If the removal of some part of the system causes the loss or disturbance of one of its functions, then it is reasonable to attribute that function to the removed part. This intuition underlies, for instance, some of our understanding of brain function which has been deduced from brain lesions. Such lesions are typically caused by an injury or disease. However, the same fundamental intuition also underlies knockout experiments that aim at the identification of the system’s mechanisms. One well-known example is given by gene knockouts.
Let us now study this intuition based on our formal model of knockout interventions which we developed in “Mechanistic modelling of knockout interventions” section. We begin with the example shown in Fig. 6. We have already discussed the knockout of node 3, which gave us the post-knockout function (11). This implies
$$\begin{aligned}&\ln \kappa _{\{1,2,4,5,6\}}(x_1,x_2, x_4, x_5, x_6; y)\nonumber \\&\quad = Z(x_1,x_2,x_4,x_5,x_6) + \phi _{\{5,6\}}(x_5, x_6; y). \end{aligned}$$
(17)
If we knock out nodes 4 and 5, the remaining contribution is from the nodes 1, 2, and 3:
$$\begin{aligned}&\ln \kappa _{\{1,2,3\}}(x_1, x_2,x_3; y) \nonumber \\&\quad = Z(x_1,x_2,x_3) + \phi _{\{1,2,3\}}(x_1, x_2, x_3; y). \end{aligned}$$
(18)
Finally, for the knockout of nodes 1 and 5, the remaining contribution comes from nodes 3 and 4:
$$\begin{aligned}&\ln \kappa _{\{3,4\}}(x_3, x_4; y) \nonumber \\&\quad = Z(x_3,x_4) + \phi _{\{3,4\}}(x_3, x_4; y). \end{aligned}$$
(19)
This shows that we can recover the interactions, up to some function of \(\varvec{x}\), given by the normalisations Z, if we know the post-knockout function of some appropriately chosen knockouts. This backs up the strategy of system identification in terms of knockout experiments. The non-uniqueness of the recovered interaction terms, due to the functions that only depend on the input \(\varvec{x}\), does not harm here. Any family of interaction terms will represent the functional modalities \(\kappa _J\), \(J \subseteq I\), equally well.
Note that the knock-outs that reveal the individual interaction terms are also not unique. There are several appropriate knockout protocols. For instance, in the above example, we could also consider the three knockouts of node 1, node 4, and node 5, leading to a linear system
$$\begin{aligned}&\ln \kappa _{\{2,3,4,5,6\}}(x_2, x_3, x_4, x_5, x_6; y) \\&\quad = Z(x_2,x_3, x_4, x_5, x_6) \\&\quad \quad +\,\phi _{\{3,4\}}(x_3, x_4 ; y) + \phi _{\{5,6\}}(x_5, x_6 ; y) , \\&\ln \kappa _{\{1,2,3,5,6\}}(x_1, x_2 , x_3, x_5, x_6; y)\\&\quad =\, Z(x_1, x_2, x_3, x_5, x_6) \\&\quad \quad +\,\phi _{\{1,2,3\}}(x_1, x_2, x_3 ; y) + \phi _{\{5,6\}}(x_5, x_6 ; y) , \\&\ln \kappa _{\{1,2,3,4,6\}}(x_1,x_2, x_3, x_4, x_6; y)\\&\quad =\,Z(x_1, x_2, x_3, x_4,x_6)\\&\quad \quad +\,\phi _{\{1,2,3\}}(x_1, x_2, x_3 ; y) + \phi _{\{3,4\}}(x_3, x_4 ; y). \end{aligned}$$
This can be easily solved, and we obtain again the interactions from the post-knockout functions. Altogether, we have shown that two different knockout protocols, each of them involving only three knockout interventions, allow us to identify the three interaction terms of the system. In general, the situation is much more complex. On the one hand, there can be many more interaction terms involved so that we will require correspondingly many knockout interventions. We cannot assume that performing these interventions will always be feasible. On the other hand, even if only a few interactions are actually involved, without prior knowledge on these interactions it is not possible to determine a correspondingly small set of knockouts that would allow us to reveal the interactions within the system. Let us assume, for the moment, that we can perform all possible knockouts and measure the corresponding post-knockout functions. In theory, the resulting set of equations can be solved in terms of the Möbius inversion. Let us be more precise: Clearly, each \(\kappa _J\) in (13) is strictly positive. Using the Möbius inversion, it is easy to see that each strictly positive family \((\kappa _J)\) has such a representation. In order to see this, we simply set
$$\begin{aligned} \phi _A (\varvec{x}_A, y) \; := \; \sum _{J \subseteq A} (-1)^{| A {\setminus } J |} \ln \kappa _J(\varvec{x}_J ; y) \, . \end{aligned}$$
(20)
Note that this representation is not unique: If an arbitrary function of \(\varvec{x}_{A}\) is added to the function \(\phi _{A}\), then the functional modalities remain unchanged. We have a one-to-one correspondence, modulo this ambiguity, between functional modalities and interactions:
$$\begin{aligned} \kappa _J, \; J \subseteq I \,\longleftrightarrow \, \phi _A, \;A \subseteq I. \end{aligned}$$
In this section, we do not interpret a knockout as a natural perturbation but as an experimental test. The intention is to reveal the inner working, the mechanisms, of the system based on its response to these unnatural perturbations. Indeed, we can identify important aspects of the interactions. This is remarkable, because these interactions provide the basis of the unperturbed function. Thus, we perturb the system in order to understand the constituents of its unperturbed function. This provides a formal basis for knockout experiments.
It is important to note that the choice of the \(\phi _A\), \(A \subseteq I\), for describing knockouts is essential. If we choose to represent a post-knockout function in terms of different interactions, say \(\psi _B\), \(B \subseteq I\), we have to translate the effect of knockouts accordingly. To be more precise, assume that we can represent the initial functions as
$$\begin{aligned} \phi _A(\varvec{x}_A ; y) \, = \, \sum _{B \subseteq A} \alpha _{A, B} \, \psi _B (\varvec{x}_B ; y), \qquad A \subseteq I. \end{aligned}$$
(21)
This implies
$$\begin{aligned} \ln \kappa _J(\varvec{x}_J ; y)\sim & {} \sum _{A \subseteq J} \phi _A(\varvec{x}_A ; y) \qquad \qquad (\text{by } (13)) \end{aligned}$$
(22)
$$\begin{aligned}= & {} \sum _{A \subseteq J} \sum _{B \subseteq A} \alpha _{A, B} \, \psi _B (\varvec{x}_B ; y) \qquad \quad \;\;(\text{by } (21)) \nonumber \\= & {} \sum _{B \subseteq J} \left( \sum _{C \subseteq J {\setminus } B} \alpha _{B \cup C, B} \right) \, \psi _B (\varvec{x}_B ; y) \nonumber \\= & {} \sum _{B \subseteq J} \beta _{J,B} \, \psi _B (\varvec{x}_B ; y) , \end{aligned}$$
(23)
where we define the coefficients \(\beta _{J,B}\) accordingly. This calculation highlights the following fact: When describing the post-knockout function \(\kappa _J\) in terms of the \(\phi _A\) we simply remove all terms \(\phi _A\) for which A is not contained in J, as described by (22). This is the basic requirement that defines the \(\phi _A\), \(A \subseteq I\). When trying to represent the post-knockout function in terms of different interactions, \(\psi _B\), \(B \subseteq I\), this rule changes. In order to see this, let us analyse the representation (23). Also in this case, the terms \(\psi _B\) do not appear whenever B is not contained in J. However, the coefficients \(\beta _{J,B}\) change as a consequence of a knockout. That means, if we have an expansion of \(\ln \kappa (\varvec{x} ; y)\), where \(\kappa \) is the unperturbed function, into terms \(\psi _B\), with weights \(\beta _{I,A}\), then simply removing weighted terms as result of a knockout is not sufficient.