Rule 110 Turing machines

Given an initial condition for rule 110, the initial condition for the Turing machine shown here is obtained as Prepend[4 list, 0] with 1's on the left and 0's on the right. The Turing machine

{{1, 2}  {2, 2, -1}, {1, 1}  {1, 1, -1}, {1, 0}  {3, 1, 1}, {2, 2}  {4, 0, -1}, {2, 1}  {1, 2, -1}, {2, 0}  {2, 1, -1}, {3, 2}  {3, 2, 1}, {3, 1}  {3, 1, 1}, {3, 0}  {1, 0, -1}, {4, 2}  {2, 2, 1}, {4, 1}  {4, 1, 1}, {4, 0}  {2, 2, -1}}

with s = 4 states and k = 3 possible colors also emulates rule 110 when started from Prepend[list + 1, 1] surrounded by 0's. The s = 3, k = 4 Turing machine

{{1 , 0}  {1, 2, 1}, {1, 1}  {2, 3, 1}, {1, 2}  {1, 0, -1}, {1, 3} {1, 1, -1}, {2, 0}  {1, 3, 1}, {2, 1}  {3, 3, 1}, {3, 0}  {1, 3, 1}, {3, 1}  {3, 2, 1}}

started from Append[list, 0] with 0's on the left and 2's on the right generates a shifted version of rule 110. Note that this Turing machine requires only 8 out of the 12 possible cases in its rules to be specified.