O'Hely Martin
Department of Mathematics, University of Queensland, Brisbane, QLD, 4072, Australia.
J Math Biol. 2006 Aug;53(2):215-30. doi: 10.1007/s00285-006-0001-6. Epub 2006 May 6.
Consider a haploid population and, within its genome, a gene whose presence is vital for the survival of any individual. Each copy of this gene is subject to mutations which destroy its function. Suppose one member of the population somehow acquires a duplicate copy of the gene, where the duplicate is fully linked to the original gene's locus. Preservation is said to occur if eventually the entire population consists of individuals descended from this one which initially carried the duplicate. The system is modelled by a finite state-space Markov process which in turn is approximated by a diffusion process, whence an explicit expression for the probability of preservation is derived. The event of preservation can be compared to the fixation of a selectively neutral gene variant initially present in a single individual, the probability of which is the reciprocal of the population size. For very weak mutation, this and the probability of preservation are equal, while as mutation becomes stronger, the preservation probability tends to double this reciprocal. This is in excellent agreement with simulation studies.
考虑一个单倍体种群，在其基因组内有一个基因，该基因的存在对任何个体的生存至关重要。这个基因的每个拷贝都可能发生突变从而破坏其功能。假设种群中的一个个体以某种方式获得了该基因的一个重复拷贝，其中这个重复拷贝与原始基因的位点完全连锁。如果最终整个种群都由最初携带该重复拷贝的这个个体的后代组成，那么就说发生了保存。该系统由一个有限状态空间的马尔可夫过程建模，而这个马尔可夫过程又由一个扩散过程近似，由此推导出保存概率的显式表达式。保存事件可以与最初存在于单个个体中的选择性中性基因变体的固定进行比较，其概率是种群大小的倒数。对于非常弱的突变，这个概率与保存概率相等，而随着突变变强，保存概率趋于这个倒数的两倍。这与模拟研究非常吻合。
