On the architecture of cell regulation networks
 Yueheng Lan^{1} and
 Igor Mezić^{2, 3}Email author
DOI: 10.1186/17520509537
© Lan and Mezić; licensee BioMed Central Ltd. 2011
Received: 7 April 2010
Accepted: 2 March 2011
Published: 2 March 2011
Abstract
Background
With the rapid development of highthroughput experiments, detecting functional modules has become increasingly important in analyzing biological networks. However, the growing size and complexity of these networks preclude structural breaking in terms of simplest units. We propose a novel graph theoretic decomposition scheme combined with dynamics consideration for probing the architecture of complex biological networks.
Results
Our approach allows us to identify two structurally important components: the "minimal production unit"(MPU) which responds quickly and robustly to external signals, and the feedback controllers which adjust the output of the MPU to desired values usually at a larger time scale. The successful application of our technique to several of the most common cell regulation networks indicates that such architectural feature could be universal. Detailed illustration and discussion are made to explain the network structures and how they are tied to biological functions.
Conclusions
The proposed scheme may be potentially applied to various largescale cell regulation networks to identify functional modules that play essential roles and thus provide handles for analyzing and understanding cell activity from basic biochemical processes.
Background
Cellular behavior, including motility, metabolism and reproduction is controlled by complex biochemical reaction networks, many of which have been identified and studied in detail [1]. These networks realize their regulatory roles through complex molecular interactions. Contemporary high throughput experiments produce unprecedented amount of data that serve to pinpoint the players and their interactions, resulting in complex chemical reaction graphs. How to analyze these intricate graphs and gain insight into the regulation mechanism employed by cell has become a central problem of molecular biology.
Much progress has been made in the analysis of functions of complex networks, no matter if they are modeled deterministically [2, 3] or stochastically [4–9]. These studies concentrate on the investigation of dynamics of given networks by checking their stability, parameter dependence, robustness and inputoutput relation. However, for largescale networks such as those commonly found in important biological processes [10, 11], the incurred computational load often severely limits our ability for performing detailed analysis. More critically, with continued experimental efforts that are revealing more details of networks' global wiring, their growing complexity has made it harder and harder to identify the underlying local functional structures and thus probe the network function.
Normal cell life involves physical or chemical activities at vast range of spatial and temporal scales and it is vital to identify characteristic structures at all scales and study their roles in relation to a particular cell function [12–17]. These key structures are called modules, the existence of which contributes almost to every aspect of the cell regulation: robustness, sensitivity, adaptivity, evolvability. Their detection and study much simplifies the analysis of complex networks since a small set of modules could come from and be a lot simpler than a collection of many entangled individual agents [18]. The simplification may be carried on by constructing modules of modules.
Recently, useful concepts distilled from statistical physics such as the smallworld and the scalefree networks [19, 20], began to see their application in gene regulation networks and lead to considerable success in unraveling the statistical nature of these networks. However, this type of statistical analysis mainly aims at gross features of networks [21] and thus ignores local structural properties and heterogeneities, which often determine the operation of a network in an essential way, since disparate network modules generally imply distinct dynamics and fit for different functional requirements [22, 23]. Nevertheless, the determination of modular structure in a large network is not straightforward since one molecular species may be involved in many different pathways with very distinct external connections. Such intercorrelation is easily underappreciated and yet has profound consequences on the organism.
In this paper we propose a new theory of architecture of biochemical networks based on control and graph theoretic analysis. In this theory, a network consists of two major modules: one is the pipeline of linear information production unit which serves to generate the required output (e.g. protein concentrations); the other is the set of feedback loops which act as controllers of the production. These two modules are identified based on the information flow in a network. Specifically, input and output nodes define a polarity of the network. Information is received at the input, processed and then sent to the output. The agents that carry on the information along the forward direction belong to the production unit. The remaining agents direct part of the information in the opposite direction and thus are elements of the feedback controller [22]. In the paper, detailed algorithm are presented for the construction of the production unit and the feedback controller.
The concept of modules has been used in modeling of biological networks for decades. The existence of this special structure is universally agreed upon but its exact definition is done on casebycase basis. Recently, modules and community structures are defined in the graph theoretic studies of many realworld networks [20, 24], based on the connectivity between nodes. Useful as it is, this type of definitions ignore the importance of controller loops. The community structure in the synchronization study involves more dynamics information but it works for a special class of networks and for particular types of equations of motion. Closely related concepts, such as "network motif" are also proposed [13, 25]. Motifs consist of a small number of nodes and appear repeatedly (more than expected from pure statistical consideration) in a network. The modules determined by our algorithm are different from all these in that we emphasize the information processing and controlling units but not simple fixed graph structures given a priori. In contrast, the decomposition procedure based on the function of the network and the associated polarity supplies the detailed structures of our modules. Different polarities may result in different decompositions and different initial conditions may define different MPUs. So our concept of modules depends on the information flow through or the function of the network.
In the following, we will use the NFκ B regulation network [26] as an example to explain our graph theoretic analysis procedure and display the generic producercontroller structure. We also analyze the chemotaxis network of E. coli, TNFα initiated apoptosis network [27], the circadian clock network in Drosophila[28] as interesting examples of the proposed architecture. Three more examples of biological networks are presented in Additional File 1 and are all found to possess the same architecture.
Results and Discussion
The NFκB regulatory network
The NFκ B regulatory pathway concerns the switching dynamics of the nuclear factor NFκ B, which regulates various genes important for pathogen or cytokine inflammation, immune response, cell proliferation and survival [29, 30]. In the cytoplasm of a resting cell, NFκ B usually binds to Iκ Bα and its activity is suppressed. Certain external signals activate the switch protein IKK which phosphorylates Iκ Bα such that NFκ B is released [31]. The free NFκ B then translocates into the nucleus and initiates the transcription of a large set of proteins, including protein Iκ Bα and protein A20. Protein Iκ Bα, once synthesized in the cytoplasm, enters the nucleus, binds to NFκ B, transports it out to the cytoplasm and thus terminates the transcription. Protein A20 deactivates IKK. Therefore, the module mainly consists of two forward proteins IKK and NFκ B and two feedback proteins Iκ Bα and A20. Also, the translocation of the proteins between the nucleus and the cytoplasm is an important biological process that realizes spatial localization of different protein species.
For any networked system described by certain dynamical equations, it is easy to write an interaction graph with the vertices representing the reacting agents and the edges directed from each agent to the ones under its influence. The interaction graph for the NFκ B model is shown in Figure 1B.
It is straightforward to write down the adjacency matrix for the interaction graph, which marks 1 at the entries corresponding to connected edges and zero otherwise. The interaction graph and the adjacency matrix neglect details of the interactions and only map out the network topology which holds true almost everywhere in the phase space and the parameter space, except for a set of measure zero [33]. This robustness confers flexibility of analysis to analyzing vastly different dynamics described by ODEs or mappings or even stochastic equations. Certain system properties, like the uniqueness of the stationary point sometimes can be deduced from pure topological consideration of network structures [34, 35]. So, understanding of structure of interaction graphs helps unveil the key elements in a complex system which possibly has uncertainties in the parameter values or is influenced by a noisy environment. Graph theoretic techniques will be developed here to enable an automatic decomposition of a biochemical network into forward and feedback modules, thus unraveling the architecture responsible for its biological function.
Controllers of the NFκB network
The horizontalvertical decomposition (HVD) of an interaction graph of a dynamical system has been discussed in a paper [33]. It is a technique that studies information flow and processing in interconnected systems. Vertically, the HVD decomposes a system into a linear series of layers, where the layer downstream is influenced by the layer upstream but not vice versa. So, the input signal propagates unidirectionally. Horizontally, the HVD decomposes each layer into independent groups with no direct connections between. In one layer, each group receives its own input from upstream layers and output the signal to downstream layers. Each group is a strongly connected component (SCC) such that a path always exists between any two nodes in the group. If each group collapsed into a point, the whole network will become cyclefree [36].
Direct application of the HVD to the interaction graph in Figure 1B results in three layers with the top and bottom layer consist of the vertex sets {x_{1}}(IKKn) and {x3,x_{15}}(IKKi, cGenmRNA), respectively. The rest of the vertices are strongly connected and belong to the middle layer. This type of structure with dominant intermediate processing unit exists in most biological and engineering networks [33, 37] as a result of omnipresent feedback loops and reversibility of many biochemical reactions. Below, we apply our cycle search and selection technique to the middle layer for further decomposition into the production unit and feedback controller.

FB_{ a } the one through vertex 4: IKKa associates with free Iκ Bα and catalyzes its decay.

FB_{ b } the one through vertex 14: Iκ Bα_{ n }captures NFκ B_{ n }to form (Iκ BαNFκ B)_{ n }, which then moves out of the nucleus.

FB_{ c } the one through vertex 12: NFκ B_{ n }promotes the production of the Iκ Bα mRNA which translocates to the cytoplasm and initiates a burst of Iκ Bα production.

FB_{ d } the one through vertices 8 and 9: NFκ B_{ n }promotes the production of the A20 mRNA and thus initiates the production of A20, which catalyzes the decay of IKK_{ a }.
This identification agrees very well with the usual recognition of feedback loops of this system in the literature [29, 30] based on biological reasoning. The correct identification of feedback loops is essential for understanding the signal processing of a network since many important cellular activities are controlled or even realized by feedback signaling [22, 23]. We emphasize that we recognized the feedback loops by an automatic procedure based on graph decomposition.
Extracting the minimal production unit
After the structured network is constructed as in Figure 2A, we proceed to the extraction of the minimal production unit (MPU). In the case of signal transduction network, the MPU is the minimal subgraph of a network that produces a response to external stimuli. The MPU is minimal in the sense that removal of any links from the subnetwork will lead to zero output. However, the response of the MPU may happen at a value that is different from what is desired in a real cell and setting that correct value is one of the roles of the feedbacks. Its identification depends both on the initial state of the system and on the signal that is of interest. Moreover, certain qualitative aspects of chemical kinetics of the network need to be considered in the course. As a matter of fact, the binary or dissociative reactions correlate certain edges that represent same reactions. For example, the associative reaction A + B→ C is depicted as A→ C ← B in the interaction graph and the two arrows represent the same reaction. In previous computation, we ignored this correlation and carried out our analysis purely from a graph theoretic point of view. A more detailed consideration needs to incorporate this correlation: these two arrows have to coexist. Below, the NFκ B network is used as an example to demonstrate the procedure of the MPU extraction in detail.
As we now only consider the forward production part to output x_{15}, the feedbacks and the associated reactions are first removed. For the NFκ B network, we remove {x_{4}, x_{8}, x_{9}, x_{12}, x_{14}} and arrive at Figure 2B. The correlation among edges has been considered as suggested by the abovementioned binary reaction, i.e., the correlated arrows will be removed or kept coincidentally. Next, all the outputs except the one we are interested in are removed. That is, {x_{3}, x_{11}} are removed. Here we see that the final MPU indeed depends on what signal we are looking at. Different output may result in different MPUs. Finally, we remove other irrelevant vertices in a recursive way according to the topology of the resulting graph and the given initial conditions. In the NFκ B example, based on Figure 2B, x_{10} is removable since it does not lie on the main information path and x_{10}(t) = 0 all the time with x_{10}(0) = 0 being given. All this being done, we produce the MPU depicted in Figure 2C.
The MPU of the NFκ B network contains the vertex set S_{ m } = {x_{1}, x_{2}, x_{5}, x_{6}, x_{7}, x_{13}, x_{14}}, while all other vertices can be regarded as functional controllers. To check if what we got in Figure 2C is indeed an MPU, we keep only the variables in the vertex set S_{ m } and their interactions in the evolution equation. Numerical simulation of this reduced set of equations produced an output curve depicted with the thick solid line in Figure 1C, which displays a fast approach to a steady state value that is much larger than the equilibrium value of the full system. It is interesting to note that the saturation value and the relaxation time are very close to those of the first oscillation peak of the full equation. The vertex set S_{ m } constitutes the MPU of the NFκ B gene regulation network, and it is the smallest subgraph that generates a quick and large response to the external signal. It can be checked that cutting any link in Figure 2C will totally disrupt the outputproducing ability. For example, if the edge (2, 5) (from x_{2} to x_{5}) is cut, the edge (2, 13) has to be cut as well because of the correlation mentioned earlier, and there will be no output signal. The vertices non in S_{ m } act as controllers to bring down the initial pulse to a desired steady value in a larger time scale. Both the short and the long time response in this network bear important biological significance [30].
Biological significance of the MPU and the feedbacks
So far, we have identified the MPU and the feedbacks. Next, we go on to discuss the biological relevance of these "modules" to the operation of NFκ B network. In this and several other networks we studied, as an important observation, we find out that the MPU is the core signal production unit which responds quickly to the external cues. In the NFκ B network, when a signal such as TNF arrives, IKK _{ n } gets immediately activated into IKK _{ a } while the deactivation of IKK _{ a } is minimized since its constitutive decay rate is small. So, the concentration of IKK _{ a } will rapidly increase until A20 is produced by the feedback loop and starts the catalyzed decay of IKKa. The forward reaction rate is thus maximized transiently and enables cell response to signals with short duration [30]. So, the network has a very sensitive and fast transient response, which is essential for certain signaling pathways [30].
The feedback structures we identified respond at a much larger time scale. Only when the concentration of NFκ B reaches a high enough value and induces significant transcriptions in the nucleus, does the negative feedback start to bring down the IKK _{ a } concentration to a steady level which is much lower than the transient peak. The feedback FB _{ b } mainly facilitates the step of clearing NFκ B out of the nucleus. FB _{ c } is to restore the concentration of Iκ Bα that has been consumed by the IKKacatalyzed decay. FB _{ d } is to deactivate IKK _{ a } by A 20 to bring down the activation level of the whole network. Thus, our structural decomposition detects forward production unit for quick reaction and feedbacks responsible for long time responses.
Like other feedback signaling from the output [4, 38], these loops bring about sensitivity and robustness to the network for fulfilling its basic function [39]. The oscillation observed in Figure 1C is a signature of trading stability for sensitivity [17]. The forward immediate amplification confers easy excitability to the network while together with the delayed feedbacks brings about oscillations. On the other hand, over long time, the reaction rates of all biochemical processes are to some extent influenced by environmental variables such as temperatures, pH values, concentrations of certain ions [40]. To function normally under different conditions, the chemical network should possess structural stability. Here the double feedbacks FB _{ c } and FB _{ d } offer extra structural stability against parameter uncertainty: if the parameter changes incur a temporary increase of the concentration of NFκ Bn, then both FB _{ c } and FB _{ d } will act to bring it down. Even if one of FB _{ c } or FB _{ d } does not function well, the other one will minimize the change of NFκ B concentration. Computation shows that when the rate of the reaction involving either FB _{ c } or FB _{ d } assumes 50% of their normal value, the output signal changes little. However, major changes in the oscillation period, amplitude and the final equilibrium value of the output x_{7} are observed when both of the previous changes are made simultaneously. Therefore, these feedbacks provide extra protections for keeping the system stable under parameter fluctuations [22].
The above procedure of searching for MPU is easily generalized to more complex networks, with possible multiple inputs and outputs which interact with each other. We will study their competition or cooperation all together instead of individually. The critical step lies in our capability of detecting feedback loops. Once the feedback controllers are found, the MPU is obtained by removing all the feedbacks and then all the dynamically inessential nodes. The observed separation of time scales, can, however, leads to further theoretical study using averaging methods or normally hyperbolic invariant manifold concept from dynamical systems. We expect to pursue this in our future studies. In what follows, we analyze the E. coli chemotaxis network and several other signaling networks. More examples are available online in the Additional File 1.
Decomposing the E. coli chemotaxis network
Figure 3B displays its feedback and forward structure upon application of graph decomposition. The first level consists of the vertex set {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} which are different methylation states of the receptor complex. External signals propagate down through x_{6}, x_{8} and finally reaches the flagellar protein x_{9}.
There is one feedback vertex x_{7} (CheBp). The minimal production unit (MPU) is obtained after all the reactions involving x_{7} are removed and is contained in the box in Figure 3B.
With the feedback through CheB _{ p } (x_{7}), the system has sensitive detection and robust adaptivity as shown with thick solid line in Figure 3C. Starting with zero value, the CheY _{ p } quickly reaches the saturation level. At t = 500s, an external stimulus  10μM concentration ligand is supplied, which induces a drop of CheY _{ p } concentration followed by an exponential decay back to the saturation value. At t = 1000s, the ligand is removed which triggers a jump of CheY _{ p } concentration but regains its stable value exponentially fast. When the feedback is removed, the MPU reaches the stable value after a quick initial rise and stays at the value no matter how the concentration of external ligand changes. The robustness is retained but the adaptivity is lost. So, in this example the feedback is essential for the system's transient response to external stimulus and maintaining the adaptivity. As in the previous example, the productioncontroller dichotomy structure guarantees the normal functioning of a cell regulation network with both parts playing irreplaceable roles. Here, the forward production reacts quickly accounting for the sensitivity of the network while the controller works in a larger time span to realize the adaptivity.
Survival and apoptotic pathways initiated by TNFα
The variables in the TNFα model
x _{1}  TNFα  x _{17}  FADD 
x _{2}  TNFR1  x _{18}  <x_{7}>/RIP1/FADD 
x _{3}  TNFα/TNFR1  x _{19}  TRADD/TRAF2/RIP1/FADD 
x _{4}  TRADD  x _{20}  Caspase8 
x _{5}  TNFα/TNFR1/TRADD  x _{21}  TRADD/TRAF2/RIP1/FADD/Caspase8 
x _{6}  TRAF2  x _{22}  Caspase8* 
x _{7}  TNFα/TNFR1/TRADD/TRAF2  x _{23}  Caspase3 
x _{8}  RIP1  x _{24}  Caspase8 * /Caspase3 
x _{9}  μ⟨x_{7}⟩/RIP1  x _{25}  Caspase3* 
x _{10}  IKK  x _{26}  DNA  frag 
x _{11}  ⟨x_{7}⟩/RIP1/IKK  x _{27}  cIAP 
x _{12}  IKK*  x _{28}  Caspase3 * /cIAP 
x _{13}  Iκ B/NFκ B  x _{29}  DNA 
x _{14}  Iκ B/NFκ B/IKK*  x _{30}  Caspase3 * /DNA 
x _{15}  Iκ BP  x _{31}  Iκ B 
x _{16}  NFκ B 
Circadian clock in Drosophila
The variables of the circadian clock model
x _{1}  Per_{ m }  x _{7}  PER · P_{ c }  x _{13}  Pdp_{ m }  x _{19}  CLK_{ c } 
x _{2}  Tim_{ m }  x _{8}  PER · P_{ n }  x _{14}  Clk_{ m }  x _{20}  CLK · CYC_{ c } 
x _{3}  PER_{ c }  x _{9}  TIM_{ n }  x _{15}  VRT_{ c }  x _{21}  CLK · CYC · P_{ c } 
x _{4}  TIM_{ c }  x _{10}  PER · TIM_{ n }  x _{16}  VRI_{ n }  x _{22}  CLK · CYC_{ n } 
x _{5}  PER · TIM_{ c }  x _{11}  SM_{ c }  x _{17}  PDP_{ c }  x _{23}  CLK · CYC · P_{ n } 
x _{6}  PER · TIM_{ f }  x _{12}  VRI_{ m }  x _{18}  PDP_{ n } 
Conclusions
In this paper, we discuss some of the universal aspects of the architecture of biochemical networks that relate to their production and feedback function. We also devise an automatic procedure for identifying the key functional modules of that architecture by applying graph theoretic methods and invoking additional dynamic information. The key ingredients of the architecture are revealed by identifying the forward production unit and the feedback controller. We successfully applied the HVD and the feedback loop searching and selection algorithm and obtained this anatomy in the NFκ B regulatory, the E. coli chemotaxis network, the TNFα pathway and the circadian network. In the Additional File 1 we show that similar structures exist in a number of other cell regulatory networks.
The dissection of large networks into functional modules greatly facilitates their analysis. The functional modules can be studied individually with welldesigned boundary conditions. The properties of the whole network are deducible by piecing together the modules in an ordered way. Henceforth, our strategy of analysis is characterized by a decomposition and recombination procedure. Current technique can be further extended to the analysis of hierarchical structures at different scales with disparate internal dynamics. In the topdown direction, the network may be broken into functional modules at different scales by the above decomposition technique. From bottom up after the property of each module is conveniently explored, a hierarchy of modules of increasing size may be built until the whole network is covered. From biological evolution point of view, it is likely that this nested structure stems from a simple core and is later wrapped with complex regulation mechanisms during evolution. So, our theory reveals the stable, potentially generic feature of a biochemical network, which can be used to explore either the intricacy in a single structure or interdependencies of a series of systems.
The detection of modular structures provides additional insight into how a regulatory network works and thus gives clear indication of key protein species and key reactions in a cascade, which finds wide applications in the drug design and synthetic biology [44]. The identification of the dominating skeleton subnetwork such as the MPU and key feedbacks in a regulatory pathway also simplifies the determination of reaction rates of in vivo biochemical reaction since the distracting unimportant reaction components have been removed from the skeleton structure [45, 46]. In all, the production and feedback dichotomy of biological networks shapes cellular signaling [22] and the current graph decomposition technique provides a convenient handle to uncover this important aspect of their architecture.
Methods
Identification of forward and feedback edges
As mentioned previously, here, we present an algorithm to identify the forward and feedback edges with given polarity, by searching and ordering important topological invariants  cycles. First, a cycle search procedure is discussed which produces all the cycle generators for a strongly connected component. Then a selection procedure is discussed which generates a partial order of the vertices and enables the detection of feedbacks in a straightforward way. Before proceeding directly to the algorithm part, we state a principle which will be used in our selection procedure.
Principle of minimum feedbacks
Very often, in complex systems, multistep processes are carried out in a wellordered sequential way with a small number of feedback controllers modulating the behavior of the system. The cascade structure with minimal number of feedback controls yields balance between robustness and evolvability. It also has the advantage of maximizing operation efficiency and minimizing energy cost. As an analogue, we propose that in order to make optimal use of resources and at the same time maintain necessary stability cells employ a minimum feedback principle: the number of feedback edges should be minimal in a cell regulation network. It seems evolutionarily advantageous to allocate only necessary resources to feedback control. As always happens in biology, there may exist other requirements which weaken this principle. Here, we just stick to this principle which produces reasonable results for all the examples we are looking into so far.
How to find a minimum set of feedback edges is an NPhard problem in graph theory but there exist approximate algorithms which could do the job relatively fast [47]. It is conceivable that the solution might not be unique. However, extra constraints may help remove some nonuniqueness. From a control theory point of view, the signal transduction network consists of two major components, the information forwarding part and the feedback controller. The forwarding part receives external signal at one end, passing and processing it along different paths, and producing an output at the other end. So, the associated information flow defines a direction on the network. The feedback component modulates the flow by sending downstream signals back to upstream nodes. The identification of these two components is essential for understanding the function of different parts of a network. The problem of searching for the minimal set of feedback arcs has to be consistent with the polarity determined by the information flow. Accordingly, we may restate the problem in an equivalent way: find an ordering of the vertices with the given polarity determined by input and output vertices, such that the number of feedback edges is minimized.
Cycle search
(1) Record all the selfloops of $G$ which are encoded by the nonzero diagonal elements of A. After removing the corresponding edges from $G$, we obtain a new graph ${G}_{1}$ and a new adjacency matrix A _{1}.
(2) Search and record a shortest cycle ${l}_{1}=\overline{[{a}_{{i}_{1}},\mathrm{...},{a}_{{i}_{k}}]}$ of A _{1} for some k > 1 by looking for the nonzero diagonal elements of the mth powers of A.
(3) The induced subgraph ${H}_{1}$ with the vertex set {a _{1},..., a_{ m } } and their connections has an adjacency matrix B _{1} which is a submatrix of A _{1}. Each nonzero element (i _{ p }, i _{p+1}) of B _{1} can be made to a cycle by connecting ${a}_{{i}_{p+1}}$ back to ${a}_{{i}_{p}}$ with part of the cycle l _{1}, e.g., by the chain of edges $[{a}_{{i}_{p+1}},{a}_{{i}_{p+2}},\mathrm{...},{a}_{{i}_{p}}]$. Initially, this step is not necessary since besides l _{1} there is no extra edge in ${H}_{1}$. However, after the collapse in step 4, there may appear multiedges between some pair of nodes. For each of those in ${H}_{1}$ but not in l _{1}, we can identify and record a new cycle.
(4) Collapse all the edges and vertices in the subgraph ${H}_{1}$ into one point P _{1}, and we obtain the updated graph ${G}_{2}$ for which a new adjacency matrix A _{2} is written down. If ${G}_{2}$ only contains P _{1}, the iteration is terminated. Otherwise, we go back to step 2 and repeat the procedure with the new graph ${G}_{2}$ and the new adjacency matrix A _{2}. Note that ${G}_{2}$ may not be a simple graph: there could be more than one edge between some pair of vertices. This is the origin of extra edges on a shortest cycle in step 3.
It is easy to show that each cycle of ${G}_{1}$ corresponds to a unique cycle either in ${H}_{1}$ or in ${G}_{2}$. Vice versa, each cycle l in ${G}_{2}$ can be identified with a unique cycle in ${G}_{1}$: if the cycle l runs through P_{1}, then its incidence vertex and exit vertex in ${H}_{1}$ can be connected by a unique path embedded in the cycle l_{1} and thus a unique cycle in ${G}_{1}$ is produced by concatenating this path to the edges contained in l; if the cycle l stands apart from P_{1}, it directly corresponds to one cycle in ${G}_{1}$. So, after the search is done, finally, we can trace backward all the cycles we have found so far in the original graph $G$ except the selfloops. In the algorithm just described, not all cycles but a set of linearly independent cycles are recorded, which by definition constitutes a cycle generator set C gen. The generators derived from the above algorithm are prime in the sense that any proper subset of a generator is not a cycle. Note that the set C gen may not be unique since the selected cycle in step 2 might not be unique. What consequences this nonuniqueness brings about is an interesting problem that deserves further investigation. However, the important point here is that all the feedback edges appear at least once in C gen.
For the NFκ B gene regulatory network, we apply the cyclesearching technique and find that the total number of cycle generators are 33 with 15 1cycles and 8 2cycles. 10 cycle generators have length greater than 2.
Selection procedure
(1) With the long cycles and the polarity determined, we first look for cycles connecting x _{2} and x _{7} and thus extract a set of forward paths that go from x _{2} to x _{7}.
(2) From the remaining long cycles, we search for the ones intersecting an extracted path at two nodes. If more than two intersections are found, we choose the two intersections that are most separated. This choice is to put as many edges as possible to the forward direction and thus to minimize the feedback ones. Using the edges on the cycle as a replacement of the edges in the path that connect the two intersections, an alternative path from x _{2} to x _{7} is constructed.
(3) We repeat the search until no more alternative paths can be generated from the available long cycles.
(4) Now, it is possible to construct a subgraph $F$ expanded by the vertices and the edges contained in these forward paths. A node in $F$ belongs to the production unit and to the feedback controller otherwise. For the middle layer of the NFκ B network, the vertex set in $F$ has been computed as V_{ f } = {x _{2} , x _{5} , x _{6} , x _{7} , x _{10} , x _{11} , x _{13}} which sit in the forward production unit and are displayed inside the rectangle in Figure 2A. The complementary vertex set consists of V_{ b } = {x _{4} , x _{8} , x _{9} , x _{12} , x _{14}} which should be included in the feedback controller.
(5) The HVD is applied to $F$ to partially order its vertices and edges. We rearrange the order of the vertices in $F$ according to the partial order. In the new order, an adjacency matrix only has subdiagonal nonzero entries, which represent forward edges. If we restore all edges in the original graph that connect nodes in $F$, the adjacency matrix may have superdiagonal entries, which are considered as feedback edges. For the NFκ B network, the collection of the feedback and the forward edges are clearly seen in the rectangle in Figure 2A. For a complex feedback controller, if needs arise, we may carry out further decomposition with our cycle search and selection algorithm. For the NFκ B network, it is not necessary since the feedback controllers are simple line graphs.
Declarations
Acknowledgements
This work was in part supported by DARPA DSO under AFOSR contract FA955007C0024. Approved for public release, distribution unlimited. This work was in part supported by AFOSR contract FA95500910141 and DARPA DSO under AFOSR contract FA955007C0024. Approved for public release, distribution unlimited.
Authors’ Affiliations
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