a t - test can be applied to the paired samples. Otherwise, the Wilcoxon signed - rank test can be used.

[0456]

[0457] The Shapiro - Wilk normality test indicates that the paired t - test results are invalid because the difference 'd' does not follow a normal distribution.

[0458]

[0459] Alternatively, the Wilcoxon signed - rank test that does not rely on the normality assumption can be used.

[0460]

[0461] 3. Linear models and generalized linear models for the third generation of the memory strain

[0462]

[0463] 4. Linear models and non - linear models for met1

[0464]

[0465]

[0466] Linear mixing cannot be fitted. Alternatively, if the difference x - y follows a normal distribution, a t-test for paired samples can be applied. Since there are more samples in the control than in the'met1' group, the first three samples from each chromosome in the control group are taken for paired comparison.

[0467]

[0468] 5. Linear model with a fixed effect for met1

[0469] The linear model with a fixed effect indicates that the reason for the failure of the linear mixed model is that the chromosome effect (when considered as a fixed effect) is not statistically significant.

[0470]

[0471] 6. Results table

[0472]

[0473]

[0474]

[0475] As can be seen from the foregoing, the present disclosure achieves at least all stated objectives.

[0476] List of reference characters

[0477] The following table of reference characters and descriptors is not exhaustive, nor is it limiting, and includes reasonable equivalents. Where possible, the elements identified by the following reference characters and / or those that are almost ubiquitous in the art may replace or supplement any element identified by another reference character.

[0478] Table 6: List of reference characters

[0479]

[0480]

[0481] Glossary

[0482] Unless otherwise defined, all technical and scientific terms used above have the same meaning as commonly understood by one of ordinary skill in the art to which the embodiments of the present disclosure belong.

[0483] The terms "a", "an", and "the" include both singular and plural referents.

[0484] The term "or" is synonymous with "and / or" and means any one member or combination of members of a particular list.

[0485] As used herein, the term "exemplary" refers to an instance, case, or illustration and does not denote a most preferred embodiment unless otherwise specified.

[0486] As used herein, the term "about" refers to a minor variation in a numerical quantity relative to any quantifiable variable. For example, accidental errors may occur through the use of typical measurement techniques or equipment, or due to differences in the preparation, source, or purity of a component.

[0487] The term "substantially" refers to a great or significant degree. Thus, in a given appropriate context, "substantially" may refer to a plurality, majority, and / or absolute majority of the quantifiable variable.

[0488] The term "substantially" encompasses both "about" and "substantially".

[0489] The term "configured" describes a structure capable of performing a task or adopting a particular configuration. The term "configured" may be used interchangeably with other similar phrases such as constructed, arranged, adjusted, prepared, etc.

[0490] Terms characterizing order, position, and / or orientation are not restrictive and are only referred to in accordance with the views presented.

[0491] In biological systems, "methylation" is enzyme-catalyzed; such methylation can be involved in the modification of heavy metals, the regulation of gene expression, the regulation of protein function, and RNA processing. In vitro methylation of tissue samples is also a method for reducing certain histological staining artifacts. The reverse of methylation is demethylation.

[0492] "DNA methylation" is the biological process by which it adds methyl groups to DNA molecules. Methylation can change the activity of DNA segments without changing the sequence. When located at gene promoters, DNA methylation can be used to inhibit gene transcription. In mammals, DNA methylation is essential for normal development and is associated with many key processes including genomic imprinting, X chromosome inactivation, transposon element suppression, aging, and carcinogenesis.

[0493] "Methylome" is the collection of nucleic acid methylation modifications in the genome of an organism or in a particular cell.

[0494] "Epigenetics" is the study of heritable phenotypic changes that do not involve alterations in the DNA sequence. Epigenetics most often involves changes that affect gene activity and expression, but the term can also be used to describe any heritable phenotypic change. Such effects on cellular and physiological phenotypic traits may result from external or environmental factors or may be part of normal development. "Epigenetics" also refers to the changes themselves: functionally relevant changes to the genome that do not involve changes in the nucleotide sequence. Examples of mechanisms that produce such changes are DNA methylation and histone modification, each of which can alter gene expression without changing the underlying DNA sequence.

[0495] In information theory, the "entropy" of a random variable that follows a discrete probability distribution is the average level of "information", "surprise", or "amount of uncertainty" inherent in the possible outcomes of that variable. Given a discrete random variable that takes values in an alphabet and is distributed according to the following distribution and according to x → [0,1]:

[0496]

[0497] where Σ represents the sum of the possible values of the variable. In different applications, different bases for the logarithm log are chosen. Base 2 represents the unit of bits; base e represents the "natural unit" nat; and base 10 represents the unit of "dit". An equivalent definition of "entropy" is the expected value of the self-information of the variable.

[0498] For a random variable x with a continuous probability distribution f(x), the entropy is estimated as:

[0499]

[0500] Then the Gibbs entropy S is given as S = -k B H(X), where k B is the Boltzmann constant.

[0501] In information theory, the "Shannon–Hartley theorem" states the maximum rate at which information can be transmitted over a communication channel of a specified bandwidth in the presence of noise. The theorem is a typical case of applying the noisy-channel coding theorem to a continuous-time analog communication channel subject to Gaussian noise. The theorem determines the Shannon channel capacity for such a communication link, i.e., the maximum amount of error-free information that can be transmitted per unit time with a specified bandwidth, assuming that the signal power is bounded and the Gaussian noise process is characterized by a known power or power spectral density in the presence of noise interference.

[0502] "The present invention" is not intended to refer to any single embodiment of a particular invention, but rather encompasses all possible embodiments as described in the specification and claims. The "scope" of the present disclosure is defined by the appended claims and the full scope of equivalents authorized by such claims. The scope of the present disclosure is further limited to include any possible modifications to any aspect and / or embodiment disclosed herein that would result in other embodiments, combinations, sub-combinations, etc. that would be obvious to a person skilled in the art.< / e> < / e>

Claims

1. A methylation system for interpreting messages in a framework of a communication system using a methylation mechanism, the methylation system comprising: A methylation group dataset related to a molecular machine; A generalized gamma distribution model for describing genome-wide methylation changes observable in the methylation group dataset, the generalized gamma distribution model comprising a probability density function f(E|β, …)) and expressed in terms of the information divergence (χ) of the methylation changes, where the information divergence (χ) (1) is proportional to the energy (E i ) dissipated per bit of information in the methylation changes, and (2) has a symmetry axiom; An instrument for calculating or measuring the entropy (S) derived from the state (i) of the methylation system, the entropy (S) being based on the information divergence (χ).

2. The methylation system according to claim 1, wherein the molecular machine is an enzyme.

3. The methylation system according to claim 1, further comprising a quantizer that summarizes the statistical physics underlying the methylation changes not induced by the methylation mechanism.

4. The methylation system according to claim 1, further comprising applying thermodynamic principles to chromatin dynamics on DNA molecules to maximize the Boltzmann entropy, thereby enabling identification of the most probable methylation density state for the methylation system.

5. The methylation system according to claim 4, wherein the most probable methylation density state for the methylation system is determined by the number of cytosine sites in the DNA molecule.

6. The methylation system according to claim 5, wherein the methylation system is constrained by the discrete probability (π i ) of observing dissipation of the energy value (E i ).

7. The methylation system according to claim 6, wherein the methylation system is constrained by the mathematical expectation value ( <e>)Constraint.

8. The methylation system according to claim 7, wherein the discrete probability (π i ) of observing genome-wide energy dissipation between 0 and the energy (E) is inversely proportional to the partition function of the methylation system including a temperature (T)-dependent scaling constant (β).

9. The methylation system according to claim 8, wherein the probability (p k-1 ) of the amount of energy (E) being dissipated in the interval ([E1, E2), …, [E k )) is proportional to the energy value (E i ). i ) is proportional to the energy value (E 10. The methylation system according to claim 9, wherein for each choice of the parameter (α) that carries information about the molecular machine and realizes the energy value (E i ), the number of times (N k-1 ) the energy (E) is in the interval ([E1, E2), …, [E k )) and the sum of the energy values (E i ) is positive. i α ) is positive.

11. The methylation system according to claim 1, wherein the molecular machine includes a capacity (C) that is related to the maximum amount of information gained per operation of the molecular machine.

12. The methylation system according to claim 11, wherein at normal human body temperature, the minimum energy dissipation per bit of information per machine operation is at least 1.784 J mol -1 .

13. For the methylation system according to claim 11, the probability of observing methylation changes will decrease as the amount of energy dissipated per bit of information processed by the molecular machine increases.

14. The methylation system according to claim 11, which further includes independent moving parts in the molecular machine participating in the operation, wherein the number of the independent moving parts is proportional to the capacity (C) of the molecular machine.

15. The methylation system according to claim 14, wherein the entropy (S) of the methylation system is output as a classical term (S 经典 ) and a contribution (S 机器 ) from the independent moving parts.

16. The methylation system according to claim 11, wherein the capacity (C) can be increased by increasing the energy (E y ) relative to the thermal noise energy (N y ).

17. The methylation system according to claim 1, further comprising an output representing the information divergence (χ) of the methylation change as a Hellinger divergence.

18. The methylation system according to claim 1, further comprising an output representing the information divergence (χ) of the methylation change as a J-divergence.

19. The methylation system according to claim 1, further comprising an output representing the information divergence (χ) of the methylation change as a total variation divergence (TV).

20. The methylation system according to claim 1, further comprising an output representing the information divergence (χ) of the methylation change as a total variation distance (TVD).

21. The methylation system according to claim 1, further comprising an output representing the information divergence (χ) of the methylation change as a cross entropy.

22. The methylation system according to claim 1, further comprising a code describing the genome-wide patterning of DNA methylation occurring at specific landmarks.

23. The methylation system according to claim 1, wherein the code describes a conditional probability (P xy ) such that if a message (x) is generated by a source, the message can be recovered at a receiving point (y).

24. The methylation system according to claim 23, wherein the transmitted message (x) is represented by the methylation levels (x,y) observed in a treatment group or a patient group at each cytosine site.

25. The methylation system according to claim 24, wherein the methylation level (x,y) is estimated as the percentage of the number of methylated (nC m ) and unmethylated (nC i ) cytosines observed at a site (i).

26. The methylation system according to claim 24, wherein the information divergence (χ) is a symmetric information divergence (χ(x,y)) with respect to the methylation levels (x and y).

27. The methylation system according to claim 1, wherein the entropy (S) is a rough estimate for different tissues / cells and is based on the information divergence (χ i ) after representing the energy values (E i ) as an information divergence (χ i ) according to the probability density function (f(E|β,…)).

28. The methylation system according to claim 1, wherein the entropy (S) is proportional to the Shannon entropy (H) at a constant temperature (T).

29. The methylation system according to claim 28, wherein the Shannon entropy (H) depends only on distribution parameters estimated from individualistic data within the methylome dataset.

30. The methylation system according to claim 28, wherein the instrument for calculating or measuring the entropy (S) can quantify methylome gain information (I m ) and can determine whether there has been information loss or information gain.

31. The methylation system according to claim 28, wherein the instrument for calculating or measuring the entropy (S) uses an empirical cumulative distribution function (ECDF).

32. The methylation system according to claim 31, wherein the instrument for calculating or measuring the entropy (S) uses the empirical cumulative function (ECDF) and a kernel density estimation algorithm with a cubic spline function to estimate the empirical density function (EDF).

33. The methylation system according to claim 31, further comprising (a) pseudo-random numbers, (b) Monte Carlo simulation and resampling, or (c) a bootstrap method or numerical method that allows the calculation of the entropy (S) (1) without estimating the G gamma parameter via a numerical integration algorithm, or (2) acceptably estimating the Gibbs or Helmholtz free energy (G,F).

34. The methylation system according to claim 1, wherein the methylome dataset is large enough such that there is an equilibrium between the methylation process and the demethylation process along each DNA molecule, and assuming that the total mass determined by the number (N) of the DNA molecules, the volume (V) of the DNA molecules, and the temperature (T) of the DNA molecules are constant, whereby the methylation system becomes a closed system.

35. The methylation system according to claim 34, further comprising an estimator of the Helmholtz free energy (F) that uses the entropy (S) to further determine the thermodynamic potential of the closed methylation system.

36. The methylation system according to claim 35, wherein the estimator is capable of estimating the entropy (S) in a single-seed descent population of six consecutive generations of Arabidopsis thaliana Col-0 isolate (accession) (wild-type control, WT), methyltransferase mutant met1, and heritable epigenetic memory (mm) and full-sibling non-memory (nm) lines derived from msh1 grown under two different growth conditions.

37. The methylation system according to claim 36, further comprising an estimator for Gibbs free energy (G), the estimator using Shannon entropy (H) to further determine the thermodynamic potential of the closed methylation system.

38. The methylation system according to any one of claims 35 to 37, wherein the estimator is an R software package, the R software package including functions for estimating entropy (S) and Helmholtz free energy (F), the functions being given by a rough estimate of Gibbs free energy (G) for different tissues / cells and the changes in Helmholtz free energy (F).

39. The methylation system according to claim 1, further comprising a device for observing phenotypic changes consistent with the entropy (S).

40. The methylation system according to claim 1, wherein the methylome dataset is related to information about human genes.

41. The methylation system according to claim 40, wherein the methylome dataset includes information about primary human pluripotent cells.

42. The methylation system according to claim 40, wherein the methylome dataset includes information about human cancer types and corresponding normal tissues selected from the group consisting of: white matter of the brain, glioma of the brain, normal breast, breast cancer, breast metastasis, normal colon, colon cancer, colon metastasis, normal lung, lung cancer, normal blood B cells, and placenta.

43. The methylation system according to claim 1, wherein the methylome dataset is related to plant genes.

44. The methylation system according to claim 43, wherein the methylome dataset includes information about transgenic null plants that have undergone changes in phenotypes selected from the group consisting of: a. flowering time; b. the overall size of the transgenic plant; and c. the color of the plant.

45. The methylation system according to claim 44, wherein the methylome dataset contains a collection of Arabidopsis thaliana methylome datasets of msh1 memory and non-memory sibling plants derived from msh1 mutants.

46. A method utilizing the thermodynamics of the DNA methylation process, the method comprising: distinguishing methylation regulatory signals from background noise; and evaluating the thermodynamics of methylation variation in a cell population by observing: the statistical physics underlying the methylation changes; and the methylation regulatory mechanism.

47. The method according to claim 46, further comprising: further evaluating the development and environmental responsiveness of a multicellular organism based on the thermodynamics of the methylation variation.

48. The method according to claim 46, wherein the evaluation does not involve intentionally altering the DNA sequence.

49. The method according to claim 46, further comprising obtaining the background noise by first obtaining a probability distribution representing the methylation variation as an information divergence of methylation levels.

50. The method according to claim 49, wherein the probability distribution is derived for a constrained scenario based on statistical physics.

51. The method according to claim 46, wherein the statistical physics is observed using a probability density function that quantitatively summarizes the statistical physics underlying methylation changes not induced by the methylation regulatory mechanism.

52. The method according to claim 46, further comprising applying the principles of thermodynamics to chromatin dynamics to increase the Boltzmann entropy, thereby resulting in a more probable methylation density state.

53. The method according to claim 46, wherein the channel capacity (C) of the methylation mechanism is observed.

54. The method according to claim 46, wherein a control including methyltransferase and demethylase activities achieved by epigenetic regulatory mechanisms is observed.

55. The method according to claim 46, wherein the statistical physics underlying the methylation changes for genome-wide energy dissipation between 0 and energy E is observed.

56. The method according to claim 46, wherein the statistical physics underlying the methylation changes for genome-wide information divergence between 0 and divergence χ is observed.

57. An extension program for a general-purpose programming language or a statistical programming language, the extension program comprising: an algorithm for analyzing the thermodynamics of methylation signals on an extension of a DNA sequence, the DNA sequence being characterized by: i. methylation information; and ii. The physicochemical information around each methylated cytosine; wherein the algorithm includes one or more functions that can: Estimate the probability of observing genome-wide energy dissipation between 0 and energy E; Estimate the probability of observing genome-wide information divergence between 0 and χ; Quantitatively summarize the statistical physics underlying methylation changes using the probability density function of DNA methylation information divergence; Calculate the conditional probability that the message recovered at the receiving point is generated by the source; Determine the thermodynamic potential of a closed system under constant temperature and constant volume.

58. The extended program according to claim 57, wherein the extended program is written in the R statistical language.

59. The extended program according to claim 57, further comprising digital signal processing (DSP) tools that can be used in programming languages other than the R statistical language.

60. The extended program according to claim 59, wherein the DNA sequence and physicochemical properties of the DNA bases are encoded into a numerical or complex signal further comprising DSP.

61. The extended program according to claim 60, wherein the DSP is applied by using the Gene Word Framework (GWF) R software package.

62. The extended program according to claim 61, wherein the methylation and physicochemical signals of the DNA bases are encoded into a binary, numerical or complex signal.

63. A computer-implemented diagnostic / predictive method, comprising: Preparing a set of differentially methylated positions (DMPs) or differentially information methylated regions (DMRs) from the methylome data of a sample of an animal or plant whose phenotypic characteristics are different from those of the wild type of the same species of animal or plant; and Using the following thermodynamic data associated with the set: i. Estimate the probability of observing genome-wide energy dissipation between 0 and energy E; ii. Quantitatively summarize the statistical physics underlying the methylation changes; iii. Calculate the conditional probability that the message recovered at the receiving point is generated by the source; and / or iv. Determine the thermodynamic potential of a closed system under constant temperature and constant volume; and Using the thermodynamic data to diagnose or predict the genetic traits of the plant or animal.

64. The computer-implemented method according to claim 63, wherein the genetic trait is related to a disease affecting the animal or plant.

65. The computer-implemented method according to claim 63, further comprising determining an accuracy or confidence level associated with the prediction based on the thermodynamics of DNA methylation in a cell population.

66. The computer-implemented method according to claim 65, further comprising maintaining an automated heuristic that records the observed thermodynamics of previously evaluated genetic traits, and wherein the determination of the accuracy or confidence level associated with the prediction based on the thermodynamics of DNA methylation in a cell population is based on at least one aspect of the recorded information.

67. A method for quantifying the entropy at a DNA cytosine site undergoing methylation changes, the method comprising: distinguishing a methylation regulatory signal from background noise; and following an algorithmic approximation that applies a polynomial or Dirichlet distribution to calculate the Gibbs entropy (S) of the DNA cytosine site undergoing methylation changes; increasing the Gibbs entropy (S), thereby resulting in a more probable methylation density state.

68. According to the method, wherein the Gibbs entropy (S) is the Boltzmann entropy.

69. The method according to claim 68, wherein an increase or decrease in the Gibbs entropy also conveys an increase or decrease in the Boltzmann entropy.< / e>

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