DSM Resolve, The Maastricht Forensic Institute, P.O. Box 18, 6160 MD Geleen, Urmonderbaan 22, 6167 RD Geleen, The Netherlands.
Forensic Sci Int Genet. 2012 Jan;6(1):17-25. doi: 10.1016/j.fsigen.2011.01.003. Epub 2011 Feb 4.
In forensic DNA profiling use is made of the well-known technique of PCR. When the amount of DNA is high, generally unambiguous profiles can be obtained, but for low copy number DNA stochastic effects can play a major role. In order to shed light on these stochastic effects, we present a simple model for the amplification process. According to the model, three possible things can happen to an individual single DNA strand in each complete cycle: successful amplification, no amplification, or amplification with the introduction of stutter. The model is developed in mathematical terms using a recursive approach: given the numbers of chains at a given cycle, the numbers in the next can be described using a multinomial probability distribution. A full set of recursive relations is derived for the expectations and (co)variances of the number of amplicon chains with no, 1 or 2 stutters. The exact mathematical solutions of this set are given, revealing the development of the expectations and (co)variances as function of the cycle number. The equations reveal that the expected number of amplicon chains without stutter grows exponentially with the cycle number, but for the chains with stutter the relation is more complex. The relative standard deviation on the numbers of chains (coefficient of variation) is inversely proportional to the square root of the expected number of DNA strands entering the amplification. As such, for high copy number DNA the stochastic effects can be ignored, but they play an important role at low concentrations. For the allelic peak, the coefficient of variation rapidly stabilizes after a few cycles, but for the chains with stutter the decrease is more slowly. Further, the ratio of the expected intensity of the stutter peak over that of the allelic peak increases linearly with the number of cycles. Stochastic models, like the one developed in the current paper, can be important in further developing interpretation rules in a Bayesian context.
在法医 DNA 分析中，广泛使用了 PCR 这一知名技术。当 DNA 含量较高时，通常可以获得明确的图谱，但对于低拷贝数 DNA，随机效应可能会起主要作用。为了阐明这些随机效应，我们提出了一个简单的扩增过程模型。根据该模型，在每个完整循环中，单个 DNA 链可能会发生三种情况：成功扩增、未扩增或扩增时出现微卫星不稳定性。该模型以数学术语表示，采用递归方法：给定特定循环中的链数，下一个循环中的链数可以使用多项概率分布来描述。我们推导出了一系列完整的递归关系，用于描述无、1 个或 2 个微卫星不稳定性的扩增子链数的期望和（协）方差。我们给出了该方程组的精确数学解，揭示了期望和（协）方差随循环数的发展。该方程组表明，无微卫星不稳定性的扩增子链数的期望呈指数增长，但对于具有微卫星不稳定性的链，其关系更为复杂。链数的相对标准偏差（变异系数）与进入扩增的 DNA 链数的平方根成反比。因此，对于高拷贝数 DNA，可以忽略随机效应，但在低浓度下，它们会起重要作用。对于等位基因峰，变异系数在几个循环后迅速稳定，但对于具有微卫星不稳定性的链，其下降速度较慢。此外，微卫星峰的预期强度与等位基因峰的比值随循环数呈线性增加。在贝叶斯背景下，像本文中提出的随机模型可以在进一步制定解释规则方面发挥重要作用。
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