Modify the "/author" rewrite rule in WordPress to read "/profile" (permalinks refresh required).

matty-profile-rewrite.php

<?php
/**
 * Plugin Name: Add "/profile" URL rewrite rule.
 * Version: 1.0.0
 * Author: Matty
 * Description: Add a rewrite rule to redirect "/profile" URLs to the appropriate author archive screen.
 */

new Matty_Profile_Rewrite();

class Matty_Profile_Rewrite {
	public function __construct () {
		add_filter( 'author_rewrite_rules', array( 'Matty_Profile_Rewrite', 'modify_author_base' ) );
	} // End __construct()

	public function modify_author_base ( $rules ) {
		$new_rules = array();
		foreach ( (array)$rules as $k => $v ) {
			$key = str_replace( 'author/', 'profile/', $k );
			$new_rules[$key] = $v;
		}
		$rules = $new_rules;
		return $rules;
	} // End modify_author_base()
} // End Class
?>