Stringent, stochastic viral population bottlenecks have been observed in the infections of many 5 viruses, but exactly how and why they occur is unclear. A critical review of recent literature prompts us 6 to propose a new hypothesis, designated Isolate, Amplify, and Select (IAS), that satisfactorily explains 7 the bottlenecks in single-stranded, positive sense (+) RNA virus infections. This new hypothesis 8 postulates that, unlike those in free-living organisms, the viral population bottlenecks are imposed by 9 viruses themselves, inside the infected cells, through virus-encoded bottleneck-enforcing proteins 10 (BNEPs) that function in a concentration-dependent manner. Most BNEPs are directly translated from 11 the (+) RNA genomes of invading viruses, so that if numerous virions of the same virus invade a cell 12 simultaneously, the bottleneck-ready concentration of BNEPs would be reached sufficiently early to 13 arrest nearly all internalized viral genomes. As a result, in these cells very few (as few as one) viral 14 genomes escape from the bottlenecks stochastically to initiate viral reproduction. Repetition of this 15 process in successively infected cells ensures the progeny genomes in each cell descend from the same 16 parental genome(s), hence isolating different mutant genomes in separate cells. This isolation precludes 17 mutant viral genomes encoding defective replication proteins from exploiting the complementing 18 proteins synthesized by sister genomes, leading to the prompt elimination of such mutants. Conversely, 19 the IAS model ensures replication proteins with beneficial mutations exclusively amplify the viral 20 genomes that harbor the very mutations. Reiteration of this process in consecutively infected cells 21 enriches such beneficial mutations in the virus pool. In conclusion, the IAS hypothesis provides a 22 compelling evolutionary model for population bottlenecks of (+) RNA viruses.