1. Introduction
In the past few decades, biology has started to integrate a comprehensive knowledge of biological systems into biochemical networks and access the benefit of using it for mathematical modeling and simulation. For example, a whole-cell mathematical model of the bacterium
Mycoplasma genitalium, which contains 525 genes, was built based on an enormous amount of experimental data and enabled the researchers to discover a new enzyme and some other suggestions [
1]. When we aim to construct a comprehensive network, statistical methods represent one of the first choices [
2,
3,
4]. However, even if we build such a large-scale network, to translate it into equations and then analyze them is still challenging, given that many parameters need to be determined in advance. Estimation of all the parameters of a whole-cell model has not yet been appropriately resolved [
5]. Moreover, analyzing its dynamics is realistically impossible. Therefore, a large-scale network is often reduced to a feasible-size network for the analysis of its dynamics [
6,
7,
8].
In this study, we developed a new method that consists of comprehensive modeling and the reduction of a comprehensive network to analyze the dynamics of regulatory mechanisms of target phenomena. First, we constructed a comprehensive network from databases and literature information. Second, to reduce the network size, we contracted a cascade structure, consisting of a sequence of unidirectional edges and nodes, and preserved the loop structures which affect the essential behavior of the network [
9,
10]. This method makes it possible to obtain a small network that can reproduce the dynamic behavior of a large network.
We modeled the differentiation of neural stem cells (NSCs) and analyzed the characteristic changes of dynamics during differentiation. NSCs replicate and differentiate into neurons, astrocytes, or oligodendrocytes [
11]. Some models simulate early differentiation or functional neurons [
12,
13], but no model enables us to simulate and analyze the dynamics of the large-scale regulatory network of neuronal differentiation. The basic helix-loop-helix-type transcription factors HES1, ASCL1, and OLIG2 show characteristic differences in their dynamics before and after differentiation [
14]. They also maintain oscillatory dynamics during cell replication. If the concentration of ASCL1 is higher than that of HES1 during the non-oscillatory state, the NSCs differentiate into neurons. If the concentration of HES1 or OLIG2 is higher than that of ASCL1, the NSCs differentiate into astrocytes or oligodendrocytes, respectively [
14]. In the current study, we constructed a comprehensive molecular-interaction network of NSC differentiation into neurons using the available data. We then developed a mathematical model that maintained the original dynamics of the network by integrating loop structures. The model could simulate the characteristic dynamic changes before and after differentiation. The model also reproduced the effect of overexpression or knockdown of the Id2 gene, which encodes an inhibitor of HES1 dimerization [
15]. We suggest that the stabilization of oscillations and characteristic dynamic changes are regulated by the combination of multiple feedback loops.
Figure 1 illustrates the analysis processes employed in this study. Our method allows the analysis of a comprehensive regulatory network by exhaustively collecting information and scaling down the network according to the rationality of dynamics without arbitrariness. On the basis of the analysis of the dynamics of neuronal differentiation, we propose that a combination of multiple loops is important for defining the major dynamics of an entire network.
4. Discussion
We generated an NSC differentiation network containing four feedback loops on the basis of publicly available data. Our digested model was constructed through cascade contraction of the comprehensive regulatory network with the preservation of feedback loops. Three types of HES1 and ASCL1 states regulated by the NOTCH concentration were consistent with the NOTCH-dependent neural differentiation suggested by Imayoshi et al. [
14]. Although experimental data show complex waveforms of HES1 and ASCL1, we simulated the main-frequency waves, which have a period of 2 to 3 h [
14], and analyzed the dynamics of the transition from the oscillatory to the non-oscillatory state qualitatively using a digested model. The results of the simulation of GSK3B, aPKC_PAR3_PAR6, and PI3K, represented as GSK3B_ca, aPKC_ca, and PI3K_ca in the digested model, respectively, are consistent with previous experimental results [
21] (
Figure 5B,C). The results of the simulation of Id2 knockdown or overexpression are also consistent with experimental results [
15]. Therefore, our digested model could adequately simulate the dynamics not only of HES1 and ASCL1, but also of other molecules. Our model suggests that three loops (HES1 negative self-feedback, positive feedback between aPKC_PAR3_PAR6 and PI3K, and negative feedback between GSK3B and HES1) are important for maintaining undifferentiated state oscillations. Our simulation result showed that inhibition of the HES1 self-feedback loop caused the disappearance of its oscillatory expression (
Figure 7A). At the same time, the ASCL1-dominant condition, which is equal to the neuronal differentiation-dominant condition, becomes narrower. This means that the inhibition of the HES1 self-feedback loop could suppress the differentiation of NSCs. A previous study experimentally showed that the inhibition of HES1 by overexpression of the id protein caused the inhibition of differentiation [
15]. Therefore, the result of our simulation was consistent with this previous experimental knowledge. We suggest that the negative-feedback loop between beta-catenin and HES1 in the comprehensive regulatory network is also important because of its greatest contribution to the characteristic dynamics (
Figure 7D). A relation between beta-catenin and HES1 plays a role in tumorigenesis [
46]. As HES1 controls cancer stem cells [
47], the negative feedback loop that has not been focused on may be related to the proliferation and differentiation of cancer stem cells. It is expected that a further experimental study, such as perturbating the loop by knock down, will reveal detailed mechanisms of neural differentiation. These findings could only be produced by using the analysis based on a large-scale regulatory network, thus highlighting the effectiveness of our approach.
We demonstrated that focusing on feedback loop structures instead of the whole network when constructing a model was sufficient for producing data in agreement with experimental results. Our approach could be applied to an analysis of various biochemical networks by simulation. By streamlining large-scale regulatory network construction, our approach could help to analyze various biological phenomena, such as cell differentiation, cell division, or pathogenesis. However, the large-scale regulatory network will probably be insufficient and heterogeneous when it is constructed using the available data alone. To overcome this false-negative problem (the relations that exist, but cannot be detected), many data-driven network reconstruction methods have been developed. These statistical approaches are mainly classified under two categories: expression-based [
48] and sequence-based [
49]. Although the methods of both categories can reveal undiscovered relations that cannot be inferred manually, the reconstructed network includes many false-positive regulations. A nonlinear model intrinsically causes complex behavior. With an increase in the number of false-positive regulations, an increase in the number of nonlinearities becomes avoidable. Based on this mathematical background, a model with a high number of false-positive regulations seemingly generates the real behavior, but is different from the real system. Therefore, it is desirable to build a mathematical model only from reliable elements. The addition of false-positive regulation to the model could have a considerable effect and complicate the conversion of a data-driven network into a mathematical model. Currently, the manual methods are better than the data-driven methods for the construction of a mathematical model.
Our large-scale regulatory network of neuronal differentiation may lack some components and thus may not completely represent neuronal differentiation. In our network, ASCL1 is directly affected by HES1; ASCL1 oscillation controls proliferation and differentiation [
14] and affects NOTCH receptors of adjacent cells via the activation of DLL [
50]. Neural differentiation is also affected by adjacent cells [
51]. We excluded this information because we focused on the dynamics of a single cell. To reveal the entire mechanism of neural differentiation, adding a path to adjacent cells, for instance via DLL, might be required. The analysis of multiple cells might provide a model that can simulate dynamics other than state transitions. The concentrations of both HES1 and ASCL1 decrease in a non-oscillatory state when an NSC differentiates into an oligodendrocyte [
14]. To simulate this transition, we need to add a signaling pathway focusing on the oligodendrocyte marker OLIG2, which oscillates with a period of 400 min in NSCs [
14]. This period is much longer than that of HES1 or ASCL1, and the dispersion of oscillation is very high. Therefore, OLIG2 regulation might involve a delay mechanism to elongate the period and a mechanism to amplify dispersion. Recently, a similar method was used to analyze oligodendrocyte differentiation [
52]. Similar to our study, the authors used a manual method to construct a network; however, they also introduced publicly available interaction data from omics databases. In comparison with our method, this approach may reduce the number of false-negative interactions. On the other hand, the study [
52] only focused on two- to four-node feedback loops. Our contraction method may detect a larger regulatory system of oligodendrocyte differentiation. The complete mechanism of neural differentiation may be simulated by integrating this information and methods. Our model can simulate the dynamics of NSC-to-neuron transition and exemplify the reverse transition by increasing the concentration of NOTCH, but differentiation is mostly irreversible. Therefore, it is difficult to validate the results of reversing from a neuron to an NSC. Some hypotheses suggest the core factors of differentiation that also inhibit reprogramming [
53] or control the mechanisms generated by neurogenic niches [
54]. Specific network structures such as positive feedbacks or micro-environmental factors may be important for hysteresis in differentiation, and a more detailed, larger network needs to be analyzed.
There exist some network reduction approaches, but some of them are limited in terms of the preservability of original dynamics [
6,
7]. Kernel Identification [
8] is one of the methods employed to preserve the network dynamics; however, the preservability is only validated with Boolean models. Because we aimed to maintain the network dynamics at the level of ordinary differential equations, we concentrated on extracting loops. We did not modify the network, except for cascades as the target of reduction, instead of applying the previous work. Practically, network reduction by Kernel Identification may produce a similar contracted network and will be advantageous when the network is larger than in this case.
To simulate more realistic behaviors of biological cells, highly crowded and inhomogeneous environments should be considered. The use of fractional derivatives has been well-evaluated in the usage of visco-elastic processes of soft materials [
55], and recently, they have started to apply more biological targets [
56,
57,
58,
59]. Our mathematical model of neural differentiation is also expected to become more realistic by fractional derivatives instead of classical integer derivatives. Although we focused on NOTCH signaling in this study, our network also includes FGF as another input signal. An analysis of the behavior of the network stimulated with FGF may show variate responses and as a result, may reveal other mechanisms of neural differentiation. Although the core network did not include a feed-forward loop, a feed-forward loop accelerates the response time of a system and achieves cell state transition rapidly. Because the cellular state transition obtained by NOTCH signaling is also known to be accelerated by a feed-forward loop [
60], a feed-forward loop may exist upstream of the core network. Our method can be used to analyze the dynamics of a new large-scale regulatory network when new information becomes available.