The list from Table 3 only accounts for transitions in the forward direction towards the attractor. By extending the corresponding lists to include the backward transitions, we generate a new rule-set that extends to a new network including all possible backward pathways as well. Note, however, that we can only include backward pathways starting with level 1 and higher. Including a backward path for level 0, i.e., the attractor, means that the simulation can leave this state and it therefore would be no longer be a steady state.

Using the extended transition sets and the same translation into updating rules, we obtain

The network structure and simulation results for this rule-set and the initial state (1, 1, 0, 0) is depicted in the middle row of Fig 3 on the left.

The network depiction demonstrates, that all paths in the network are now enabled. I.e., the state (1, 1, 0, 0) can also transition into (1, 1, 1, 0), which means that the product ES is generated. We observe, that the simulation now both consumes enzyme and creates the complex ES before the attractor is reached. The dynamics, depicted in the middle row of Fig 3 on the right, match the ground truth from Fig 2 well. We can also see, that it takes noticeably longer for all simulations to reach their overall attractor. This makes sense, since backward paths also enable simulations to loiter in loops.

Reduction of logic-rule search space with experimental data (ES-E). Both networks described above are created by automatically mapping initial states to their corresponding attractor without any additional knowledge. Due to the construction of our method, however, it is straightforward to include expert knowledge into the dynamics as well.

Let us for example look back at the construction of our first network. We have noted that in this case we omit the pathway for the creation of the complex ES. We are, however, aware that this part is a necessary step of the dynamics. In this example, we therefore propose to start with the forward-pathway network and add the transitions for (1, 1, 0, 0) to (1, 1, 1, 0), as well as the resulting consumption of E, namely the transition from (1, 1, 1, 0) to (0, 1, 1, 0) to the corresponding transition list.

The resulting ruleset is

and in Fig 3 bottom, we see the resulting network (left) and the corresponding simulation for the initial state (1, 1, 0, 0) (right). Since we only added the absolute minimum necessary to create ES, most of the loops from the backward pathways model are omitted and the simulation reaches the steady state in a similar time frame as the simulation with the forward pathways only while also capturing some of the dynamics of the ES creation and E consumption.

Note, that in this case we manually added transitions to the network we judged feasible. We also provide the option to exclude transitions that the user is certain are biologically unfeasible.

Our implementation enables the user to start with either the forward-path network, or the full backward path including network from which transitions can be added or removed as seen fit. Note, however, that not all removals are valid to keep the network dynamics: adding and/or removing random transitions could result in the following problems:

adding a transition that leads directly away from the attractor will result in a loss of this attractor as a steady state

adding a transition could create a pathway to the wrong attractor

removing a transition could make it impossible for a state to reach its attractor

Since the full backward path network includes all possible pathways between all nodes in the network, for an unknown process, we recommend to start with the full backward path network and start strategically removing transitions from there. This way, we can be certain that the necessary network connections are present, while we only need to assure that point 3. of the list is not violated. Note, however, that due to the many loops that are created in this network, more steps are required by the asynchronous updating scheme before equilibrium is achieved.

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