DReSS: A difference measurement based on reachability between state spaces of Boolean networks

Researches on dynamical features of biological systems are mostly based on fixed network structure. However, both biological factors and data factors can cause structural perturbations to biological regulatory networks. There are researches focus on the influence of such structural perturbations to the systems’ dynamical features. Reachability is one of the most important dynamical features, which describe whether a state can automatically evolve into another state. However, there is still no method can quantitively describe the reachability differences of two state spaces caused by structural perturbations. DReSS, Difference based on Reachability between State Spaces, is proposed in this research to solve this problem. First, basic properties of DReSS such as non-negativity, symmetry and subadditivity are proved based on the definition. And two more indexes, diagDReSS and iDReSS are proposed based on the definition of DReSS. Second, typical examples like DReSS = 0 or 1 are shown to explain the meaning of DReSS family, and the differences between DReSS and traditional graph distance are shown based on the calculation steps of DReSS. Finally, differences of DReSS distribution between real biological regulatory network and random networks are compared. Multiple interaction positions in real biological regulatory network show significant different DReSS value with those in random networks while none of them show significant different diagDReSS value, which illustrates that the structural perturbations tend to affect reachability inside and between attractor basins rather than to affect attractor set itself. Author summary Boolean network is a kind of networks which is widely used to model biological regulatory systems. There are structural perturbations in biological systems based on both biological factors and data-related factors. We propose a measurement called DReSS to describe the difference between state spaces of Boolean networks, which can be used to evaluate the influence of specific structural perturbations of a network to its state space quantitively. We can use DReSS to detect the sensitive interactions in a regulatory network, where structural perturbations can influence its state space significantly. We proved properties of DReSS including non-negativity, symmetry and subadditivity, and gave examples to explain the meaning of some special DReSS values. Finally, we present an example of using DReSS to detect sensitive vertexes in yeast cell cycle regulatory network. DReSS can provide a new perspective on how different interactions affect the state space of a specific regulatory network differently.