Biological systems are nothing like anything we would ever engineer, and to understand them we must remove our "anthropocentric engineer goggles" and look at them for what they are. Analogies between biological systems and computers, software, machines, etc. are very loose analogies meant to illustrate a point. Never take these analogies too seriously.
DNA is not a program-- it is a molecule, and one that may very well do things at the quantum level that are biologically important.
The brain is not a neural network. It is an interconnected colony of living cells of a variety of types, and it has been shown that all types of cells in the brain are involved in cognition.
We are nowhere near anything with the parallelism or information density of the brain. Getting close would take an advancement in computer technology of the magnitude of the transition from vacuum tubes in individual boxes to a 32nm Core i7. Nature has had billions of years, and it is way ahead of us. There's a nascent field called quantum biology that suggests that the brain may very well be a quantum computer, so maybe quantum computers are the vacuum tubes->ICs scale transition I am speaking of.
I think it's now possible with a big rack of multi-core machines in an MPI cluster to approach the capabilities of a fruit fly's brain.
Cells are not machines. I don't think we have the language to really describe what they are perfectly, but the closest I can come is "stochastic quantum probability field device." A bacterial flagellum is not a "motor," it is a quantum-scale chemo-electro-motive... well... our language breaks down. Like Feynman said, don't tell stories. Just speak literally and then use math.
Actually, come to think of it, there's one engineering analogy that might work for biology. Biology is quantum-scale nanotechnology. Yeah, that's pretty close.
(On a related tangent, I've found that many engineers are sympathetic to intelligent design type arguments against evolution. This is because they try to think about biology like engineers and take these machine analogies literally. It just doesn't work like that.)