Equilibrium Time, Permutation, Multiscale and Modified Multiscale Entropies for Low-High Infection Level Intracellular Viral Reaction Kinetics

Kinetics Monte Carlo simulation has been done for solving Master equation for intracellular viral reaction kinetics. There is scaling relationship between reaction equilibrium time and initial population of template species in intracellular viral reaction kinetics. Kinetics Monte Carlo result shows that mathematical presentation between initial population of template species and reaction equilibrium time is ( a = 163.1, b = -0.1429 ), where N , N are initial population of template species and reaction equilibrium time respectively. Kinetics Monte Carlo shows that increasing initial population of template species decreases the reaction equilibrium time. Initial population for template species with range 1 are called low, medium and high infection level in intracellular viral kinetics reaction respectively. Entropy generation has been considered in low, intermediate and high infection level of intracellular viral reaction kinetics in during dynamical population. Permutation, multiscaling and modified multiscaling entropies have been calculated for species, genome, structural protein, and template species. Dependency of permutation entropy on permutation order is small in high infection level. At short time scale, convergency of permutation entropy occurs with medium permutation order value. In the big time scale, permutation entropy ( ) H n scales with permutation order n as a scaling relation ( ) n . Three different trends for low, medium and high infection level observed for multiscaling entropy of template species versus scaling factor. Non-monotonic behavior for permutation entropy versus time could be observed for structural protein species. scaling entropy for shows that there is linear relation relationship versus scaling factor for initial population of template species at range

initial population of template species and reaction equilibrium time is ()

Introduction
Lyapunov exponent, entropy and fractal dimension are essential parameters which are so useful for recognition chaotic, random and complex time series data analysis [6,13].
Many computer algorithms, numerical and mathematical methods have been developed for complexity measure for noisy, chaotic and random data in past 20 years ago. During the last three decades many interesting methods has been proposed for investigation vibrational and dynamical changes based on analyzing metric of nearest neighbors in phase space method [10]. Analyzing vibrational and dynamical changes of data based on nearest neighbors in phase space method is very calculation consuming time [10].
Permutation entropy is presented as a very effective and fast method for complexity measure and analyzing chaotic time series data and for analyzing speech signal.
Permutation entropy for analyzing of time series of heart interbeat signal has been applied for recognition healthy and heart failure people [15].
Due to regular working of heart interbeat signal of sick person, permutation entropy of healthy people is higher than those sick person [15].
Pathologies such as cardiac arrhythmias like atrial fibrillation are related to the high statistical fluctuation and uncorrelated noise [19,8,9].
A virus is made up of a core of genetic material, either DNA or RNA, surrounded by a protective coat called a capsid. Computer programming could explain many complexities behavior of living cells [9].
Our aim of work is understanding of complexity of dynamics of intracellular dynamics help us to predict the behavior of viruses in during of growth process. Entropy value in during time evolution for dynamical of complex bio-chemical reaction will be explored.

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Natural complexity of intracellular reaction dynamics in low-high infection level will be examined with different entropy approach and ability of them for recognition dynamical complexity. Initial population of template effect on entropy trend in during dynamical process will be explored. We will investigate reaction equilibrium time as function of initial population of template.

Intracellular Viral Kinetics
The Viral nucleic acids were classified as genomic (gen) or template (temp). The genome may be DNA and RNA which are positive-strand, negative strand or some other type.
According to equations (1-5) we have two ways for changing the genome. The first way, the genome converts to template which has catalytic role for synthesizing every Viral component [14]. The second way it packages within structural proteins to form progeny virus with different probabilities. It is very important to notice that the synthesis of structural protein requires that the Viral DNA be transcribed to mRNA and mRNA must be translated to generate the structural protein. At first when the population of template is very high structural protein is synthesized for incorporation into progeny particle. We can show the mechanism of viral kinetics according following reaction: [14] In this mechanism A, B, C, D, E and X are genome, structural protein, virus, degradation template, degradation or secretion, and template, respectively.

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The rate constant and initial molecular populations are taken from reference as a following

Stochastic Algorithm and Simulation
Another way to investigate the kinetics of a small system is stochastic simulation. Up to now several authors have applied the stochastic algorithms [7]. In recent years stochastic modeling has been emerged as a physically more realistic alternative for modeling of the vivo reactions [7,17,16]. Stochastic and random diffusion model can be described for autocatalytic reaction model such as Lotka-Volterra reaction-diffusion or predator-prey system [1]. Let X be the time of the event. By a constant hazard we mean that: where 0   is a constant whose value may be calculated as, If N  and 0 t   therefore equation 9 will convert to exp( ) Consequently, whenever we consider a time dependent event with constant hazard  , in Gillespie algorithm [7]. We can conclude that the time distribution is an exponential function. By choosing two uniform random numbers and within the interval [0,1] and by definition of two following expressions, we may write: We can define the permutation entropy per symbol [15,3] In our calculation the order of permutation entropy is taken n=4.

. Multiscaling Entropy
Another approach for investigation of complexity of biological signal is multiscaling

Result and discussion
Preprint of modified current paper could be found at references [18].

Reaction equilibrium time of viral kinetics model as a function of template number:
Kinetics Monte Carlo approach is used for simulation of intracellular viral reaction kinetics via Gillespie algorithm. Stochastic dynamics for investigation number of template particle is investigated as a function of time. Reaction equilibrium time after 1000 times of stochastic simulation is obtained as a function of initial template population.
On the basis of Figure 1 there is scaling relation between the reaction equilibrium of time and the number of template particle. Fitting of Figure 1 with function () shows that reaction equilibrium time decreases very fast for the great initial number of template species. Figure 1 shows reaction equilibrium time for large 9 number of template species is small. Owing to this fact, increasing the number of template species decreases the fluctuation population of template, then system approaches to stationary sate at the short time.

Permutation entropy Result
Stochastic simulations have been done for intracellular viral reaction kinetics via

3.3.1.b. Multi scaling entropy for structural protein
Multi scaling entropy calculation has been extended for structural protein versus scaling factor. Multi scaling entropy result for structural protein shows that there is linear relation relationship versus scaling factor for initial population of template species at range 1-8.
14 One peak in multi scaling entropy as a function of scaling factor for structural protein species could be observed for initial template species at population range 9 Temp 10  . Result of multiscaling entropy for structural protein species at initial template population 2, 3 and 10 numbers are presented at Figure.7.b and Figure7.c respectively.

3.3.1.c. Multi scaling entropy for Template
Multi scaling entropy has been calculated for low, medium and high infection level of template species versus scaling factor. For low, medium, and high infection level multi scaling entropy trend is completely different to each other. Result of multi scaling entropy versus scaling factor for low, medium and high infection level for dynamical template species has been presented at Figure 8.a, b, c respectively. On the basis of   Figure.8.c.

Comparison of multiscaling entropy versus scaling factor for intermediate infection level
for template, structural protein and genome species has been done and its result has been presented at Figure.8.d. On the basis of Figure.8.d there is a following order for multi scaling entropy: template> structural protein > genome.

Scaling entropy with including variation standard deviation of population
For computation of multiscaling entropy in part 3.3.1 standard deviation was constant and r parameter was set r = 0.15SD. As a matter of fact for each scaling factor in during of coarse graining method, standard deviation is not constant. [12] Multiscaling entropy based on variation of standard deviation (modified multi scaling entropy) from Gillespie algorithm for stochastic dynamics is calculated for each scaling factor parameter. For each scaling factor standard deviation is calculated; therefore, r is set as 0.15SD. As a result, modified multiscaling entropy is calculated as a function of scaling factor.
Computation result of modified multiscaling entropy for genome species for each scaling factor is shown in Figure 9.a. On the basis of Figure.9.a modified multiscaling entropy changes with scaling factor linearly for all initial template population. Modified multiscaling entropy value increases versus scaling factor for low-medium infection level and for high infection level, multiscaling entropy decreases again.
Modified multiscaling entropy of structural protein shows that modified multiscaling entropy increases with scaling factor linearly for initial template population1 Temp 8  . Result of modified multi scaling entropy for structural protein with initial condition Temp=1 has been presented at Figure.9.b. When number of initial template species increases to 10, then one maximum peak observed in modified multiscaling entropy versus scaling factor for structural protein species.
Modified multi scaling entropy calculation has been extended for template species for different initial population of template species and its result is presented at Figure.9.d. On the basis of Figure.9.d modified multi scaling entropy decreases with scaling factor monotonically. It is worthwhile to notice that there is no regular trend for multi scaling entropy versus scaling factor for template species; however modified multiscaling entropy of template has a regular trend for different initial number of template species.
Comparison of modified multiscaling entropy versus scaling factor has been done for three kinds of species, namely genome, structural protein and template. Result of comparison for modified multiscaling entropy has been shown in Figure.9.e. On the basis of Figure.9.e there is a following order for modified multiscaling entropy Template> Structural Protein > genome.

Initial Template effect on permutation entropy, multiscaling entropy and modified multiscaling entropy
Permutation entropy for three kind of species namely genome, structural protein, template has been calculated for different initial template population. On the basis of permutation entropy result, there is monotonic behavior for permutation entropy value versus initial population of template species in all time. Calculation of multiscaling entropy has been extended as a function of initial template species. Multiscaling entropy shows that there is a non-monotonic behavior for entropy versus initial template population for all species namely gen, structural protein and template. Generally modified multiscaling entropy is monotonic versus scaling factor for genome, structural protein and template species. Result of permutation entropy, multiscaling entropy and modified multiscaling entropy versus initial population of template for template species is presented at Figure.10.a, b, c respectively. On the basis of Fig.10 for template species three is non monotonic behavior for both multiscaling entropy and modified multiscaling entropy versus initial template population.

Conclusion
Kinetics Monte Carlo simulation has been done for intracellular viral reaction kinetics. Permutation entropy in presence and absence of chaotic noise is not able to predict complexity for dynamical population of intracellular viral reaction kinetics however multiscaling and modified multiscaling entropy able to predict the natural complexity for dynamical population of intracellular viral reaction kinetics.

Conflict of Interest:
The author declare that he has no conflict of interest.