Exploiting product molecule number to consider reaction rate fluctuation in elementary reactions

In many chemical reactions, reaction rate fluctuation is inevitable. Reaction rates are different whenever chemical reaction occurs due to their dependence on the number of reaction events or the product number. As such, understanding the impact of rate fluctuation on product number counting statistics is of the utmost importance when developing a quantitative explanation of chemical reactions. In this work, we present a master equation that describes reaction rates as a function of product number and time. Our equal reveals the relationship between the reaction rate and product number fluctuation. Product number counting statistics uncovers a stochastic property of the product number; product number directly manipulates the reaction rate. Specifically, we find that product number shows super-Poisson characteristics when the product number increases, inducing an increase in the reaction rate. While, on the other hand, when the product number shows sub-Poisson characteristics with an increase in the product number, this is induced by a decrease in the reaction rate. Furthermore, our analysis exploits reaction rate fluctuation, enabling the quantification of the deviation of an elementary reaction process from a renewal process.

Most chemical reactions involve reaction rate fluctuation which is adaptation and response to changing environments.Several studies have focused on quantifying the rate coefficient in enzyme reactions [1][2][3][4] and have shown that this rate coefficient is a random variable, not a constant.Motivated by these observations, Lim.et.al [5] introduce the concept of a vibrant reaction process with the rate coefficient modeled as a stochastic variable coupled to reaction environments, proposing a method for considering rate coefficient fluctuations.
Extending the concept to single enzyme reactions, Park.et.al [6] suggest a new statistical kinetics that characterizes the dynamics of enzyme activity.Recently, the Chemical Fluctuation Theorem (CFT) governing general birth-death process was recently shown to successfully and quantitatively provide an accurate description of gene expression [7].These examples make it is obvious that rate coefficient fluctuates due to reaction environment and differs whenever reaction occurs, demonstrating that reaction rate varied with the number of reaction events, or product number.
The master equation [8] is perhaps the most common approach used to explain product number fluctuation caused by reaction events.The conventional master equation assumes that the rate coefficient of elementary reactions are constants that remain unchanged, regardless of product number, and under a homogeneous reaction environment, the conventional master equation is, in fact, applicable because the rate coefficient does indeed remain constant.
However, in a heterogeneous reaction environment, this rate coefficient does not remain constant because it varies with the product number, or the number of reaction events in an elementary reaction.
In the current work, we propose an alternative description for reaction rate fluctuation induced by reaction events or product number and present our new theoretical results.By describing reaction rates as a function of product number and time, we obtain a new master equation, and we derive the mean and variance of the product number from this new equation.
Our result yields the relationship between the fluctuating reaction rate and product number counting statistics.We are additionally able to obtain the probability density function of the time required to produce n product molecules.Using this probability density function, we are also able to estimate the deviation of an elementary reaction from a renewal process as the product number increases.
To consider a realistic reaction system that cannot be accurately described by the conventional master equation, we first derive a new type of master equation for an elementary reaction, A B → .Our equation defines a reaction rate coefficient that fluctuates with time and product number; that is, in this definition, our reaction rate coefficient is a function of time and product number.We begin with two assumptions.We first assume that within the sufficiently small time interval, ( ) , the probability of a single reaction occurring can be approximated as ( ) n t h λ when the product number is n at time t, where ( ) n t λ is a real nonnegative function and represents the reaction rate.Our second assumption is that the probability of two or more reactions occurring in time interval h is zero.We can then denote the probability of the product number being n at time t by ( ) n P t , which satisfies the normalization condition, 0 ( ) 1 . With these assumptions in place, we can approximate the probability of the product number being n at time t+h by multiplying the probability of the product number being n at time t by the probability of no reaction occurring in the time interval, (t, t+h), and then multiply the probability of the product number being n-1 at time t by the probability of a single reaction occurring in time interval (t, t+h).This results in

(
) Using Eq. ( 2) to examine the counting statistics of the product number, we can take the first moment of the product number, given by 0 ( ) ( ) . We can then obtain 0 ( ) ( ) where 0 ( ) ( ) ( ) . Taking the derivative of Eq. ( 3) with respect to t, we then obtain a result in good agreement with our expectation, which is that a change in the mean product number with respect to time is the same as the mean of the reaction rate, ( ) Product number fluctuation can be exploited to obtain valuable information about the stochastic property of a reaction process.We can extract the stochastic property of an elementary reaction process by analyzing the variance of the product number, Similarly, one can discover the second moment, 2 ( ) n t 〈 〉 , of the product number and obtain the Mandel's Q parameter of the product number as follows, are the correlation coefficient of the product number, n , and the reaction rate, ( ) n t λ , respectively.
( ) n t σ and ( ) n t λ σ are the standard deviations of the product number and the reaction rate, respectively.For detailed derivations of Eqs. ( 3) and ( 4), see Supplementary Method II.From Eq. ( 4), we can note that the correlation coefficient, ( ) t ρ , of the product number, n, and the reaction rate, ( ) n t λ , determine the product number's stochastic property.When the reaction rate is not correlated with the product number or when the reaction rate shows no dependence on the product number, the product number counting statistics show Poisson characteristics, ( ) 0 n Q t = , due to the fact that ( ) 0 t ρ = .We confirm this to be the case in Supplementary Method III.When the reaction rate, ( ) n t λ , is positively correlated with the product number or when ( ) It is well-known that, at long times, the Mandel's Q parameter for the product number produced by a renewal reaction process is the same as the randomness parameter (the difference between the variance over the mean square and unity) in the waiting time required to produce one molecule [9].When we determine how an elementary reaction process deviates from a renewal process, the important quantity we must pay special attention to is the probability density function of the time required to produce n product molecules, which we denote as ⋅⋅⋅ .The non-renewal property is defined mathematically as n g s = for a renewal process.In other words, ˆ( ) 1 n g s ≠ signifies a non-renewal process.
We are now also able to estimate the deviation of an elementary reaction from a renewal process as the product number increases. ) ) Let us refer to ˆ( ) 1 n g s − as a "non-renewal quotient" that approaches zero when an elementary reaction gets closer to a renewal process.We can then say that a chemical reaction is a nonrenewal process when the non-renewal quotient is not equal to zero.We confirm this by investigating three models of ( ) n t λ in Fig 2, and as can be easily seen, as the product number increases, the non-renewal quotient becomes farther and farther from zero.In addition, the nonrenewal quotient deviates farther from zero as the Laplace variable, s, in ˆ( ) n g s becomes larger with the fixed number of product molecules, n.The large Laplace variable s corresponds to short times in the time domain, while the small Laplace variable s corresponds to long times in the time domain.For the three given models in Fig. 2, the non-renewal character becomes more pronounced when the waiting time required to produce product molecules is shorter.We are then enabled to quantify the deviation of an elementary reaction from a renewal process even if only the reaction rate, ( ) n t λ , is known.
We leave for future research studying a birth-death process with reaction rates fluctuating with product number.Birth-death processes are relevant to many fields that study the populations of systems, such as queueing models and gene expression [10][11][12].The general results derived in this work serve as useful tools for investigating birth-death processes.
In summary, the current work suggests a method which considers fluctuation in the reaction rate by making use of the product number.In order to do so, we develop new type of master equation for elementary reactions, and using this equation, we investigate the product number counting statistics and the nonrenewal quotient of an elementary reaction.We find that when the reaction rate increases with the product number, the product number counting statistics shows super-Poissonian characteristics, while a decrease in the product number results in the product number counting statistics showing sub-Poissonian characteristics.If the reaction rate is known, we can use the non-renewal quotient developed here and determine the extent to which an elementary reaction can be defined as a non-renewal process.The ability to determine the stochastic property of the product number fluctuation, Eq. ( 4), and nonrenewal quotient, Eq. ( 5), when only the reaction rate is known is a significant achievement that expands our understanding of elementary reactions.We will introduce a more useful formula than Eq.(M1-1) when we try to calculate the mean and variance of the number of product molecules.Because Eq. (M1-1) for ( ) n P t has a hierarchical structure, the largest trouble in Eq. (M1-1) is the time-consuming effort needed to be done in order to obtain the mean and variance of the number of product molecules.We rederive ( ) n P t just as a function of the reaction rate when the number of product is n at time t.
We begin with Eq. (M1-1).The so-called survival probability ( ) Integrating by parts on the right side of Eq. (M1-2), we obtain where ( ) ( ) ( ) . 2 ( ) P t can be expressed by using Eq.(M1-3). ( P t e q t e q e d q t e q t e τ τ τ λ τ τ ( ) n P t cannot be defined when n is a negative integer.If we assume that the number of product molecules is zero at time zero, the mean number of the product molecules is the definite integral from zero to time t of ( ) n λ τ with respect to τ , Eq. (3).
To find the variance of the number of product molecules, we first must find its second moment.Applying 2 0 n n ∞ = ∑ to the both sides of Eq. ( 2), then we get: We use Eq.(M2-1) to obtain Eq. (M2-2).By integrating the both sides of Eq. (M2-2) with respect to time t, one obtains Using Eq. (M2-3), we find that the variance the number of product molecules is given by ( ) The integrand in the right-hand side of Eq. (M2-5) is the covariance of n and ( ) We are going to form a conclusion: t λ is an increasing function of the product number, the counting statistics of the product number bear a super-, (Fig.1(a)), while on the other hand, when ( ) n t λ is inversely correlated with the product number or when ( ) n t λ is a decreasing function of the product number, we instead see sub-Poission characteristics, ( ) 0 n Q t < , due to the fact that ( ) 0 t ρ < (Fig. 1(b)).It is important because when the reaction rate is an oscillating function of the product number, the product number counting statistics can be Poisson, super-Poisson, or sub-Poisson.For this case, it is necessary to calculate the definite integral on the right-hand side of Eq. (4) in order for us to determine which product number fluctuation shows stochastic properties (Fig. 1(c)).Further details on a faster mean by which to obtain Eqs.(3) and (4) can be found in Supplementary Method IV.Comparing with Eq. (4) and the chemical fluctuation theorem gives the relation between the time correlation function (TCF) of the production rate, and the probability density function of reaction time interval is obtained (Supplementary Method V).
be interpreted as the probability of the product number being less than or equal to n in the time interval [0, t] (see Supplementary Method IV).

Fig1..
Fig1.Product number fluctuation with the reaction rates, ( ) n t λ , which is a function of the

.
In similar way, we obtain

(
To put it another way, we can express ˆ( ) n g s as at time t increases as the number of product molecules n increases.The