Abstract
The X-level Approach Reaction Noise Estimator (XARNES) method has been developed previously to study reaction noise in well mixed reaction volumes. The method is a typical moment closure method and it works by closing the infinite hierarchy of equations that describe moments of the particle number distribution function. This is done by using correlation forms which describe correlation effects in a strict mathematical way. The variable X is used to specify which correlation effects (forms) are included in the description. Previously, it was argued, in a rather informal way, that the method should work well in situations where the particle number distribution function is Poisson-like. Numerical tests confirmed this. It was shown that the predictive power of the method increases, i.e. the agreement between the theory and simulations improves, if X is increased. In here, these features of the method are explained by using rigorous mathematical reasoning. Three derivative matching theoremsare proven which show that the observed numerical behavior is generic to the method.
Highlights
Noise is an integral part of the workings of the living cell biochemistry [1]
The equations would be identical if not for the fact that the related equations in [10,11] are for zero-centered moments. This remarkable coincidence where the same set of coefficients is obtained in two different ways is rather intriguing. This strongly suggests that the X-level Approach Reaction Noise Estimator (XARNES) method might have advantageous properties as it comes to the derivative matching between the exact and the approximate moments computed for a suitable initial condition
The three theorems explain the mechanism behind the numerically observed fact that the XARNES method works well if the particle number distribution function is close to the Poisson distribution
Summary
Noise is an integral part of the workings of the living cell biochemistry [1]. There are many types of noise and this work focuses on the intrinsic noise. This remarkable coincidence where the same set of coefficients is obtained in two different ways is rather intriguing This strongly suggests that the XARNES method might have advantageous properties as it comes to the derivative matching between the exact and the approximate moments computed for a suitable initial condition. It will be shown by employing a strict mathematical analysis, in the same way as done in [10,11], that this is the case for the Poisson initial condition. (19) and (27) at time t = t0 where the particle number distribution function is strictly given by the uncorrelated multivariate Poisson distribution In such a case all correlation forms are zero except the ones specified by vectors ei, i = 1, . What is left to show is that the action of the operator onthe remaining second term results in zero
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