Modeling gene regulatory network motifs using statecharts

Gene Regulatory Network motifs

GRN motifs are pattern of interconnections occurring in real GRNs with a frequence that is significantly higher than that in a randomly generated GRN.

Their high frequency suggests that they play an important role in the GRN function and can, thus, be considered as its building blocks.

The functional role of most common GRN motifs has been extensively studied in some organisms, such as E. coli and other model organisms [19].

The simple regulation motif

The simple regulation motif is one of the most basic interaction patterns. It is composed of two genes X, Y, where X regulates Y and the interaction is mediated by a signal S _X . The signal can act as an inducer molecule that binds X or can represent a modification of X which activates it. Since the regulation of X on Y is either activation or repression and S _X can mediate the regulation with either presence or absence, four possible types of motifs can be described.

A simple regulation motif is coherent if both the effects are of the same polarity, i.e. activation of Y in presence of S _X (s1 in Figure 1A) or repression of Y in absence of S _X (s2). It is incoherent if the effects are of different polarity, i.e. repression of Y in presence of S _X (s3 in Figure 1A) or activation of Y in absence of S _X (s4).

Statecharts

SCs extend state transition diagrams by adding concurrency (i.e., the capability of representing a state as made up by smaller components all active at the same time) and hierarchy (i.e., the possibility of representing a state with a set of more detailed substates). The hierarchical structuring capabilities of SCs allow one to model systems at different levels of detail, while concurrency is useful for modeling multiple, mostly independent, portions of a system. Moreover, SCs are compositional, that is, they can be defined in terms of other SCs, thus making the specifications more reusable.

These additional features, if correctly exploited, provide a solution to the scalability problems of other computational modeling techniques like, e.g., those based on boolean networks and Petri nets, whose effectiveness rapidly decreases when applied to larger systems [3].

We now summarize some of the SCs features that we believe are essential to understand their potential. Please refer to [18] for more complete and detailed information.

A SC is composed of states and of transitions between states. A state is composite, if it contains other states, and is simple, otherwise. A composite state is parallel if its sub-states are executed concurrently, and is exclusive if exactly one of its sub-states is executed. The overall state of a SC is given by all the atomic states currently under execution.

Transitions are used to specify how a system evolves changing its internal state according to the external stimuli. They can be labeled by events which trigger their activation and the consequent change of state of the system, conditions for their applicability, and actions to be performed during their execution.

SCs have an intuitive graphical representation: see Figure 2A showing a SC modeling the movement and feeding of an organism by means of two concurrent substates.

SCs have very good software tool support [22–27], which can be used to generate source code (e.g. in Java) whose execution corresponds to the SCs semantics, and to interactively simulate the system execution. SCs have been extensively studied in software and systems engineering, and have demonstrated to be particularly well-suited for modeling and designing reactive systems, that is, systems which evolve reacting to internal or external events, or changed conditions. In the case of GRNs these events can be, for example, the introduction or removal of a protein or of another component.

SCs have also been successfully used to model pancreatic organogenesis in the embryonic mouse [28], cell fate specification during C. elegans vulval development [29], and T-cell development in the thymus [30].

Shin and Nourani have used SCs to model GRN motifs [17]. In their approach, each element (gene, protein, signal) can be in one of the two states: "on", which means that the gene is expressed or that the protein is present and active, and "off", which means that the gene is not expressed or that the protein is not present or present in its inactive form.

Moreover, activating interactions in GRNs are translated to transitions from the "off" state to the "on" state for the gene being activated. Similarly, inhibiting interactions correspond to transitions from the "on" state to the "off" state.

Their SCs model of the coherent simple regulation motifs s1 and s2 is shown in Figure 2B, which in their approach represents also the autoregulation motifs.

Model of simple regulation

Our models of the simple regulation motifs s1 and s2 are shown in Figure 3A left and right. In both cases, all the elements involved in the regulation, the genes X and Y and the signal S _X are modeled as concurrent states, and, for each of them, we use two states for modeling its presence (and absence). The activation and deactivation of the regulated gene are modeled by two transitions connecting its presence states, which are triggered according to the truth value of logical formulas depending on the presence of the gene X and the signal S _X . Note that in the logical formulas the green symbol ∨ represents the logical connective OR while the orange symbol ∧ the logical connective AND. Note also that in the logical formulas for any element X, the expression X = 1 is abbreviated as X and the expression X = 0 is abbreviated as $\bar{X}$.

Our approach for modeling simple regulation is non-ambiguous, because motifs s1 and s2 are represented by two different SCs. See again Figure 3A for our model and compare it with the ambiguity deriving from Shin and Nourani model shown in Figure 2B, where the same SC is used to describe both s1 and s2. Mapping different motifs onto the same SC is a potential source of problems when the mapping is inverted (i.e., from the SC to motifs) because it is not clear whether the SC should be mapped on both the original motifs (thus, possibly leading to an over-specification) or it should be mapped on only one of them.

Moreover the Shin and Nourani model for coherent simple regulations shown in Figure 2B is incomplete, because it implicitly assumes that the regulating gene X is always expressed. But ignoring the situation where X is not expressed can be significant if, for example, the same gene has a repression role in other parts of the network. If we try to solve their incompleteness problem by adding another state for X = 0 then we have to duplicate the states for Y = 0 and Y = 1, thereby obtaining the SC of Figure 3B and losing the scalability advantage of SCs.

In fact, their model does not fully exploit the concurrency features of SC. This determines sub-optimality, because it does not allow to reduce the size of the system. Their method is therefore not scalable: the complexity of their models grows faster than their size. Moreover, since the states of the regulated gene are modeled as substates of the regulating gene, and not as concurrent states, it is not possible to model networks containing genes which reciprocally regulate each other (see the model of feedback loop presented below). Note that these problems of [17] just described with reference to coherent simple regulations also affect the modeling of the other, more complex, motifs.

Similar considerations also apply to the modeling of the incoherent simple regulation motifs s3 and s4, whose SCs models with our approach are shown in Figure 3C.

Model of feedback loop

The feedback loop motif is not addressed by the modeling approach defined by Shin and Nourani [17] and we will shortly prove that it cannot be. We first note that the authors themselves observe in the "Further Discussion" section of their paper [17] that feedback loop motif is not part of their modeling scheme and that they intend to incorporate it in the future. We observe that this is not possible in their method, because it requires the states of the regulated gene to be substates of the states of the regulating gene. Since in the feedback loop motif X and Y act as both regulated and regulating genes, this requirement cannot be fulfilled.

Our modeling approach does not have this limitation because, as already mentioned, the genes and the signals are modeled as concurrent states.

The double-positive feedback loop motif has two genes X and Y which reciprocally activate each other. The model for this motif can easily be obtained from the model of the coherent simple regulation motif s1 (previously shown in Figure 3A) by adding the states for the signal S _Y and the transitions between the states for the gene X which correspond to the regulation of the gene X by Y. The resulting model and the motif are shown in Figure 4A.

From now we shall discuss also the temporal behavior of each SCs model representing a given in vitro motif so as to determine how closely each model is able to reproduce the corresponding in vitro behavior. Note that since a SC is a discrete model the state of the regulated gene at time instant t + 1 depends on the state of its regulating gene at time instant t. Also note that the results of this investigation are a priori limited by the fact that since our SCs models are boolean any behavior requiring more than two values in the domain cannot be reproduced.

The temporal behavior of the SCs model of the double-positive feedback loop motif is shown in the diagrams reported in Figure 4B. In particular, when X and Y are initially both present or both absent, it exhibits the "joint bistability" behavior [31], that is X and Y are either both always "off" or both always "on", as shown in Figure 4B (left and middle). But, as you can see in Figure 4B (right), when the initial state for X and Y is different, the temporal behavior, due to the approximation of the boolean domain where only two values are available, is not able to escape from the oscillating pattern to fall into one of the two steady states that are known from the in vitro experiments [5, 31].

Our approach allows us also to build the model for the double-negative feedback loop motif, where the two genes X and Y reciprocally repress each other (see Figure 5A).

Also in this case, our SCs model is able to reproduce the temporal behavior of the motif, that is, X always "on" and Y always "off", or viceversa (this is called "exclusive bistability" in [31]). The corresponding diagrams are reported in Figure 5B (left and middle). Once again, the roughness of the boolean model does not allow the temporal behavior to be attracted into one of the two steady states when the initial states of X and Y are the same, see Figure 5B (right).

For completeness, we also show the SCs model of the negative feedback loop motif (Figure 6A), and the diagram of its temporal behavior (Figure 6B), where the oscillatory behavior known for this kind of motif [32] is reproduced. Some variations of this motif exhibit a damped oscillatory behavior: as said above, the roughness of the boolean model does not allow our modeling approach to reproduce it. We are working on an extension to overcome these limitations.

Model of coherent feedforward loop

The c1 FFL motif with the AND combination of X's and Y's regulations on Z has been used as a model of the arabinose system in E. coli. This motif, already illustrated in Figure 1C and reported for convenience in Figure 7A (top), can be modeled in our approach by using the SC of Figure 7A (bottom), which, despite its discrete nature, is able to exhibit the same temporal behavior of the in vitro system, consisting in (i) a delayed activation of Z after the activation of X, and (ii) an immediate de-activation of Z when X disappears (such a behavior is called "sign-sensitive delay" in [13]).

A diagrammatic representation of the temporal behavior of the considered SCs model is reported in Figure 7B, where it can be observed (right) that there is no delay in the deactivation of Z (Z and Y become both inactive at time instant t = 3 immediately after X disappears at time instant t = 2), but its activation (left) is delayed (only Y is active in the time instant t = 3 right after X appears at time instant t = 2, and Z becomes active only in the step after Y's activation, that is at time instant t = 4).

Model of incoherent feedforward loop

The i1 FFL motif (once again, with the AND combination of X's and Y's regulations on Z) has been used as a model of the galactose system in E. coli [33] where it produces an impulsive behavior in the regulated gene which first rises very quickly and afterwards soon goes down.

The i1 FFL motif, already illustrated in Figure 1C and reported for convenience in Figure 8A (top), is modeled by using the SC of Figure 8A (bottom) which can reproduce pulse-like dynamics, as shown in the temporal diagram presented in Figure 8B. Soon after X becomes active at time instant t = 2 (left), also Z gets activated at time instant t = 3 together with Y but, after one more time step, the repressive action of Y deactivates Z at time instant t = 4. Of course, the approximation of the boolean domain only allows a unit time impulse, but that is is enough to show that our SCs model is able to reproduce the dynamic behavior typical of this motif. When X becomes inactive at time instant t = 2 (right) there is no effect on Z which remains inactive, while Y becomes inactive in the next step at time instant t = 3.

On the other side, our SCs model is not able to express the response acceleration dynamics of the i1 FFL motif with respect to simple regulation [33], as previously said in the discussion of the intrinsic limitation of the boolean domain. We are currently working on the extension of our SCs-based approach to the more general case of a many-valued discrete domain.

Model of autoregulation

The negative autoregulation motif is a very common and widely studied pattern of regulation.

Experimental results [34] have shown that it behaves as an accelerator of the gene response (with respect to the simple regulation motif), in presence of a high initial concentration of the self-regulating gene. The opposite behavior is exhibited by the positive autoregulation motif which slows down the production of the gene [35].

Our models for the negative autoregulation motif (see Figure 9A) and the positive autoregulation motif (see Figure 9B) are inherently boolean: therefore they do not have the means of reproducing the acceleration and deceleration which can be observed in vitro. The diagrams of their temporal behavior are shown in Figure 9C (left) and (right), respectively. As already mentioned, we plan to extend our modeling approach to take into account these aspects.

On the other side, note that Shin and Nourani have observed in [17] that with their modeling approach both negative and positive autoregulation are identical to simple regulation in logical domain (see in [17] their Figures 2 and 3 and their discussion of autoregulation). But as you can see by comparing our SCs models for simple regulation (Figures 3A and 3C) to our SCs models for negative and positive autoregulations (to the right in both Figures 9A and 9B), our modeling approach allows to fully distinguish, in the logical domain, the various cases. This is true even if we build with our approach the SCs models for exactly the same autoregulation motifs considered by Shin and Nourani in [17] (shown in Figure 10A) where Y is regulated by the AND combination of itself and an additional activating gene X. Such SCs models are presented for completeness in Figures 10B (positive autoregulation) and 10C (negative) and the temporal dynamics of Y when X is expressed is the same shown in Figure 9C.