We present simulations of non-enzymatic template-directed RNA synthesis that incorporate primer extension, ligation, melting, and reannealing. Strand growth occurs over multiple heating/cooling cycles, producing strands of several hundred nucleotides in length, starting with random oligomers of 4 to 10 nucleotides. A strand typically grows by only 1 or 2 nucleotides in each cycle. Therefore, a strand is copied from many different templates, not from one specific complementary strand. A diverse sequence mixture is produced, and there is no exact copying of sequences, even if single base additions are fully accurate (no mutational errors). It has been proposed that RNA systems may contain a virtual circular genome, in which sequences partially overlap in a way that is mutually catalytic. We show that virtual circles do not emerge naturally in our simulations, and that a system initiated with a virtual circle can only maintain itself if there are no mutational errors and there is no input of new sequences formed by random polymerization. Furthermore, if a virtual sequence and its complement contain repeated short words, new sequences can be produced that were not on the original virtual circle. Therefore the virtual circle sequence cannot maintain itself. Functional sequences with secondary structures contain complementary words on opposite sides of stem regions. Both these words are repeated in the complementary sequence; hence, functional sequences cannot be encoded on a virtual circle. Additionally, we consider sequence replication in populations of protocells. We suppose that functional ribozymes benefit the cell which contains them. Nevertheless, scrambling of sequences occurs, and the functional sequence is not maintained, even when under positive selection.