  Chromatin fibers composed of DNA and proteins fold into consecutive loops to
form rod-shaped chromosome in mitosis. Although the loop growth dynamics is
investigated in several studies, its detailed processes are unclear. Here, we
describe the time evolution of the loop length for thermal-driven loop growth
processes as an iterative map by calculating physical quantities involved in
the processes. We quantify the energy during the chromatin loop formation by
calculating the free energies of unlooped and looped chromatins using the
Domb-Joyce model of a lattice polymer chain incorporating the bending
elasticity for thermal-driven loop growth processes. The excluded volume
interaction among loops is integrated by employing the mean-field theory. We
compare the loop formation energy with the thermal energy and evaluate the
growth of loop length via thermal fluctuation. By assuming the dependence of
the excluded volume parameter on the loop length, we construct an iterative map
for the loop growth dynamics. The map demonstrates that the growth length of
the loop for a single reaction cycle decreases with time to reach the condensin
size, where the loop growth dynamics can be less stochastic and be regarded as
direct power stroke of condensin as a kind of motor proteins.
