Computational and Mathematical Methods in Medicine

Volume 2017, Article ID 7925106, 8 pages

https://doi.org/10.1155/2017/7925106

## Exponentially Modified Peak Functions in Biomedical Sciences and Related Disciplines

Saint-Petersburg State University, Saint Petersburg, Russia

Correspondence should be addressed to A. Golubev; ur.relbmar@vblgxl

Received 26 January 2017; Accepted 11 May 2017; Published 5 June 2017

Academic Editor: Hans A. Braun

Copyright © 2017 A. Golubev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In many cases relevant to biomedicine, a variable time, which features a certain distribution, is required for objects of interest to pass from an initial to an intermediate state, out of which they exit at random to a final state. In such cases, the distribution of variable times between exiting the initial and entering the final state must conform to the convolution of the first distribution and a negative exponential distribution. A common example is the exponentially modified Gaussian (EMG), which is widely used in chromatography for peak analysis and is long known as ex-Gaussian in psychophysiology, where it is applied to times from stimulus to response. In molecular and cell biology, EMG, compared with commonly used simple distributions, such as lognormal, gamma, and Wald, provides better fits to the variabilities of times between consecutive cell divisions and transcriptional bursts and has more straightforwardly interpreted parameters. However, since the range of definition of the Gaussian component of EMG is unlimited, data approximation with EMG may extend to the negative domain. This extension may seem negligible when the coefficient of variance of the Gaussian component is small but becomes considerable when the coefficient increases. Therefore, although in many cases an EMG may be an acceptable approximation of data, an exponentially modified nonnegative peak function, such as gamma-distribution, can make more sense in physical terms. In the present short review, EMG and exponentially modified gamma-distribution (EMGD) are discussed with regard to their applicability to data on cell cycle, gene expression, physiological responses to stimuli, and other cases, some of which may be interpreted as decision-making. In practical fitting terms, EMG and EMGD are equivalent in outperforming other functions; however, when the coefficient of variance of the Gaussian component of EMG is greater than ca. 0.4, EMGD is preferable.

*“Essentially, all models are wrong, but some are useful.”*

Box G. E. P. and Draper N. R. (1987), Empirical Model Building and Response Surfaces, John Wiley & Sons, NY, p. 424.

#### 1. Introduction

The normal (Gaussian) and the exponential are probably the most widely known distributions and quite ubiquitous, too. No wonder that situations are possible where they are expected to meet each other. The resulting composite distributions have been suggested to be relevant, for example, to times between consecutive cell divisions [1–4] in cell biology and to times from stimulus to response [5–8] in psychophysiology.

Generally speaking, when the times of the passages of certain type objects from their initial to their final state are composed of variable times of their transition to an intermediate state and of their dwelling in the intermediate state, out of which they exit at random to their final state, then the random variable that represents the overall passage time of any such object is the sum of two independent random variables, the transit time and the dwell time, and the distribution of the overall passage times is defined as the convolution of the distributions of its summands [9]. The convolution of a Gaussian distribution and a negative exponential distribution is known as the exponentially modified Gaussian (EMG). Its generic plot is shown in Figure 1, and the approximations of empirical datasets with EMG are shown in Figures 2, 3, and 5.