Gene regulatory networks modelling using a dynamic evolutionary hybrid
- Ioannis A Maraziotis^{1},
- Andrei Dragomir^{2} and
- Dimitris Thanos^{1}Email author
https://doi.org/10.1186/1471-2105-11-140
© Maraziotis et al; licensee BioMed Central Ltd. 2010
Received: 25 September 2009
Accepted: 18 March 2010
Published: 18 March 2010
Abstract
Background
Inference of gene regulatory networks is a key goal in the quest for understanding fundamental cellular processes and revealing underlying relations among genes. With the availability of gene expression data, computational methods aiming at regulatory networks reconstruction are facing challenges posed by the data's high dimensionality, temporal dynamics or measurement noise. We propose an approach based on a novel multi-layer evolutionary trained neuro-fuzzy recurrent network (ENFRN) that is able to select potential regulators of target genes and describe their regulation type.
Results
The recurrent, self-organizing structure and evolutionary training of our network yield an optimized pool of regulatory relations, while its fuzzy nature avoids noise-related problems. Furthermore, we are able to assign scores for each regulation, highlighting the confidence in the retrieved relations. The approach was tested by applying it to several benchmark datasets of yeast, managing to acquire biologically validated relations among genes.
Conclusions
The results demonstrate the effectiveness of the ENFRN in retrieving biologically valid regulatory relations and providing meaningful insights for better understanding the dynamics of gene regulatory networks.
The algorithms and methods described in this paper have been implemented in a Matlab toolbox and are available from: http://bioserver-1.bioacademy.gr/DataRepository/Project_ENFRN_GRN/.
Keywords
Background
Fundamental biological processes including cell differentiation, metabolism, the cell cycle or signal transduction are governed by the expression of genes. Gene products and especially the amounts and temporal patterns in which these products are expressed in the cell enable cell survival and numerous cellular functions. A holistic view on the regulatory mechanisms governing the dynamics of genes expression requires the understanding of the networking and interactions among genes and their products, since it is known that genes act in a collective manner to achieve biological functions [1, 2]. By understanding the complex relations within these gene regulatory networks (GRN) we can highlight inhibitory or excitatory interactions, as well as how intracellular or extracellular factors affect gene products or dysregulate cellular processes.
The recent advances in biotechnology allowed simultaneous measurements of mRNA production from virtually all genes within genomes through high-throughput technologies such as the microarrays [3]. This resulted in massive amounts of data that proved to be a significant challenge for traditional analysis techniques due to the inherent combinatorial problems arising from the large number of variables. Advanced computational models are required to fill in the knowledge gap. These models have to cope with peculiarities of the data such as measurement noise. Additional challenges are posed by the modelling task, such as the under-determinism caused by the large number of variables compared to the available number of experiments. Also, since groups of genes are commonly acting to activate/inhibit a regulated gene, the computational search space grows to a much larger potential search space of virtually countless combinations [4].
Initial computational approaches for inferring gene regulatory networks from gene expression data consisted of classical techniques. Boolean networks are binary models that consider the expression of a gene to be either on or off, and model the effects of other genes on a specific target gene through a Boolean function. A major drawback is the fact they do not consider intermediate levels of gene expression, hence resulting in information loss [5–7]. Furthermore, they assume transitions between genes activation states to be synchronous, which is biologically implausible [8]. Bayesian networks (BN) are a mixture of probability calculus and graph theory, and attempt to model gene regulatory networks as directed acyclic graphs in which the nodes represent the genes and the edges connecting the nodes represent regulatory interactions, which are encoded via conditional dependences learned from the data. Although effective in dealing with noise, incompleteness and stochastic aspects of gene regulation, they fail to consider temporal dynamic aspects that are an important part of regulatory networks modelling [9]. Dynamic Bayesian networks (DBN) evolved feedback loops to effectively deal with the temporal aspects of regulatory networks but their benefits are hindered by the high computational cost required for learning the conditional dependencies in the cases where large numbers of genes are involved [10]. Linear additive regulation models [11] revealed certain linear relations in regulatory systems but failed to capture nonlinear dynamics aspects of genes regulation. Other approaches included differential equations, which besides the high computational costs have also the disadvantage of being sensitive to noisy data (as is the case of data resulting from microarray experiments) [2, 12].
Recent approaches tried to overcome the drawback of traditional methods in several ways [13]. Keedwell and Narayanan [4] used a hybrid neuro-genetic algorithm to mitigate the under-determinism problem. Their method combines genetic algorithms (GA) with a single layer artificial neural-network (ANN), where each chromosome of the GA selects from the whole data set a small number of regulating genes and the ANN is used to determine how the expression levels of these input genes affect another gene's expression. However, the lack of a recurrent structure and the training method of the ANN may pose serious problems when modelling the complex temporal dynamics of gene expression regulation and renders the method vulnerable to local minima traps, respectively.
Some of the most effective approaches towards problems regarding temporal information processing are the recurrent neural networks (RNNs) [14] and recurrent fuzzy neural networks (RFNNs) [15]. Recurrent networks, in general, can deal with temporal and spatial/temporal problems by adapting self loops and backward connections to their topologies and structures, both of which are used to memorize past information. Recently, Xu et al[8] used an RNN combined with Particle Swarm Optimization (PSO) to capture the complex nonlinear dynamics of gene regulatory networks. Although RNNs are generally efficient for temporal sequence production problems, it has been shown that recurrent fuzzy networks outperform recurrent neural networks [16, 17] in problems that involve concurrent spatial and temporal mapping like the one of regulatory networks reconstruction. Additionally, fuzzy-based approaches are better candidates in dealing with the uncertainties of modelling noisy data, due to their high-level, human-like reasoning [18, 19]. In [18] regulatory interactions among genes are extracted in the form of fuzzy IF-THEN rules by searching for statistically significant fuzzy dependency relationships among genes. In [19], Sokhansanj et al perform an exhaustive search for possible rules describing gene interactions (they use the same IF-THEN formalism), under the framework of a linear fuzzy logic scheme that restricts the search space. However, both methods have the drawback of requiring prior data discretization, while [19] has the additional disadvantage of not considering temporal information.
In this study, we propose a novel multi-layer evolutionary trained neuro-fuzzy recurrent network (ENFRN) applied to the problem of GRN reconstruction, which addresses the major drawbacks of currently existing computational methods. Our choice was driven by the benefits, in terms of computational power, that neural network based methods provide [4, 14, 15]. The self-organized nature of ENFRN algorithm is able to produce an adaptive number of temporal fuzzy rules that describe the relationships between the input (regulating) genes and the output (regulated) gene. Related to that, another advantage of our approach is that it overcomes the need of prior data discretization, a characteristic of many computational methods which often leads to information loss [18–21]. The dynamic mapping capabilities emerging from the recurrent structure of ENFRN and the incorporation of fuzzy logic drive the construction of easily interpretable fuzzy rules of the form: 'IF gene x is highly expressed at time t THEN its dependent/target gene y will be lowly expressed at time t+1'. The evolutionary training, based on the PSO framework, tries to avoid the drawbacks of classical neural networks training algorithms. Additionally, we are approaching the under-determinism problem by selecting the most suitable set of regulatory genes via a time-effective procedure embedded in the construction phase of ENFRN. Also, besides determining the regulatory relations among genes, our method can determine the type of the regulation (activation or repression) and at the same time assign a score, which might be used as a measure of confidence in the retrieved regulation. Experiments on real data sets derived from microarray experiments on Saccharomyces Cerevisiae prove the ability of the proposed method to capture biologically established gene regulatory interactions, outperforming at the same time other computational methods.
Results and Discussion
We applied our method on Saccharomyces Cerevisiae gene expression data obtained from cell cycle microarray experiments to identify regulatory interactions and construct putative regulatory networks. The cell cycle process consists of four phases: G1 (in which the cell grows and, under appropriate conditions, commits to division), S (in which the DNA is synthesized and chromosomes replicated), G2 (a 'gap' between S and M), and M (in which chromosomes are separated and the cell is divided). After the M phase, the cell enters the G1 phase, hence completing a 'cycle' [22].
The datasets we employed originate from a study containing measurements of virtually all yeast genes (6178) across 77 time points (experiments). In Spellman et al.'s experiments, yeast cell cultures were synchronized with different methods including αlpha factors arrest (two complete cycles, sampled each 7 min, totally 18 samples), temperature arrest of temperature sensitive mutants (cdc15, three complete cycles, sampled each 10 min, totally 24 samples) and elutriation synchronization (one complete cycle, sampled each 30 min, totally 14 samples) [23]. This data set includes also a subset from a previous experiment in which the cells were synchronized by temperature arrest of a cdc28 allele (17 samples taken at 10 min intervals) [24], as well as 4 samples initially intended for a study on cyclins CLN3p and CLB2p.
It must be noted at this point that all the results in this study were obtained using cdc15 as training set for the ENFRN models, while alpha and cdc28 subsets as testing data sets. The choice for this training set was based on the fact that it had the largest amount of time points. Alpha subset was chosen because it originates from the same experimental setting to the training data set and possesses the largest amount of data after cdc15. At the same time, it allows us to provide comparison to other methods that have used the same data set. Cdc28 was chosen to provide a means of testing the method against data originating from a different experimental setting than that of the training. For reasons of simplicity and coherence we will present in the following results based on alpha data set (used as testing set). Similar results were obtained when using cdc28 subset as testing set. Full details on the results of cdc28, as well as general settings of the parameters of the algorithm and additional information regarding both test data sets, are provided in 'Additional file 1'.
Information regarding the extracted interactions based on alpha test data set.
a/a | Regulator | Target | Type | Composite Score | Genetic Interaction | Physical Interaction |
---|---|---|---|---|---|---|
1. | HTB1 | HTA1 | + | 0.49589 | 1 | 1 |
2. | HTA1 | HHF1 | + | 0.44116 | 1 | 1 |
3. | HHF1 | HTA2 | - | 0.58690 | 0 | 1 |
4. | HTB1 | HHF1 | + | 0.59115 | 1 | 1 |
5. | HHF1 | HTB2 | + | 0.53736 | 0 | 1 |
6. | HHF2 | HTA1 | - | 0.54367 | 1 | 1 |
7. | HHF2 | HTA2 | - | 0.46783 | 0 | 1 |
8. | HHF2 | HHF1 | + | 0.48136 | 1 | 1 |
9. | HHT1 | HTA1 | + | 0.56520 | 1 | 1 |
10. | HHT1 | HTB1 | - | 0.59864 | 0 | 1 |
11. | HHT1 | HTB2 | + | 0.52335 | 0 | 1 |
12. | HHT1 | HHF2 | - | 0.59846 | 1 | 1 |
13. | HTA1 | HHT2 | + | 0.58984 | 1 | 1 |
14. | HHT2 | HHF1 | + | 0.52567 | 1 | 1 |
15. | HHF2 | HHT2 | + | 0.53911 | 1 | 1 |
16. | HHT2 | HHT1 | + | 0.55456 | 1 | 1 |
Information regarding the increase of efficiency and simultaneous decrease of complexity throughout the various training phases of ENFRN
Composite Score Values in ENFRN Structures | Number of Rules and Output Nodes in ENFRN Structures | ||||||||
---|---|---|---|---|---|---|---|---|---|
a/a | Regulator | Target | Initial | Simplified | |||||
Initial | Simplified | Trained | Rules | Output | Rules | Output | |||
1 | HTB1 | HTA1 | 0.899 | 0.935 | 0.533 | 12 | 9 | 12 | 9 |
2 | HTA1 | HHF1 | 1.056 | 0.988 | 0.435 | 11 | 8 | 9 | 8 |
3 | HHF1 | HTA2 | 0.644 | 0.698 | 0.595 | 12 | 10 | 11 | 10 |
4 | HTB1 | HHF1 | 0.842 | 0.815 | 0.536 | 12 | 8 | 10 | 8 |
5 | HHF1 | HTB2 | 0.654 | 0.642 | 0.560 | 12 | 8 | 10 | 8 |
6 | HHF2 | HTA1 | 0.741 | 0.773 | 0.618 | 14 | 10 | 12 | 10 |
7 | HHF2 | HTA2 | 0.657 | 0.669 | 0.543 | 14 | 10 | 12 | 10 |
8 | HHF2 | HHF1 | 0.748 | 0.785 | 0.503 | 14 | 9 | 12 | 8 |
9 | HHT1 | HTA1 | 0.898 | 0.916 | 0.664 | 12 | 9 | 10 | 9 |
10 | HHT1 | HTB1 | 1.095 | 1.097 | 0.811 | 12 | 10 | 10 | 9 |
11 | HHT1 | HTB2 | 0.632 | 0.660 | 0.567 | 12 | 8 | 10 | 8 |
12 | HHT1 | HHF2 | 0.841 | 0.791 | 0.755 | 12 | 9 | 10 | 9 |
13 | HTA1 | HHT2 | 1.233 | 1.188 | 0.606 | 11 | 9 | 8 | 8 |
14 | HHT2 | HHF1 | 0.658 | 0.625 | 0.567 | 14 | 9 | 14 | 9 |
15 | HHF2 | HHT2 | 0.890 | 0.881 | 0.589 | 14 | 10 | 13 | 10 |
16 | HHT2 | HHT1 | 0.962 | 0.975 | 0.749 | 14 | 10 | 10 | 9 |
This table represents interactions detected by the three computational models, ENFRN, BN and DBN.
Model Derived Interaction | |||||||
---|---|---|---|---|---|---|---|
a/a | Regulator | Target | Composite Score | Type | ENFRN | DBN | BN |
1. | FUS3 | FAR1 | 0.35898 | T | √ | - | - |
2. | FAR1 | CLN2 | 0.35138 | T | √ | - | - |
3. | SIC1 | CLB5 | 0.53303 | T | √ | - | - |
4. | CLB5 | CLB6 | 0.40858 | T | √ | - | - |
5. | CLB6 | CLB5 | 0.53303 | T | √ | - | √ |
6. | CLN2 | CLN1 | 0.54337 | T | √ | - | √ |
7. | CLN3 | CDC28 | 0.34730 | T | √ | - | - |
8. | CLN1 | SIC1 | 0.54336 | T | √ | - | √ |
9. | CDC28 | CLB6 | 0.41569 | T | √ | - | - |
10. | CLB6 | CDC28 | 0.39661 | T | √ | - | - |
11. | CDC28 | CLN1 | 0.54336 | T | √ | - | - |
12. | CLN3 | SWI4 | 0.33336 | T | √ | √ | - |
13. | CDC20 | CDC28 | 0.49565 | T | √ | - | - |
14. | CDC28 | CLB5 | 0.53309 | T | √ | - | - |
15. | SWI4 | CLN1 | 0.38757 | T | √ | √ | - |
16. | CLN2 | SIC1 | - | T | - | √ | - |
17. | SIC1 | CLN2 | - | T | - | √ | - |
18. | CLN1 | CLN2 | - | T | - | √ | - |
19. | CLN2 | CLB6 | 0.38983 | I | √ | √ | - |
20. | CDC20 | CDC6 | 0.40848 | I | √ | √ | √ |
21. | SIC1 | CDC6 | 0.55806 | I | √ | - | - |
22. | FAR1 | SIC1 | - | I | - | - | - |
23. | FAR1 | FUS3 | - | I | - | √ | √ |
24. | FUS3 | CDC28 | - | I | - | √ | |
25. | CLN1 | FAR1 | - | I | - | - | √ |
26. | CLN1 | CLB5 | - | I | - | - | √ |
27. | CLB6 | MBP1 | 0.37033 | F | √ | - | - |
28. | SWI6 | CLN3 | 0.56508 | F | √ | - | - |
29. | FUS3 | CLN3 | 0.57180 | F | √ | - | - |
30. | CDC6 | SIC1 | - | F | - | √ | - |
31. | CDC6 | CLB6 | - | F | - | √ | - |
32. | CDC6 | CLB5 | - | F | - | √ | - |
33. | CDC20 | FAR1 | - | F | - | √ | - |
34. | SIC1 | FAR1 | - | F | - | - | √ |
35. | CDC6 | FUS3 | - | F | - | - | √ |
36. | FAR1 | CLN3 | - | F | - | - | √ |
37. | CLB5 | MBP1 | - | F | - | - | √ |
38. | SIC1 | MBP1 | - | F | - | - | √ |
39. | MBP1 | SIC1 | - | F | - | - | √ |
40. | CDC6 | CLN3 | - | F | - | - | √ |
We next proceed evaluating our approach on larger datasets. Hence, the third set we tested, consisted of 19 genes which include DNA polymerases (POL1, POL2, POL12, and POL30), DNA helicase (HPR5), type B cyclin genes (CLB5 and CLB6), DNA primases (PRI1 and PRI2), radiation sensitive genes (RAD53 and RAD54), repair related genes (MSH2, MSH6, and PMS1), replication protein A encoding gene (RFA3), DNA replication initiation factor (CDC45), securin gene (PDS1), nucleosome assembly factor (ASF1), and a subunit of the cohesin complex (MCD1). These genes play an important role in the process of cell cycle conducting important cellular activities such as DNA replication initiation, DNA damage-induced checkpoint arrest, DNA damage repair, formation of mitotic spindle, and so on. As observed in Figure 2, ENFRN retrieved 21 regulatory interactions among 17 genes (there was no interaction found for RAD54 and POL1). Bold edges denote retrieved interactions that are backed by genetic interaction while dashed edges correspond to those having physical interaction counterparts in experimental databases (Biogrid and SGD). Thin edges denote interactions that could not be found neither in Biogrid nor SGD. For 16 out of 21 interactions there exists biological experimental evidence stored in the Biogrid and Saccharomyces Genome Database (SGD). This is more than double the number of valid interactions the method of Chen [29] found using the same group of genes. Of particular interest is the upregulation of cyclin CLB5 involved in DNA replication during S phase by the securing gene PDS1. This regulation is substantially supported by experimental evidence at both genetic and protein level [30, 31], however it has not been retrieved by Chen et al's method. The composite score of ENFRN for this interaction was relatively low (0.36518), thus underlining the high confidence of the retrieved regulation. An interesting case is that of PRI2-POL12, which we inferred as a valid interaction. Although we could not find existing biological experimental information regarding the respective genetic interaction, there was significant evidence of physical interaction among the protein products of this pair of genes from a series of Affinity Capture-MS experiments [32, 33]. It must be noted that this interaction was marked as a valid genetic interaction by Chen et al[28], although their cited evidence highlighted it as a protein-protein interaction via Tandem Affinity Purification experiments [32, 34]. A complete list of our interactions, their composite scores and experimental information can be found in 'Additional file 1'.
Our results highlight the crucial role of cyclins in the cell cycle, of which the most typical is CLB2 as reported by others [29, 35]. Accordingly, ENFRN identified the upregulating interaction among CLB1 and CLB2 [29]. Worth to be noted is also the activation of the NDD1 by the CLB2 that takes place in the G2 phase of the cycle, which is part of the larger scale interactions between CLB2 and the SFF complex (consisting of FKH1, FKH2 and NDD1-directly connected in our network) [29, 35]. MCM1, a DNA binding protein that works with the SFF complex, has been placed in direct interaction with FKH2, thus highlighting its role within the complex.
The important role of the cyclin-dependent kinase SIC1 is captured by its close interaction with the G1-cyclins CLN1,3. Additionally, the documented inhibition of SIC1 by the SFF component FKH2 and the inhibition of SWI5 by the same FKH2 were correctly retrieved [29, 36]. The control of the cyclins CLN1,2 by the SBF complex (via SWI4 activation of CLN1) [37, 38] is reflected by the direct interconnection among these genes. Also, the suggested upstream activation of CLN1 by CLN3 at the beginning of the cell cycle [39] was promptly retrieved by our method.
In a number of cases the microarray data show transcriptional co-regulation patterns and although the gene products are not physically interacting, this tight genetic connection might suggest that genes sustain indirect connections through other gene products [40]. Computational methods usually accommodate this scenario by considering indirect connections through one or even two intermediate nodes [28, 29, 40]. A large number of such interactions can be identified in our network, such as the indirect activation of CLB2 by the SBF and MBF complexes (SWI4 and MBP1, through PCL7) [35]. Another example is the loop where the transcription of SWE1 activated by CLN3 takes place in our network indirectly, through genes CLN1 and YHP1 [29]. Additional information on the interactions and their composite scores is presented in 'Additional file 1'.
As a final comment we should notice that many of the gene regulatory interactions take place exclusively at a specific phase of the cell-cycle, the pair of interacting genes remaining silent for the rest of the cycle. Problems arise in such cases from the reduced availability of experimental samples (e.g. G2 phase of the cycle is represented in the experimental dataset we used by only 2 samples). Despite the great difficulties deriving from this fact, we have proven that the proposed ENFRN-based method managed to acquire biologically validated relations among genes from about 60% to even 100% (depending on the data set) percent of the extracted interactions.
Conclusions
Fundamental processes occurring in organisms are carried out through complex networks of regulatory interactions among genes and their products. The inference of gene regulatory networks based on experimental data obtained from microarrays has become an important way to understand these regulatory mechanisms. Herein we describe a novel evolutionary trained neuro-fuzzy recurrent network to model the regulatory networks and reveal interactions between genes. Our model is able to identify potential regulators of genes through a time efficient process. The recurrent structure of ENFRN ensures that the dynamics of gene expression are properly retained while, at the same time, the self-organizing properties of the method automatically account for the discretization process. The self-organization combined with the inherent fuzzy logic successfully manage to deal with the important amount of noise present in microarray data and at the same time provide an efficient representation of the interactions through fuzzy rules.
One of the key characteristics of the proposed approach is that by incorporating ENFRNs neural network-specific processing and learning capabilities the method can be applied (after the training process is concluded) to unseen samples arriving from microarray data of different experimental settings in order to test whether the same regulatory interactions are valid within the new data set.
The proposed ENFRN-based method successfully extracted relations that in their majority were in accordance with biologically proven regulatory interactions, outperforming other computational approaches. In certain cases our algorithm picked up interactions that could not be retrieved in the lists of experimental interactions currently existing in databases. It might be possible that some of these interactions are valid but currently unknown, since the available experimental data is still far from complete at present. However, those interactions and especially the ones assigned a high degree of confidence should be further examined.
Methods
Evolutionary Recurrent Neuro Fuzzy Network
This section presents the proposed method, its architectural structure together with key aspects of its functioning: dynamic mapping capability, temporal information storage and fuzzy inference system.
Unlike other neuro-fuzzy network architectures, where the network structure is fixed and the rules should be assigned a priori, there are no fuzzy rules initially in the architecture we are presenting; they are constructed during learning, in a self organized manner. The two learning phases (the structure and parameter learning) are used to accomplish this task. The structure learning phase is responsible for the generation of fuzzy IF-THEN rules as well as the judgment of the feedback configuration, and the parameter learning phase for tuning the free parameters of each dynamic rule (such as the shapes and positions of membership functions). These are accomplished through repeated training on the input-output patterns. The way the input space is partitioned determines the number of rules.
Under the framework of the GRNs reconstruction that this paper addresses, the input and output spaces correspond to the expression profiles of the input and output genes, respectively. A fuzzy rule derived by ENFRN is time-dependent and is realized via linguistic labels corresponding to fuzzy sets (represented in this study by Gaussian membership functions) that describe the variations of the expression profiles of the input-output genes. Those labels are unified in a rule via a fuzzy AND operation:
IF g_{1}(t-1) is low AND g_{2}(t-1) is high AND ... g_{ N }(t-1) is low THEN g_{ o }(t) is high
where low, medium, and high correspond to linguistic labels represented by fuzzy sets and partition the input and output space. The left hand side of a rule such as the one described above corresponds to a certain cluster of the expression profiles of the input genes (regulators), while the right hand side on a corresponding partition of the output space (target genes).
Given the scale and complexity of the data, the number of possible rules describing the gene expression data at hand is kept under constraint by employing an aligned clustering-based partition method for the input space, meaning that both input and output variables may have a different number of fuzzy sets describing them [16, 41]. Additionally, we enforce a setting in which rules with different preconditions may have the same consequent part. By incorporating the clustering scheme, we are able to tackle the problem of rule set combinatorial explosion that appears when using other neuro-fuzzy approaches [42].
ENFRN Architecture
where x_{ i }is the value of the i- th input (gene expression) value.
where β_{ ij }denotes the link weight for the feedback unit and φ_{ ij }is the j-th membership function of the i-th input variable. As it can be deduced from (3) each linguistic node, acts also as a memory unit storing past information for ENFRN concerning the input.
where D_{ i }= diag(1/σ_{i 1}, 1/σ_{i 2}, ..., 1/σ_{ in }), c_{ i }= (c_{i 1}, c_{i 2}, ..., c_{ in })^{T}. The output of every one of the rule nodes corresponds to the firing strength of a rule; meaning how well fitted is the corresponding rule, such as to adequately describe the current input pattern.
As can be deduced from the equation above, the output for every one of the nodes of the current layer is the sum of the output of the rule nodes that have as consequent part the current node.
w_{ ij }are the widths of the fuzzy sets of the output layer. Each one of the nodes corresponds to the value that ENFRN has calculated to be the predicted output value for the corresponding output variable.
Partitioning of input and output space
The creation of a new rule within ENFRN corresponds to the creation of a new cluster in the input space. Therefore the way the input space is partitioned (clustered determines the number of fuzzy rules created. Thus, the number of rules created by ENFRN is problem depended, the more complex a problem is, the greater becomes the number of rules.
where y_{ k }is the value computed by the model based on the current input pattern x_{ k }while y_{ k }^{ d }is the desired output for the same pattern.
The first criterion checks whether the error of the model is greater than a specific value, indicating that under the current state of the network we cannot handle the new pattern without a large value of error. The second checks to see whether the pattern is 'close' enough to a cluster, so that it can become a member of it, or if a new cluster has to be created for it.
where δ is a constant deciding the overlap of the clusters, d_{ min }is the distance of the closest cluster (from the current pattern), and c_{ i }, σ_{ i }are the center and width for the membership function of each input variable. Similar methods have been used before, such as [41, 43]. To reduce the number of fuzzy sets of each input variable and to avoid the existence of redundant fuzzy sets, we check the similarities between them in each input dimension. For the similarity measure of two fuzzy sets we use the formula previously derived in [44], which concerns bell - shaped membership functions.
Finally, ENFRN has to determine if a new output cluster must be created. A cluster in the output space is the consequent of a rule. We have already mentioned that one of the characteristics of ENFRN is that more than one rule can be connected to the same consequent. As a result, the creation of a cluster in the input space does not necessarily mean a subsequent creation of a cluster in the output space. It depends on the incoming pattern, since the newly created cluster in the input space could be connected to an already existing cluster of the output space. The model decides whether or not to create a new output cluster based on the two criteria described above (Eq. 11).
PSO for ENFRN Optimization and Learning
Particles move through the problem space following a current of optimum particles using Eqs. (14), (15). The process is then iterated a fixed number of times or until a predetermined minimum error is achieved where hopefully, given a sufficient number of particles the swarm will converge towards an optimized solution [45].
Structure Optimization
As we have already mentioned, during the creation of the initial ENFRN structure we do not spent much time determining the centers and widths of fuzzy sets so as to find a perfect cluster. This might drive ENFRN, especially in cases where we have difficult problems such as the one we study here, to create a rather large structure containing many rules that could even be redundant.
Hence, at this phase of the learning algorithm of ENFRN we have developed a scheme for deleting some of the rules along with their corresponding output nodes (in cases where they are not connected to other rule nodes), if they are either redundant or the patterns they describe could be efficiently represented by other rules. Therefore, the objective of this part of the learning process has a dual goal of decreasing the redundancy and simplifying the model. For the optimization of the ENFRN structure we have employed a discrete version of the PSO algorithm.
To optimize the number of rule nodes in the initial network structure we use a version of Binary PSO (BPSO) algorithm. Potential solutions to our optimization problem are encoded through particles that under BPSO are symbolized as fixed size binary strings X_{ i }= (X_{i 1}, X_{i 2}, ..., X_{ iM }), where X_{ ij }∈ {0,1}. Given a list of rule nodes, R = (R_{1}, R_{2}, ..., R_{ M }) the first element of X_{ i }, X_{i 1}corresponds to the first rule R_{1}, the second to the second rule R_{2} and so forth. A value of zero at the coordinate associated to a rule node, indicates that the corresponding rule is not selected, while unity indicates that the rule is selected.
where MSE is the well known mean square error, M is the initial number of rules, as given by the structure learning phase and N is the number of rule nodes present in the simplified structure. We can derive from the formulation of the problem that we assume the number of rule nodes described by each one of the particles to be lower or at the most the same with the initial number of rule nodes (therefore it is 0<N<M → 0 < M-N → M-N>0 → M-N+1>1). It is obvious from equation (16) that the closer a particle arrives to an optimized solution in terms of small value of the error with a minimum number of rule nodes, the smaller the fitness value is obtained. Hence, BPSO attempts to minimize the function (16).
Where the parameters c_{1}, c_{2}, n_{1}, n_{2} and W_{ I }are the same as the one for equation (14) of PSO.
that is the well known sigmoid function.
Parameters Learning
In this section we describe the algorithmic methodology we follow to train the parameters of the ENFRN. Several variations of the family of back propagation algorithms (e.g. Back Propagation through time) have been used to train recurrent multilayer neural and fuzzy neural networks by means of error propagation via variational calculus [46]. Even though they are widely used, these algorithms have many drawbacks. For instance, their success depends upon the quality of the training data, as well as the initial values of the weights of the network. Additionally, they are known to be easily getting trapped in local minima, losing this way the global minimum of the error function [47].
where k is the number of input variables, N_{ o }is the number of output nodes, m_{ ij }and σ_{ ij }represent the centers and widths of the fuzzy sets describing the input variables, β_{ r }corresponds to the centers of the fuzzy sets of the output variable, while w_{ ij }are the recurrent weights of the 2^{nd} layer of ENFRN.
where T is the number of time points in the dataset, y_{ i }represents prediction of the output based on input value as calculated ENFRN and y_{ i }^{ d }is the actual value given by the dataset. Hence the purpose of applying PSO in this final phase of ENFRN learning scheme is to minimize (21) by fine-tuning the parameters of ENFRN structure.
ENFRN reconstructs GRNs
In this section we present the approach through which, given an ENFRN structure trained and optimized as described above, we can determine if the input regulates the output, the kind of the regulation and also provide a quantitative measure (which we call regulation score) specifying how well this task is performed. Additionally, we propose our approach on handling the important and computationally expensive problem of determining a set of possible regulators for a certain target gene.
From ENFRN Structure to Regulation Type
A set of fuzzy rules arriving from a certain ENFRN structure is employed to determine the regulation type (i.e. up, down, or even no regulation) that the input imposes to the output based on a given data set. Each of the ENFRN rules has a prerequisite part consisting of a number of fuzzy sets (matching the number of input variables). These fuzzy sets correspond to linguistic labels describing the values of the input variable. In our case the linguistic labels correspond to a certain expression level of the input gene (e.g. high, medium-high, medium, etc). The same applies for the output.
Throughout this study, we follow the main principle that for a certain gene x to be considered as a possible regulator of some other gene y, then x must have a previous expression status alteration when compared to y. An expression status alteration (both for input and output genes) occurs when the linguistic label describing the expression status of a gene at a specific time point t is different than the value of the linguistic label of the same gene at time point t-1.
As it can be deduced from (22) and (23) we follow a first order Markov process where we search for an alteration at the expression levels of all input and output variables/genes between 2 consecutive time points. However, due to the recurrent nature of the ENFRN, the proposed approach actually takes under consideration all of the previous time dependencies during the construction and learning processing phases.
If there is more than one input variable in the Input Layer then those input variables/genes are connected thought a fuzzy AND operation at the Rule Layer. Hence, all of the input genes must have a simultaneous change in their expression profiles. Therefore, in the case of multiple inputs, if the alteration of the input variables is not simultaneous for any time point of the experiment, the output of RS indicates that the input genes do not regulate the target (Figure 6-D).
Determining potential regulators
Irrespective of the algorithmic methodology followed for determining the possible regulation of a set of genes to some target gene(s), the major problem for gene networks reconstruction is the computational complexity rising from the enormous number of candidate regulators (i.e. feature space) that should be considered. This fact makes the application of many data-driven computational models prohibitive for accurate prediction of large scale modelling of regulatory networks. Various tactics [4, 8, 48] have been used to overcome this significant problem. One of those techniques is to apply clustering [48] towards the construction of co-expressed clusters of genes, then further analysis proceeds using just the centres (i.e. mean) of those clusters. This approach however imposes a very serious amount of information loss since only general trends concerning groups of genes can be inferred and not real biological associations among pairs of genes.
Another approach is to employ heuristic or evolutionary schemes, like PSO, for the determination of the best regulators for a certain target gene [8]. Specifically, starting from some gene population, BPSO feeds randomly various input subsets of the initial population to the main method used (e.g. ANN) to determine the regulation. Finally, BPSO concludes on the best regulators using as fitness function value the output of the main method [4]. The application of such algorithms at this point of the GRNs reconstruction actually leaves the problem intact. Indeed, considering as input to those algorithms large set of genes would induce severe computational complexity. The problem is magnified if we consider that based on the stochastic nature of those algorithms the whole process must be repeated several times.
where n is a constant ranging between 0 and 1, indicating the degree of importance we set for the network MSE value and regulation score RS (eq. 22). In this study the value of n is set to 0.6.
In the second stage, a set of all possible binary 2^{ Nk }configurations (where 1 represents the presence and 0 the absence of each one of the N_{ k }genes as input to the ENFRN) of the N_{ k }candidate genes is created. This choice is based on evidence from previous studies that genes might act individually as regulators of another gene or they might act concurrently with a group of genes to activate or repress a third-party gene or to achieve a specific biologic function [4]. Each one of these subsets of genes is inserted as input to the ENFRN and subsequently the best N_{ B }configurations are selected, in terms of network error. The two stages of the methodology described so far are based on the first phase of the ENFRN learning scheme (presented in section Partitioning of Input and Output Space), thus making use of the significant computational efficiency of this phase that allows a very large number of genes to be examined as possible regulators. In the last stage of the procedure, the N_{ B }selected subsets of candidate genes are optimized using the second and third phase of the ENFRN learning process and the final N_{ F }configurations are identified. These will be set as the potential regulators of the output gene. It must be underlined at this point that the procedure might yield a single potential regulator gene or the whole set of (maximum) N_{ k }genes, if indeed the expression patterns of these N_{ k }genes indicate them to be plausible regulators of the gene selected as output. The connectivity within the regulatory network is generally problem-dependent, however, based on biologically relevant evidence from previous studies we set the maximum number N_{ k }of possible regulators to 5 [4], unless otherwise stated.
Declarations
Authors’ Affiliations
References
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