Turing power of induced HGRNs

Determine whether the subclass of induced hybrid gene regulatory networks, defined for networks with two discrete levels per gene whose celerities are given by c(α)j = ∑_{i: αi=1} ρ(i,j) − ∑_{i: αi=0} ρ(i,j) for a complete weighted directed influence graph H = (V,E,ρ), is Turing-powerful; that is, ascertain whether such induced HGRNs can simulate arbitrary Turing machines via their hybrid trajectories.

Background

The paper proves that general hybrid gene regulatory networks (HGRNs) are Turing-powerful by constructing a simulation of a deterministic Turing machine using an HGRN with appropriately designed celerities and multiple discrete levels per gene.

Later, the authors introduce a subclass called induced HGRNs, where each gene has two discrete levels and celerities are linear combinations determined by a complete weighted directed influence graph. The open problem asks whether this restricted, more structured subclass retains Turing power.