A Random Walk Model to Evaluate the Post Transcriptional Process in Rett Syndrome

We have proposed a random walk model to model the post-transcription process. We have chosen mature microRNA sequences related to individuals with Rett syndrome. We have analyzed the micro-RNA sequences by transcribing the nucleotides into random fractals. We have reported our results from the exponent, fractal dimension, entropy, and Binder Cumulant. Our analyzes allowed us to report that the system is sensitive to the order of purines and pyrimidines, presenting different measurements from observable physics even for equal purine and pyrimidine proportions


Introduction
A mental disorder is a syndrome in an individual characterized by clinically significant disorders in cognition, emotional regulation, or behavior that reflect a dysfunction in the psychological, biological, or developmental processes underlying mental functioning.
Mental disorders are commonly associated with distress or disability in social, occupational or other important activities.
The diagnosis of a mental illness must be of clinical utility, leading clinicians to determine diagnoses, treatment plans and potential new treatments. There are specific criteria that must be met to diagnose mental disorders such as the following: background validators (similar genetic markers, family traits, temperament, and environmental exposure), competing validators (similar neural substrates, biomarkers, emotional and cognitive processing) and predictive validators (same clinical course -clinical history -and same response to treatment).
Neurodevelopmental disorders are a group of conditions that begin in the developmental period. These developmental disorders typically manifest themselves before the child enters school, characterized by manifestations of personal, social or other occupational activity disabilities. The spectrum of developmental disabilities ranges from specific learning limitations to deficiencies in social skills or intelligence [1-6].
In particular, Rett syndrome is a neurodevelopmental disorder of genetic etiology, resulting from mutations in the MECP2 (methyl-CpG-binding protein 2) gene found on the X chromosome. Observation of this disease is unrelated to a specific ethnic group. It is prevalent in females, presenting one case every 10,000-20,000.
The molecular rationale is that mutated MECP2 encodes a defective protein, incapacitating it to properly perform its biological function.
Thus, genes that should be inactive during specific phases of neuron development remain active, culminating in damage to the central nervous system [7][8][9][10][11][12][13][14]. The aim of our work is to construct in analogy to the protein coding process. We chose a set of Rett syndrome-related mRNAs. We have used these RNA sequences to construct fractals using discrete random walks as a tool.

Materials and Methods
Inspired by the memory class random walk model, we have proposed a random walk model to construct the random fractal in analogy with the protein coding process [15][16][17][18]. In this model the length of the sequences comprises the total time of the fractal construction. At each instant of time a nucellotide is read and a step is performed by the random walker. Each step performed at of time is. RNA is called ribonucleic acid. It is a linear polymeric molecule, made up of smaller units called nucleotides, nucleotides receive a dichotomous classification into purines (A and G) and py-

rimidines (C and U).
Random tours have well-defined stochastic rules. It occurs as follows: the walker walks one step to the right or one step to the left, with the stochastic evolution equation given by for the instant of time. When deciding to take a step to the right, that is, the variable σ (t+1) assumes value by stepping left, assume value -1.
The memory is formed of a set of random variables. σt' for the time t'<t. The microscopic dynamics occur as follows: (a) in the time: t+1, the time: t' is randomly chosen with uniform probability 1⁄t of the set 1,2, 3,...,t.
In the time: t=0, the walker is in positionX_0, moves to the right when reading a purine or to the left when reading a pyrimidine, assuming the value, σ 1 =+1 and σ 1 =-1, respectively. In the instant of time: t, the walker moves to the right when reading a purine or to the left when reading a pyrimidine, assuming the value: σ t' =+1 e σ t' =-1, respectively. Walker position is quantified by the stochastic evolution equation: We have performed numerical experiments using them to obtain measurements of the following observable physics such as skewness, kurtosis, Hurst exponent, fractal dimension, information entropy, and fourth order Binder cumulative.
Skewness measures how much a distribution is deviated to the right or left from the distribution mean. It is measured by following the formula: Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. Kurtosis is quantified according to the following stochastic equation: ( ) The Hurst's exponent serves to classify Anomalous diffusion can be classified as superdiffusive (H>1⁄2), normal diffusive(H=1⁄2) and subdifusive(H<1⁄2) The fractal dimension (D) is a measure of roughness while(H) is a quantity that classifies the diffusion regimes. The fractal dimension is related to the Hurst exponent by the following equation: in which δ is the dimension in Euclidean space in this case (δ=1). The information entropy measures the overall effect of bit variation associated with random walks. Information entropy is described by the equation: in which Fj (x,t) is the probability distribution of finding random walkers in the position x and in the time: t [19].
Binder cumulant is a quantity widely used in the analysis of phase transition systems [20].
We have measured these observable physics based on the RNA sequence data in the (Table 1).

Results
We have performed measurements of skewness, kurtosis,, Hurst exponent, fractal dimension, entropy of information and fourth order Binder cumulant. In the (Figures 1-4), we have presented the measurements of skewness, Hurst exponent, fractal dimension and entropy of information as a function of the label assigned to each RNA sequence presented in the (Table 1). In the (Figure 1    In the (Figure 4), we have presented typical measures of information entropy. Entropy measurements are related to the Hurst exponent and the fractal dimension of the system. We have noted that the larger (smaller) the measurements observed for the Hurst exponent (the fractal dimension), the smaller the entropy variations. The largest entropy variation occurred for the sequence with ID hsa-miR-296-3p, being equal abits, which has a percentage of 39.13% purines and 60.87% pyrimidines. For information entropy, we have observed three typical behaviors in the construction of random walks, which are presented In the ( Figure 5). In general, entropy from zero variation, ie at the beginning of nucleotide reading and grows to a point where it remains with a fixed range of bits. Entropy variations occur in three typical ways presented In the (Figure 5a, 5b & 5c), showing the results of entropy variation reading the RNA sequences with ID dre--mir-146a, hsa-miR-122-3p and hsa-miR-296-3p, respectively. For the dre-mir-146a, hsa-miR-122-3p and hsa-miR-296-3p sequences the entropy exhibits sigmoid behavior with saturation points equal to and bits, accompanied by larger fractal dimension values equal to, and respectively. The dre-mir-146a sequence shows smoother growth for entropy, while the hsa-miR-122-3p sequence has more abrupt growth, as can be seen from the curve In the (Figure 5b).
The entropy of the hsa-miR-296-3p sequence grows faster than the sequences of (Figure 5a & 5b), We have also noticed that even though curve (Figure 5b) has a higher saturation point, curve ( Figure   5a) grows faster but stabilizes first. This fact can be explained by looking at the results In the figure 6. In the (Figure 6), the ordinary Binder cumulative measurements are presented. The term responsible for such variations is the (⟨x 2 ⟩) 2 , because according to our observations the kurtosis is invariant for all the averages that we have made, and it is equal to ⟨x 4 ⟩=214.748. Note that, for curve ( Figure   6b), the Binder cumulant grows to saturation point, while curves ( Figure 6a & 6c), have points the cumulant has a lower concave region. , then grows reaching the saturation points each. Besides, we have noticed that the bigger this region is concave, the greater the initial entropy variation, as can be seen in curves (Figure 5a

Discussion
We have studied Rett syndrome from a stochastic perspective, performing a process of constructing fractal objects through an analogy between protein coding and random class walks with memory.
This method was used to be used as a tool for analysis and diagnosis of Rett syndrome, observing some RNA sequences related to the post-transcription process responsible for protein coding. We observed that typical skewness measurements are not related in this problem as averages of Hurst exponent and fractal dimension.