Strand displacement & seesaw gates. The primitives are built and characterised — Qian & Winfree's seesaw gates made working DNA logic and a 4-neuron Hopfield net in a tube (2011).
DNA neural nets. Cherry & Qian (2018) classified handwritten digits in solution — real, but hours-slow, one-shot, and hard to reset or scale.
Practical / in-vivo molecular computers. Reusable, fast, in-cell DNA computation — open. Today it's a beautiful proof, not a workhorse.
Two strands — an input and a weight — drift in solution. They react only when they bump into each other, and the rate of bumping is set by the law of mass action: rate = k · [input] · [weight] A product of concentrations is a multiply, done by diffusion. Turn the weight strand up and the same input produces more output: a tunable synapse made of nucleotides.
Summing is easy — many reactions feed one output strand and its concentration adds them up. The hard part, the nonlinearity, is a real trick: a threshold strand that greedily consumes signal until it's used up, after which the output restores sharply. The result is a sigmoid in chemistry — below the threshold, almost nothing; above it, a clean ON. That seesaw gate is a DNA neuron.
Set the weight concentrations and the molecular neuron learns AND/OR; one gate is one cut, so XOR needs a second layer of gates fed by the first's restored output — depth, in cascaded reactions. The gift is parallelism and place: the computation happens everywhere in the volume at once, in the same medium as biology.
Honest place in the family: DNA won't out-run silicon at anything you'd time with a clock. Its case is where it runs — in a droplet, in tissue, in the body — doing a small classification with no hardware at all. A real perceptron you could, in principle, swallow.