There is growing consensus among neuroscientists that neural circuits critical for survival are the result of genomic decompression processes. We introduce SynaptoGen, a novel computational framework--member of the Connectome Models family--bringing synthetic biological intelligence closer, facilitating neural biological agent development through precise genetic control of synaptogenesis. SynaptoGen is the first model of its kind offering mechanistic explanation of synaptic multiplicity based on genetic expression and protein interaction probabilities. The framework connects genetic factors through a differentiable function, working as a neural network where synaptic weights equal average numbers of synapses between neurons, multiplied by conductance, derived from genetic profiles. Differentiability enables gradient-based optimization, allowing generation of genetic expression patterns producing pre-wired biological agents for specific tasks. Validation in simulated synaptogenesis scenarios shows agents successfully solving four reinforcement learning benchmarks, consistently surpassing control baselines. Despite gaps in biological realism requiring mitigation, this framework has potential to accelerate synthetic biological intelligence research.