The phrase "be permeated with" means to be completely filled or saturated with something. However, there are several synonyms that can be used to express the same idea. For example, the words "imbued," "infused," and "penetrated" can all be used in place of "permeated with" to describe a situation where something has deeply influenced or saturated a place or person. Other synonyms include "suffused," "pervaded," and "saturated." All of these words carry a similar connotation of something filling or saturating a space, and can be used interchangeably to add variety and interest to your writing.