https://pubmed.ncbi.nlm.nih.gov/30237357/
Programmable protein circuits in living cells
However, they are also self-assembling, self-repairing, wet, massively parallel, etc. Artificial biological systems are in a sense programmable, but with the difficulties of a physical system that can be error-prone and extremely complex.
What OP seems to be talking about is systems modelling, where you model the network of genes and proteins (genomics and proteomics) using tools partly derived from modelling of computer networks.
In other words, a bioinformatician is to a computer scientist what a computational biologist is to a data scientist.
Using biological systems to do computing is more in the realm of "biological computers".