7.1 INTRODUCTION

Even the simplest of organisms can have an extremely complex network [1] spanning many levels of interactions from cellular to gene, to protein, and beyond. It is possible to model biological networks using gene regulatory networks (GRNs), which are groups of genes that interact through the production of their proteins. How these GRNs are constructed for large biological networks and how transcription factors are able to regulate the expression of thousands of genes in response to environmental changes is a fundamental problem in biology [2]. Furthermore, the reverse engineering of biological networks from expression data and the inference of the complexity in the networks is a current problem in computational and biological sciences [3]. The challenges of network reconstruction increase with the size of the network and suffer “underdeterminism,” where there are insufficient data to build statistical models for the number of genes in the network [4]. Hase et al. [5] observed that in order to infer large-scale networks, it is vital to know a priori information of high performance algorithms so methods can be selected based on the details of the problem. However, systematic tests are difficult for reconstruction algorithms as the availability of real gene expression data is limited. As a result, many methods use synthetic data to benchmark their algorithms by comparing results to the known network. Artificial data sets, however, lack the level of complexity and uncertainty that real experimental data sets contain. Moreover, when using experimental data, in general, one may not know what the “real” network is. A greater emphasis is needed on the inference of unknown networks and the use of experimental data as opposed to benchmarking algorithms with artificially generated gene expression profiles.

Because of the lack of available data and the curse of dimensionality, the complexity of network reconstruction increases significantly with the number of genes. For larger networks, simplifications may be required as the number of genes means that a complete model capturing all biological interactions is computationally infeasible. When considering all possible interactions, even a small number of genes can lead to complex network dynamics [6] and an interaction topology that spans many levels, as illustrated in Figure 7.1. Because of this high complexity, collaborations between mathematicians, computer scientists, and biologist attempt to model the biological networks of large organisms such as some yeast (2000 genes) [7, 8], Escherichia coli (5000 genes) [9], Streptomyces coelicolor (8000 genes) [10, 11], flies (13,525 genes) [12], and humans (22,287 genes) [12].

7.1.3 The Life Cycle of Streptomyces coelicolor

Streptomyces are soil dwelling gram-positive bacteria that have genomes of nearly 8000 protein-coding genes [10]. An important aspect of these bacteria, other than their unusually large genome, is their high G+C content, of 72.4%, and their multi-cellular life cycle [10]. The growth of Streptomyces progresses from a vegetative mycelium (normal growth) to the formation of aerial mycelium (tall structures) and eventually releasing exospores in the propagative phase. Vegetative mycelium is growth related and involves all the normal cellular activities associated with cell growth and division. Streptomyces undergo a morphological switch from primary metabolism to secondary metabolism, which is non-growth linked and is non-essential but many important activities occur during this phase that help the bacterium to survive. After this morphological change, Streptomyces produce “secondary metabolites” including a range of antibiotics [55–58] that are the majority of those used in human and veterinary medicines [59, 60]. Furthermore, Streptomyces also produce many other medicinal compounds including anti-parasitic agents, anticancer and anti-tumor drugs, anti-fungals, anti-hypertensive, herbicides, and immunosuppressants [11, 61].

Depletion of environmental nutrients is widely believed to contribute to the production of antibiotics as the onset of secondary metabolites coincides with the morphological differentiation from mycelium growth to developing aerial hyphae, indicating that these two processes are governed by overlapping regulatory networks [55, 60, 62–64]. The onset of secondary metabolism in Streptomyces is due to phosphate starvation as illustrated in Figure 7.2. Here, we can see that a decrease in phosphate (purple) coincides with an increase in biomass (brown) resulting in the formation of aerial mycelium between 20 and 35 h. The point at which Streptomyces begin producing secondary metabolites is generally referred to as the stationary phase [55] because the biomass remains constant. This begins at 35 h as shown in Figure. 7.2 and is the point where antibiotics begin to be produced (thick red, blue, and dark blue lines). The switch to secondary metabolism and the production of antibiotics is a tightly regulated process triggered by nutrition depletion [65] and is designed to kill off competing microorganisms [60]. The bacteria themselves have co-evolved systems to aid resistance to the antibiotics that they produce [59]; however, they can also be lethal to their producers [65]. The production of antibiotics provides protection for the bacteria during phosphate starvation, so it can harvest phosphate from within the bacteria [60] such as from cellular material [66] and the cell walls [67].

7.2 REGULATORY NETWORKS AND GENE EXPRESSION DATA

Our method uses real biological data from microarray experiments, which give genome-wide expression profiles for all genes in S. coelicolor. Using these data we are able to build a model of the gene regulations that can reproduce the experimental expression profiles and predict regulatory interactions, that is, activating or repressive. Here we use the time-series data from the SysMO consortium [69] containing 32 time points for each gene in varying time intervals from 20 to 60 h with phosphate starvation occurring at 35 h. This is a much larger time-series data set compared to many data sets available [90] and, thus, can show trends in gene expression giving more insight into the regulatory interaction than steady-state data.

7.2.2 Data Normalization

The level of gene expression can vary in scale from gene to gene as demonstrated in Figure 7.3. When modeling networks of many genes this can require large range for the parameters in the connection equations to fit the expression profiles. For an optimization problem this can lead to enormous search spaces for all genes, which may result in sub-optimal solutions. In order to examine this potential problem we

investigate the use of unprocessed “raw” gene expression data compared to gene expression levels that are normalized. Note that by unprocessed here we mean the data published in Ref. [69]; data processing and normalization in this work refers to our setup of the data and has no reference to the experimentation and analysis conducted by Lamarche et al. [69]. Here we normalize the data so that the area of each expression profile is unity:

(7.1)

thus all expression profiles are on the same scale and a universal parameter range exists for all connections. This allows comparisons between parameters across the genes, for example, comparing regulation strength of one gene with another. To normalize the expression profiles, each of the gene expression measurements is divided by the area of the expression profile,

(7.2)

where

(7.3)

This leads to all gene expression profiles within the sub-network, being on the same scale as illustrated in Figure 7.4. By providing a universal parameter range for all genes we have localized the search space compared to the raw data method.

7.3 OPTIMIZATION USING EVOLUTIONARY ALGORITHMS

As biological networks are often large [26], particularly for more complex organisms, sophisticated reconstruction techniques are required. Because of their success in a large range of optimization problems, evolutionary algorithms (EAs) have been used to reconstruct GRNs from time-series data [25, 53, 75, 100]. In addition, EAs have the advantage of not requiring detailed prior knowledge of the system, but also have the flexibility to add biological information during inference [53]. As optimization algorithms require only an evaluation of candidate solutions, they are a powerful tool for modeling complex problems in biology [101]. In order to fully reconstruct a GRN, one must identify both the topology and the parameterization of the connections. This results in a vast search space even for small networks, which can contain a complex structure and set of dynamics [6]. Their ability to deal with very complex and underdetermined problems and their high levels of flexibility make optimization algorithms an attractive method for network inference [38, 101].

Evolutionary algorithms have been widely used to evolve the parameters and structure of models [22, 53, 103, 104] including GRNs [13, 15, 19] and have been used by the authors previously for the PyrR network in S. ceolicolor [105]. Here, we use non-dominant sorting genetic algorithm (NSGA)-II [106] for our optimization as it is one of the most widely used multi-objective optimization algorithms [107]. Additionally, NSGA-II can be setup for a single objective problem [13] providing consistency between our single objective and multi-objective representations of the problem. This real-coded multi-objective genetic algorithm is able to handle continuous optimization problems through the use of simulated binary cross-over and polynomial mutation [108]. It is also an elitist algorithm that is suited to network reconstruction as it retains good solutions and has been demonstrated to aid convergence for continuous optimization problems [13, 15, 102, 105, 108, 109].

7.4 MODELING GENE EXPRESSION

One common method of modeling GRNs dynamics is to use differential equations [75, 111] as they can provide detailed temporal information about the system. A common representation of gene regulation is through the use of Hill functions [38, 112], which are nonlinear equations that are derived from Michaelis–Menten enzymatic kinetics [21, Appendix A]. Hill functions modeling the regulation of gene x_i by gene x_j can be activating, where protein production is promoted in the target gene, or repressive, where production is suppressed. The mathematical form for activating and repressive Hill functions is given by

(7.4)

respectively. Here θ is the reaction threshold and n is the Hill coefficient that determines how steep the interaction curve is. The state of gene x_i, regulated by gene x_j, at time t is then given by

(7.5)

where is the time derivative of x, ω is the regulation weight, and γ is the rate of degradation of the protein produced by x. We can use an optimization algorithm to determine the parameters in Equations (7.4) and (7.5) based on gene expression time series to determine the connection parameters and characteristics such as the regulation type.

7.4.1 Single Objective Setup

In order to optimize the network parameters based on the state of the genes, we define an objective function as an error minimization problem. Here we calculate the sum squared error (SSE) between the model prediction and the experimental gene expression profile. As our objective function is to be minimized, the EA will vary the network parameters to provide the best fit to the expression profiles. The SSE is calculated over all N time points for all M genes and is rewritten, for convenience, as

(7.6)

where g_i(t) and x_i(t) are the values of the experimental gene expression level and the model gene expression level, respectively. The SSE given in Equation (7.6) determines the fit of the entire network to the expression data of all genes. We use Equation (7.5) to model all connections. Here, we have four free regulation parameters and one Boolean value to determine the regulation type, giving a total of five parameters for each connection in the network. The regulation parameter is used to determine if the connection is activating or repressive, H^a_ij or H^r_ij in Equation (7.4). This is a common objective for biological network reconstruction [75, 77, 113] and can be applied to both real and Boolean networks [26].

7.4.2 Multi-Objective Setup

In Section 7.4.1 we detailed a single objective optimization representation of the problem of reverse engineering regulatory networks from expression data. However, the use of multiple objectives can provide a better exploration of the parameter space in GRN reconstruction [114], which is vital for large-scale inference problems. Additionally, the benefits of multiobjectivization [115], where a single objective problem is represented with multiple objectives, may enhance the optimization process. This is achieved either by using an additional objective to guide the search though the parameter space or by decomposing the original objective into multiple objectives that are sub-sets of the original. Therefore, this representation does not necessarily yield conflicting objectives unlike multi-objective problems in general. This MOS may solve the problem more efficiently than the single objective configuration. A local optimal solution in the single objective setup corresponds to one of the two objectives in the MOS, and as it is a local optimum; by definition, the other objective value must be worse. This solution will therefore be located in the top right region of the objective space and, as we have a minimization problem, correspond to a poor candidate solution. Therefore, this solution will be dominated by other solutions during the non-dominated sorting in NSGA-II, and is unlikely to remain in the population over successive generations. The MOS may be less prone to local optima than the single objective equivalent based on this feature. The ability to escape local optima becomes increasingly important as the number of genes, and problem dimension, increases. Based on this, here we also investigate several MOSs to compare with the typical SOS commonly used.

As in the single objective case, we optimize the four network parameters, however we define two separate, but not necessarily conflicting, objectives to be optimized simultaneously. Both objectives are to minimize Equation (7.6) using Equation (7.5); however, the first objective models all connections using an activating regulation and the second objective models them all as repressive regulations, Equation (7.4). Thus, we have a multi-objective problem with the following objectives:

(7.7)

(7.8)

where x^a_i and x^r_i are the connections to gene x_i using H^a_ij and H^r_ij, respectively, in Equation (7.4). Here, the SSE per gene is summed for M genes in the network for only activating regulations, F₁, and only repressive regulations, F₂. The SSE for a gene in the network is given as the sum of the squared difference between the gene expression profile g(t) and the model x(t) over N time points. In this setup, a given set of network parameters, ω, θ, n, and γ, are evaluated using both objectives (7.7) and (7.8) producing a Pareto front of possible solutions for the problem; see Figure 7.5. In general, for multi-objective optimization problems the objectives are conflicting resulting in tradeoff solutions that cannot be compared without user preference or system constraints [116]. In this case, however, the problem is actually a single objective problem with the goal to minimize the SSE between the model and the data for all network genes. Therefore, we can select the best solution from the Pareto front based on the objective values and do not have to add additional preferences or constraints. More specifically, our “preference” for best fit to the data for each gene is implicit in this setup; thus, solutions can be selected from the front automatically. By arranging this problem as a multi-objective problem, we do not have to optimize the regulation parameter in the system directly. The final network produced by this method is not restricted to one regulation type and can be a mix of regulations.

To select the best solution, we search the Pareto front to obtain the best parameter values for each connection. As defined in Equations (7.7) and (7.8), the two objectives are determined though the sum of the SSE for each gene in the network. Thus, by retaining the individual gene SSEs, we are able to search the Pareto front to obtain the solution (parameter set) that corresponds to the lowest SSE for each gene. Each solution on the Pareto front has two objective values, each of which corresponds to summed sequence of individual gene SSEs (produced from a specific parameter set), Equations (7.7) and (7.8). Once the optimization process has finished, for each gene, the Pareto front is searched for the parameter set that yields to best objective. Figure 7.5(a) illustrates the Pareto front and Figure 7.5(b) details how each solution corresponds to a summation of individual SSEs for each connection. For each gene, in turn, the front is searched to obtain the solution corresponding to the minimum SSE from either F₁ (activating connection, f^a_i) or F₂ (repressive connection, f^r_i). That is for each gene we evaluate f^a_i < f^r_i and select the minimum f_i that indirectly determines the regulation type. The result is a network containing mixed regulation types that are each selected based on the lowest SSE.

This method has the potential to conduct a better search of the parameter space through its representation as a multi-objective problem. The overhead of searching the Pareto front, performed after the optimization, even for a large number of individuals and genes, is negligible compared to the optimization time. This multi-objective optimization process removes the need for the direct optimization of the regulation type and, therefore, reduces the number of parameters in the problem. Reducing the number of parameters from five to four for each of the n connections results in a large reduction in the search space and could therefore enhance convergence to the global optimum. Here we use 100 individuals and 1000 generations in the optimization stage.

7.4.3 Decoupled Approach

Biological systems are known to be sparse [30] therefore we assume that the connections within the network are independent and treat them as separate, a process known as decoupling [87, 88]. This reduces the dimensionality of the single objective optimization problem from one 5n problem, where n is the number of connections, to n problems each with a dimension of five in the SOS. Here the number five corresponds to the four network parameters and the one Boolean regulation type parameter. Similarly for the MOS, a 4n problem becomes n lots of four dimension problems. Furthermore, due to the decoupling, this technique can also benefit from the use of parallel computation, where each connection can be optimized on a separate CPU. This process can significantly reducing the computational runtime for large networks, though is not implemented here. The objective function here remains the same as for the full network optimization case with the exception of the summation over the number of genes. In this setup Equation (7.6) becomes

(7.9)

Here the total network SSE, , is calculated after the optimization process is completed by summing the individual connections’ objective value from Equation (7.9),

(7.10)

for all connections.

In this single objective case x_i(t) is based on one of the equations in Equation (7.4) depending on the connection parameter determined by the optimizer as in the full network case. For the multi-objective optimization setup, however, the decoupled method differs slightly. In the multi-objective decoupled arrangement, each connection is optimized in a similar way to the full network case:

(7.11)

(7.12)

However, the selection of solution from the Pareto front is much simpler. In this method, we have a Pareto front for each connection due to the problem being decoupled. Therefore, as we have a minimization problem, the lowest objective value for each connection is always one of the extreme solutions. This is clearly illustrated in Figure 7.6. For each connection we simply evaluate f^a_i < f^r_i to obtain the optimal parameter set and regulation type, see Figure 7.6. Here we use 100 individuals and 100 generations as this setup converges much faster than the full network setup thus requiring fewer generations in the optimization.

7.5 RESULTS

7.5.1 Comparing Objectives from Un-Normalized Data

For the case of the un-normalized (raw) data, each of the gene expression profiles within a single network potentially lies on a different scale; thus, determining a network error is not so straightforward. Comparing the SSE between two genes within the same network may be meaningless if the expression profiles vary in scale, with larger scale expression values tending to have larger SSEs. Therefore, if we take the ratio of SSE (δx) and the area of the expression profile () for each gene, we are left with a dimensionless quantity, δx′_i, that has removed the difference in expression scale:

(7.13)

For each gene, the dimensionless SSEs (δx′_i) can be summed to give the total error for the network, ΔX:

(7.14)

where i is the ith network gene and M is the total number of genes within the network. This normalization of the objective functions not only provides a meaningful network error measurement, but it also enables comparison between the different networks and arrangements. We can now compare the raw data setup with the normalized data arrangement directly. No further processing is required for the normalized data case, as each expression profile has the same area; see Equation (7.1). Therefore, Equation (7.13) reduces to δx′_i = δx_i and the total network error would remain the sum of the errors in the individual genes.

7.5.2 Full Network Optimization

Figure 7.7 shows the box plots of the total network SSE for 50 randomly initialized optimizations of each of the networks. All raw data here are post-processed using Equation (7.14), thus enabling a direct comparison between raw and normalized setup performances.

Raw versus Normalized Data

For the SOS, the use of normalized data yields much lower network errors compared with the raw data. The difference is two orders of magnitude across the entire range of network sizes. This illustrates the benefits of using normalized data as it has produced much better objective values. The SOS normalized distributions are bigger compared to the raw SOS cases for the larger networks; however, the difference between the distributions is still large. For the MOS, the same outcome is seen as for the SOS case. In addition, it also reduces the distribution range for the smaller sub-networks. Figure 7.7 clearly shows that the use of normalized data is beneficial and improves both the SOS and the MOS.

Single versus Multiple Objective Setup

For the raw data, the MOS case produces a large spread of network errors over the 50 simulations indicated by the long box plots, which overlap with the SOS distributions for the three smallest sub-networks. The MOS performs worse than SOS over the whole range of network sizes and appears to diverge with increasing network size. Similar behavior is seen when using the normalized data and comparing the SOS and MOS cases. Both the raw and normalized data SOS results appear to be approximately constant with increase network size, whereas both the MOS experiments increase with network size. The MOS normalized case, however, does not exhibit the larger distributions of network errors as in the SOS case.

The results here indicate that the use of normalized data can improve performance on modeling gene expression for large-scale networks compared to raw data. This improvement is due to the localization of the search space for all genes. Localization enhances convergence of the optimizer and results in lower errors in each gene and, therefore, the whole network. These results also show that an SOS is better than an MOS for both raw and normalized data over the network sizes tested. The MOS with normalized data does outperform the SOS raw arrangement; however, trends indicate that the SOS raw arrangement would perform better than the MOS normalized case for larger networks that tested here. The normalized data set with the SOS performs the best of all the methods tested. Despite the increase in distribution of errors with increasing network size, the trend indicates that increasing network size results in a small increase in the average network error.

7.5.3 Decoupled Network Optimization

Figure 7.8 shows the results from the decoupled approach for the sub-networks modeled. The use of the decoupled methodology leads to much narrower distributions compared to the full network arrangement for both raw and normalized data sets and for both SOS and MOS. In the full network case, some distributions span an order of magnitude, where the largest ones in the decoupled approach are approximately half an order of magnitude, and are the minority. Almost all distributions are so narrow that they are not fully observable in Figure 7.8.

Raw versus Normalized Data

A significant difference between using raw and normalized data is evident in Figure 7.8 where distributions based on data sets are separated by approximately 2.5 orders of magnitude. The use of normalized data has led to a reduction in network error as in the full optimization case. Additionally, the large difference between the RedD network and the other sub-network has greatly been reduced, though it has also led to a slightly wider distribution (taking the scale into account) over the 50 simulations. Also for the SOS, the normalized data have also produced a wider distribution of the PyrR network (relative to the other sub-networks) compared to the raw data. However this distributions is still around 2 orders of magnitude lower than the raw data case.

Single versus Multiple Objective Setup

For the raw data set, the difference between the SOS and MOS is small. For clarity, the distributions shown in Figure 7.8 have been plotted in difference scale in Figures 7.9(a) and 7.9(b) for the raw and normalized data, respectively. The largest ones for sub-networks show that the MOS results in lower total network errors than the SOS, though this difference is small; see Figure 7.9(a). Over the range of sub-network sizes, the decoupled method has removed the poor performance of the MOS and, thus, has led to comparable results to the SOS. For the normalized data, we can see that the MOS produces narrower distributions than the SOS with lower average values in all sub-networks. Moreover, the average value of the MOS distributions is lower than the minimum SOS distribution value for all sub-networks except for RedD. This indicates significant differences between these methodologies, and even in the exception RedD, the median of MOS is below the lower quartile of the SOS. Furthermore, the maximum (excluding outliers) of the MOS is approximately equal to the average of the SOS. This is shown in Figure 7.9(b). From this figure it is clear the MOS with normalized data produces the best performance as it results in the lowest total network error. Moreover, for the four largest sub-networks, the MOS produces narrower distributions that are approximately 15% lower in total network error. This appears to be a stable trend and may indicate that the MOS produces lower network errors for larger sub-networks than SOS in general.

7.6 DISCUSSION

Using benchmark synthetic data sets is useful in comparing an algorithm’s performance [40, 43, 53]; however, it may not be a true representation of the algorithm’s performance on real data sets. Currently, there are very few studies that use real data and models are not often based on time-series data. Topological models are useful in identifying motifs and functionalities of networks as well as comparing similarities between different networks and organisms. However, dynamic models are vital in our understanding of gene regulation and interactions within a network. Here, we have investigated several techniques and setups for modeling dynamic GRN interactions based on gene expression data. We have investigated the effect of an MOS and observe a performance hindrance when compared to an SOS for a full network optimization technique. This hindrance has less effect when using normalized data, but is still present. When using a decoupled optimization technique, the MOS is a form of multiobjectivization and is comparable in performance to the SOS when using raw data, and performs significantly better when using normalized data. These results show that the use of multiobjectivization can enhance the optimization search through objective decomposition, whereas a multi-objective representation (that is not a form of multiobjectivization) may hinder performance.

We have demonstrated that the use of normalized data leads to reductions in network error compared with raw gene expression data. This reduction is seen in all arrangements, after correcting for gene expression scale to compare setups and network sizes. The improvement through the use of normalized data is due to the localization of the objective search space. By having a universal parameter range, due to all expression profiles being on the same scale, the optimizer can search the objective space much more efficiently than for the raw data case. For the normalized data setup, the optimal solutions for all connections in a network lie within a local region of the parameter space, whereas this is not the case when using raw data. Moreover, the parameter space when using raw data may be vast with optimal connection values far apart for different genes within a network due to expression scale. The normalization of the gene expression data is a straightforward task that can be applied to data sets prior to optimization and has a significant effect on the results as indicated here.

It is clear that in the full network cases the MOS is not providing optimal solutions to the problem. This is a consequence of the methodology, as solutions on the front represent a collection of individual gene SSEs, which are searched to provide the parameter values for each connection. The selection during the optimization is based on the lowest total SSE for a network, that is, the sum of the individual connection SSEs, yielding solutions with low SSE for all connections rather than for individuals. As a result, the optimal parameter values for each connection are not necessarily selected as the overall network SSE may be high. Although, in principle, this sounds analogous to the SOS, the difference lies in the selection during optimization, where solutions are evaluated based on their total SSE and then post-optimization individual connection parameters are selected. This problem of sub-optimality increases with sub-network size leading to a higher rate of increase in SSEs for larger networks in the MOS compared to the SOS.

The performance enhancement using the decoupled case is due to the reduction in the parameter search space. This method improved performance in terms of lower network error and reduced variance between simulations compared to the full network case. The reduced variance is evident by the narrowing of the distributions over the 50 simulations as shown in Figure 7.8 compared to that shown in Figure 7.7. Additionally, we observe a reduction in the number of outliers (simulation results greater/less than 1.5 × median) over the 50 simulations. This is a result of the reduced variance of the distribution and can be attributed to the increased convergence power of the decoupled arrangement to optimal solutions. For the raw data, the SOS and MOS perform comparably over the range of network sizes; however, for the normalized data we observe a clear improvement of the MOS over the SOS. The best combination of objective setup and data type observed here is the MOS decoupled arrangement with normalized data. This produces the lowest network errors for all of the sub-networks investigated here and demonstrated high reproducibility of the results through low variance. Moreover, this methodology is significantly better than the next best combination, SOS decoupled configuration with normalized data, over all network size.

For the largest network used here (332 genes) the computational runtimes for the full network optimization are approximately 20 and 30 min for the SOS and MOS, respectively (for both the raw and normalized data). For the decoupled arrangements, the SOS runtime was approximately 15 min (for both raw and normalized data) and for the MOS, simulations took 20 and 10 min for the raw and normalized data, respectively. Therefore, the MOS method with normalized data has the shortest runtime in addition to providing the best solutions to the problem.

In this work, we have only considered networks where genes are regulated by only one gene. In general, genes may be targeted by more than one regulator that will add an additional layer of complexity to the problem by increasing the search space. However, we have shown here that this simplification can be effective for reconstructing large networks (up to 332 genes) from time-series data. As biological networks are known to be sparse [30], this is a good first approximation and may be sufficient for the majority of networks. Large networks, such as those used here, are likely to contain genes that have additional regulators, thus an extension of this method could be to add additional regulatory connections for genes that the initial method is unable to fit. This can easily be implemented after the method used here and applied to genes where the SSE between the model and data expression profiles is above a threshold. Extending the method this way would only increase the problem complexity when necessary and avoid overfitting by maintaining biological sparsity.

7.7 CONCLUSIONS

The results here suggest that normalized data enhance convergence of this optimization problem and result in consistent solutions, that is, small variance, being obtained over numerous runs. This effect is independent of objective methodology and observed in SOS and MOS for both the full and decoupled optimization arrangements. Furthermore, the use of a decoupled approach also improves consistency between optimization runs; thus, the results of each simulation can be taken with a high degree of confidence as the variance is low in the majority of cases. Additionally, we observe a reduction in outlier solutions when using normalized data. We examine all comparable methods for optimization and conclude that the best performance is obtained by using a decoupled approach with normalized expression data in conjunction with the novel multi-objective technique developed here. This configuration performs better than the single objective alternative for all network sizes, and exhibits a significant reduction in network SSE compared to the full network optimization setup. In Ref. [53], the authors noted that the application for multiobjectivization to larger networks still requires investigation. Here we have demonstrated the effectiveness of such an approach to enhance optimization across various networks that appears independent of increasing network sizes.

We have developed a novel MOS for modeling the dynamic behavior of genes in a given network and demonstrated that a decoupled arrangement multi-objective can outperform a standard single objective approach. Our novel configuration of a decoupled multi-objective optimizer setup performs comparably to a single objective case when using raw gene expression data; however, when using normalized gene expression data, it consistently performs better with increasing network size. The improved performance with increasing network size occurs from a slight reduction in dimensionality for each connection, reduced from five to four variables. Here, the small improvements over each connection lead to greater performance over larger networks. This novel setup shows little dependence on increasing network size and thus can be applied to large-scale network reconstruction problems. In addition, this methodology, in general, exhibits little variation over numerous independent simulations and thus provides consistently good solutions to this problem. Our method is able to escape local optima as, due to the representation of the problem, these solutions are dominated during the optimizations stage, see Section 7.4.2, resulting in narrow distributions of optimized solutions over a number of runs. Furthermore, this multi-objective approach has the smallest computational runtime of the methodologies tested here and can be applied to other networks and data sets.

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