For a deeper understanding of how historically analogies between living cells and computers have been made, see videos here and here by Sydney Brenner.

For a detailed analogy between living cells and computers, see:
Danchin A (2008) Bacteria as computers making computers. FEMS Microbiology Reviews 33: 3–26

For an advocacy of the living cells as Turing machines by Sydney Brenner, see:
Brenner S (2012) Turing centenary: Life’s code script. Nature 482: 461

For a historical perspective on the concept of information in biology, see:
Cobb M (2013) 1953: when genes became “information”. Cell 153:503-506

For an analysis of various state machines, see figure 3 in:
Benenson Y (2012) Biomolecular computing systems: principles, progress and potential. Nat Rev Genetics 13: 455–468

For a better understanding of the links between languages and state machines and how they are powerful tools for expressing biological messages, see:
Searls DB (2002) The language of genes. Nature 420: 211–217

Regarding biological computations and algorithms and examples thereof, see:
Navlakha S, Bar-Joseph Z (2011) Algorithms in nature: the convergence of systems biology and computational thinking. Mol Syst Biol 7: 546
Cardelli L, Hernansaiz-Ballesteros RD, Dalchau N, Csikasz-Nagy A (2017) Efficient switches in biology and computer science. Plos Comput Biol 13: e1005100
Zeng L, Skinner SO, Zong C, Sippy J, Feiss M, Golding I (2010) Decision making at a subcellular level determines the outcome of bacteriophage infection. Cell 141: 682–691

For an incursion into the relationships between information and energy and the possibility to create molecular devices making use of information, see:
Serreli V, Lee CF, Kay ER, Leigh DA (2007) A molecular information ratchet. Nature 445: 523–527
Lutz E, Ciliberto S (2015) Information: from Maxwell’s demon to Landauer’s eraser. Physics Today 68: 30

For an application of Maxwell’s demon to biological problems, see:
Binder P, Danchin A (2011) Life’s demons: information and order in biology. What subcellular machines gather and process the information necessary to sustain life? EMBO Reports 12: 495–499

Chromatin reading and chromatin writing, transcription, bacterial chemotaxis or DNA recombination enzyme assembly have been framed as a Turing Machine. For details, see:
Bryant B (2012) Chromatin computation. PLoS One 7: e35703
Lan G, Tu Y (2016) Information processing in bacteria: memory, computation, and statistical physics: a key issues review. Reports on progress in physics. Physical Society 79: 052601
Bar-Ziv R, Tlusty T, Libchaber A (2002) Protein-DNA computation by stochastic assembly cascade. Proc Natl Acad Sci U S A 99: 11589–11592

Computer scientist Charles Bennett compared the RNA polymerase with the read/write head of a Turing Machine and came up with considerations on energy dissipation during proofreading that shed new light on the physical dimension of the concept of information as a fifth category of Nature. For details, see:
Bennett CH (1973) Logical reversibility of computation. Ibm J Res Dev 17: 525–532

For a view on evolutionary algorithmics and robotics, and the proposition that real robots or hardware models can be a testing ground for biologists to investigate questions and hypothesis, for instance reproducing the selective pressure driving the evolution of biological
organisms, see:
Eiben AE, Smith, JE (2003) Introduction to evolutionary computing. Berlin: Springer
Eiben AE, Smith, JE (2015) From evolutionary computation to the evolution of things. Nature 521: 476–482