We apply elements of control theory 16 to develop a generalized model of the network dynamics. A regulatory network (Fig. 1) is represented as a bipartite graph with two types of nodes: observable nodes reproducing measurable characteristics (e.g. gene expression levels), and non-observable, or control, nodes controlling the interactions between the observable nodes. Each control node i can be modelled as:

where F_i is a functional reproducing behaviour, Y_I(·), of a set of observable nodes I based on signals, Y_O(·), from a, possibly different, set of observable nodes O, and W_i is a vector of "internal" parameters of control node i. Note that some non-trivial behaviour can be assigned to the observable nodes as well. It may account for instrumental distortions, specifics of image processing, normalization, etc.

The goal of the network reconstruction is to identify parameters W_i encoding for the interactions between the observable nodes. For that, functional F_i in (1) has to be further developed. It is frequently assumed that the cooperative regulatory contribution from different observable nodes is a sum of the contributions from each node, so that equation (1) can be written as:

where n is the number of observable nodes, y_i(t) is the measured response of observable node i, F_ij is a functional characterized by a set of parameters W_ij converting measured profile, y_j(t), at node j to measured profile, y_i(t), at node i, and b_i(t, t₀) is the output of non-regulated observable node i. We consider pair-wise controls F_ij as linear, continuous, time-invariant, finite-dimensional, single input-single output control systems that can be modelled using the state-space formalism:

where X_ij(t) is the state vector and A_ij is the state matrix, y_j(t) is the input value and B_ij is the input vector, y_j(t) is the output value and C_ij is the output vector. We also assume that F_ij are in a steady state prior to the input perturbation y_j(t) starting at time t = t₀, that is X_ij(t₀) = 0. Integrating (3) and combining n regulatory inputs as in (2) yields

with w_ij(t) = C_ijexp(tA_ij)B_ij representing the influence of node j on the regulation of node i. Although every link (control node) is unique and should be modelled in a specific way, little prior knowledge on molecular interactions does not allow us to postulate specific models for every link. Therefore, we are looking for universal models that can approximate any control node.

The LODE regulatory model is widely used in the network reconstruction 689. It can be obtained from (4), if we set w_ij(t) = const = w_ij and b_i(t, t₀) = const×t = b_it:

This model approximates system relaxation into a steady state after a small perturbation. However, it is difficult to confirm that perturbations are small enough to justify model (5).

Equation (4) allows us to create a number of less restrictive models that can cover broader spectrum of dynamical behaviours. These models can integrate prior knowledge or can be further refined in experimental data analysis. In this report, we use the following representations for w_ij(t):

where L is the number of terms, u_{l, ij} are the coefficients encoding for the regulation of node i by node j and τ_l are the characteristics times that can be either set as prior values or estimated from experimental data. The background functions b_i(t, t₀) can also be developed, but we will keep them constant as, with little data, more complicated models for b_i(t, t₀) can fit the data without identifying any link.

We have devised a library of eight models (Table 1) to be tested and compared. Rationale for using the selected kernel functions is given in [Additional file 1].

Discussion on the parameter identifiability for the developed models can be found in [Additional file 2].

Network reconstruction is done by fitting the developed models to experimental data. Among different fitting strategies 17, the forward selection (FS) algorithm has shown reasonable performance, in particular for sparse networks, and therefore, it has been adopted in this paper. We refer to 18 for the details on the implementation of the FS algorithm. A more robust modification of the FS algorithm has also been tested as described in [Additional file 3].

We can use prior knowledge on the nodes' interactions to select the best network reconstruction model from the pre-defined library (Table 1). We look for the kernel function w_ij(t) that reconstructs the prior links with the highest accuracy. The description of the adaptive model selection (AMS) algorithm can be found in [Additional file 4].

We compared the performances of the eight kernel functions from Table 1 as well as the LODE regulatory model (5) using simulated and experimental data. Three artificial systems were used for testing: the oscillating network in E. coli, called repressilator 19, the mitogen-activated protein kinase (MAPK) cascade 20 and the glycolysis pathway in yeast 21. We also used the yeast (Saccharomyces cerevisiae) cell cycle microarray time-series data 22 to demonstrate applicability of the developed approach to real experimental data. The positive predictive value (PPV) and sensitivity (Se) were applied to estimate the performance. Further details on the artificial and real systems used for testing and description of the testing procedure can be found in [Additional file 5].

The dependencies of PPV on the total number of links are presented in Fig. 2. The Se values at 50 generated links are collected in Table 2. Among the three artificial systems, the choice of a model was the most critical for the E. coli repressilator. In this case, the best reconstruction was achieved with the bi-exponential E3 model. The LODE model performed better than random reconstruction but still worse than E3. All tested kernels were significantly better than random link assignment for the MAPK cascade. All kernels also outperformed the LODE model in this case. However, there is still a notable (and statistically significant) difference between the kernels. The yeast glycolysis network (Fig. 2c) was the most difficult to reconstruct because many times series were similar and hardly distinguishable by the reconstruction algorithm. Nevertheless, several models (P1, P2, E2, E3, and I2) demonstrated the performance different from random. The LODE model could not outperform the random prediction in this case.

For the yeast cell cycle time-series data, the polynomial models (P1 and P2) were the most powerful. For the alpha dataset and for the elu dataset, P1 had the highest performance whereas P2 was the most accurate for cdc15. Note that, in each case, the best performing models (P1 and P2) also outperformed the LODE model. Comparing different experiments, we see that cdc15 led to less accurate predictions. This indicates that this experiment requires more elaborated reconstruction models or more representative datasets.

From Fig. 2 and Table 2, we can conclude that the "optimal" models were different for the artificial and real systems. The obtained results suggest that no unique model exists to ensure reasonable performance for different systems and therefore the most appropriate models should be searched for each system.

We applied the AMS algorithm [Additional file 4] to the same three artificial systems and three experimental datasets. As at each run the prior links were different, the selected model might also be different. Therefore, we counted number of times each model from Table 1 was selected in the 100 runs. The results for 2 and 10 prior links are shown in Fig. 3. We found that the higher performing models from Fig. 2 were selected more often than the lower performing ones. Moreover, reasonable model recognition could be already achieved with only two prior links. As expected, the increase in the number of prior links led to better model identification.

However, in some cases with two prior links, the AMS algorithm relatively often selected the models that were rather poor as judged by the results presented in Fig. 2. For example, for the artificial yeast glycolysis pathway or real alpha dataset, the bi-exponential E3 model was selected almost as often as other, better performing, models. This indicates that the E3 model was more adequate just for certain links and not for any link in the networks. Therefore, we can conclude that the network reconstruction model should be link-specific, that is different models may be assigned to different links.

As the AMS algorithm may select poor performing models, the overall performance of the network reconstruction is lower than for the best performing model. However, even with as small as two prior links, AMS is already better than random model selection, as illustrated in Fig. 4. If the performance of different models is not very different (as for the MAPK cascade), the prediction of the AMS algorithm is close to random. If, however, a certain model demonstrates clear advantage (as, for example, for the E. coli repressilator), the AMS algorithm can identify this model leading to the performance substantially higher than by random selection.

The performance of the AMS algorithm using independent set of artificial data described in 5 is presented in [Additional file 6].