A three dimensional continuous space Monte Carlo computer simulation study is presented to discuss the extension of flexible, linear polymer chains due to the presence of equally flexible side chains. We consider the enhancement of the persistence length of bottle-brush structures in an athermal solution due to steric interactions between the side chains. The largest structure studied consists of a backbone of 100 beads with 50 side chains of 20 beads each. The persistence length λ is evaluated in two different ways using the radius of gyration of the backbone and the bond angle correlation function, respectively. A correct description of the backbone conformations is shown to require at least two characteristic lengths. At a small length scale the backbone behaves flexible; the extension occurs at a larger length scale. There is a strong indication that the ratio between the persistence length and the diameter, which is the determining factor for lyotropic behavior of conventional semiflexible chains, levels off as a function of the side chain length. The value of this ratio is, moreover, too small to induce lyotropic behavior along this line. Recent experimental observations of lyotropic behavior of polymacromonomers are discussed in terms of these findings.