  The mean square size of topologically interlocked molecules (TIMs) is
presented as a linear combination of contributions from the backbone and
subcomponents. Using scaling analyses and extensive molecular dynamics
simulations of polycatenanes, as a typical example of TIMs, we show that the
effective exponent $\nu(m)$ for the size dependence of the backbone on the
monomer number of subcomponent $m$ is asymptotic to a value $\nu$
(approximately 0.588 in good solvents) with a correction of $m^{-0.47}$, which
is the same as for the covalently linked polymer. However, the effective
exponent for the size dependence of subcomponents on $m$ is asymptotic to the
same value $\nu$ but with a new correction of $m^{-1.0}$. The different
corrections to the scaling on the backbone and subcomponent structure induce a
surprising double asymptotic behavior for the architecture of the TIMs. The
scaling model that takes into account the double asymptotic behavior is in good
quantitative agreement with the simulation result that the effective exponent
for the size dependence of TIMs on $m$ increases with the subcomponent number
$n$. The full scaling functional form of the size dependence on $m$ and $n$ for
polycatenanes in a good solvent is well described by a simple sum of two
limiting behaviors with different corrections.
