  We study methylation dynamics of the chemoreceptors as an {\sl E.coli} cell
moves around in a spatially varying chemo-attractant environment. We consider
attractant concentration with strong and weak spatial gradient. During the
uphill and downhill motion of the cell along the gradient, we measure the
temporal variation of average methylation level of the receptor clusters. Our
numerical simulations show that the methylation dynamics depends sensitively on
the size of the receptor clusters and also on the strength of the gradient. At
short times after the beginning of a run, the methylation dynamics is mainly
controlled by short runs which are generally associated with high receptor
activity. This results in demethylation at short times. But for intermediate or
large times, long runs play an important role and depending on receptor
cooperativity or gradient strength, the qualitative variation of methylation
can be completely different in this time regime. For weak gradient, both for
uphill and downhill runs, after the initial demethylation, we find methylation
level increases steadily with time for all cluster sizes. Similar qualitative
behavior is observed for strong gradient during uphill runs as well. However,
the methylation dynamics for downhill runs in strong gradient show highly
non-trivial dependence on the receptor cluster size. We explain this behavior
as a result of interplay between the sensing and adaptation modules of the
signaling network.
