Abstract
Biological networks play a key role in determining biological function and therefore, an understanding of their structure and dynamics is of central interest in systems biology. In Boolean models of such networks, the status of each molecule is either “on” or “off” and along with the molecules interact with each other, their individual status changes from “on” to “off” or viceversa and the system of molecules in the network collectively go through a sequence of changes in state. This sequence of changes is termed a biological process. In this paper, we examine the common perception that events in biomolecular networks occur sequentially, in a cascadelike manner, and ask whether this is likely to be an inherent property. In further investigations of the budding and fission yeast cellcycle, we identify two generic dynamical rules. A Boolean system that complies with these rules will automatically have a certain robustness. By considering the biological requirements in robustness and designability, we show that those Boolean dynamical systems, compared to an arbitrary dynamical system, statistically present the characteristics of cascadeness and sequentiality, as observed in the budding and fission yeast cell cycle. These results suggest that cascadelike behavior might be an intrinsic property of biological processes.
Introduction
Biologists often view cellular processes as a chain of events, especially within small subsystems in the cell. For instance, an external signal turns on gene A, which then turns on gene B, which turns on gene C, and so on. This view of molecular events is inherently cascadelike, suggesting a chain of causality occurring over intervals of time, where each particular change involves only a few of the molecules in the subsystem. In this paper, we will investigate this phenomenon and ask the following question: is there any inherent reason for biological processes to be cascadelike?
To answer this question, we model biological system with Boolean methods. In the last decade, Boolean models have been applied widely in systems biology^{1,2}. In these models, time is discrete and the whole dynamical process is divided into several time steps, then the status of a particular molecule i at any given time step is either “on” (active, or in high concentration) or “off” (inactive, or in low concentration), and in the next time step the status of i is determined by the statuses of the molecules interacting with i in the present time step. This method was first introduced by Kauffman^{3}. After that, Bornholdt and other researchers published significant work on random Boolean threshold networks^{4,5,6,7}. Differing from the random Boolean networks used by Kauffman, the concept of activation, inhibition and selfdegradation introduced in the threshold model give the model more biological significance. Indeed, several biological systems have already been modeled by the such threshold networks and those models appear to explain biological network phenomena reasonably well^{8,9,10}.
As an example, let us consider the table in Fig. 1A. It represents a system of 11 molecules involved in regulating the budding yeast cellcycle^{8}. At the start, only molecules Cln3, Cdh1, and Sic1 are active; at each time step the status of individual molecule is changed under dominant inhibition rule (which is one of the threshold rules, details of which are shown in the “Methods” section) and finally, after 10 steps the system settles into a “steady state” or attractor. We use the term “Boolean process” to denote the sequence of these booleanized states. The sequence can be conveniently represented as a binary matrix, as shown in the table. In the following, we may use the term “process” or “binary matrix” to substitute for “Boolean process” when not ambiguous. Based on the literature of known reactions between 11 molecules, a handcrafted^{8} biological network that determines the causality of events is shown in Fig. 1B. As it turns out, this is not the only network that explains the process—the network shown in Fig. 1C is an example; it is a minimal (fewest possible edges) network that suffices. For a given Boolean process, the number of networks that can explain the process is termed its designability. As a design principle, designability was first observed in the field of the protein folding^{11}. And there is also evidence that biological processes have high designability^{12}.
Next, looking at the diagonal in the table in Fig. 1A, one can see that most of the “1” elements in the table are clustered along the diagonal, except for molecules Cdh1 and Sic1 (which are active in offdiagonal entries). This is a typical example of the striking cascadelike sequence of events. In some ways, a “mostly diagonal” table is an approximate representation of sequentiality: the molecules towards the top left are active at the beginning of the process, and turn on others, which activate others and so on. We use the term cascade to refer to a Boolean process whose activity is concentrated approximately along the diagonal. Thus, a quantitative definition is proposed to quantify the “cascadeness” (how close to a cascade is a Boolean process?). Now our main question can be formulated as: is it true that most biological processes are like the cellcycle examples, possessing high cascadeness? And if so, what is the reason? The main contribution of this paper is some evidence that design principles such as designability^{12} and robustness (defined below) naturally lead to cascadelike processes.
We now examine the two design principles in more detail:
Robustness. Robustness is a widely desired property in biological systems. One way to quantify robustness is to assess the stability of a biological system to perturbations. In this paper, we consider whether a biological process is robust to perturbations, and how robustness requirements shape a process. Observing the budding and fission yeast cellcycle, we find that: beyond dominant inhibition rule, both of them obey two additional dynamical rules which are robustnessrelated:
– Reliable transition requirement. Consider a molecule i that goes from “off” (inactive) in step t to “on” (active) in step because some other molecule j, which was active at t, stimulated this state transition for i. It is also possible that another activated molecule k at t will inhibit molecule j. Now let’s see what things will happen if the above case occurs. In our Boolean model, since all molecules update their statuses simultaneously, in this case, though the status of molecule j will be “off” in step , the status of molecule i is still “on” in step according to Eq. (1). But, in practice, a real biological system is continuous, with noninteger concentrations of biomolecules, and therefore Eq. (1) is only a simplified discrete description of it. Thus, if the inhibition of k to j is very fast, j does not have enough time to activate i. To avoid this type of instability, molecule j should remain active for enough time. In other words, to make a transition reliable, a stable activation/inhibition is needed, otherwise, the transition is “brittle” and the corresponding system is not robust and will likely be eliminated in evolution.
– Convergence requirement. The second type of robustness results from considering the trajectory of returning to the steady state after a small perturbation. It is desirable that the perturbed trajectory of states remain “close” to the original biological process, as confirmed by the previous research on trajectory perturbation for particular biological networks^{8,13}. A very different trajectory, even if it returns to the same attractor, would suggest that the original process had no function other than the attractor; this is certainly not the case for carefully staged events in processes like the cellcycle, where a drastically different sequence of events would not result in the normal cycle. To apply this robustness requirement, we define a new term: middle state “”, which represents all states between S_{t} and S_{t+1} (see precise definition in “Methods”). This definition naturally leads to the rule of “middle state to middle state” (see details in “Methods”) and fortunately, two cellcycle systems indeed comply with this added rule.
Both of these highlevel characterizations of robustness are given precise mathematical definition in the “Methods” section.
Designability. As proposed in^{12}, biological processes are likely to be highly designable. The reason maybe is that the processes with high designability can allow for exploration through mutations and this property is desirable for biological systems. In this paper, for a feasible process (the processes with nonzero designability), we extend the notion of designability in two ways: (1) its minimality m is defined to be the number of interactions in the minimal network—a network that can realize that process with the smallest number of interactions (edges). These minimal networks have been noticed in the literature^{14,15,16,17}. In our opinion, a process with smaller m indicates that the core functionality of the process can be achieved with a few interactions, suggesting that process can be efficiently designed. (2) Next, we can trace the contribution to designability from each node (molecule). By identifying which one contributes the least, the minimal contribution is termed as the minimal individual designability . In fact, in our Boolean model, the designability D is a product of individual molecular designabilities , so the smallest is exactly . The idea is to examine whether biological networks tend to have molecules that contribute little to designability, and might therefore have been discarded by evolution.
Once we are able to mathematically characterize and computationally evaluate robustness and designability, it becomes possible to answer the questions initially raised. Our results show that the cellcycle processes are robust and highly designable indeed. Also, if we require a Boolean process is robust and highly designable, then statistically, it must be a cascadelike process and the corresponding network is sequential, providing theoretically validity to the common perception. In the remainder of the paper, we describe the particular quantitative results of these conclusions, as well as the mathematical and methodological details (an extension of the Boolean model and technique described in^{13}).
Results
Feasible processes are rare
First, let us consider the feasibility of a process. We ask the question: how likely is it that an arbitrary Boolean process has a network solution? We examined this question for both the case with robustness constraints (Eqs (2, 3, 4, 5, 6, 7)) and without (only Eq. (1)). Several billion such processes are generated and for each of them we checked the feasibility. Results indicated that feasibility was very sensitive to the mean activity value a, which is the ratio of “1” in the binary matrix (see Fig. 2). Another observation is that there is a narrow range of a for which feasible processes are likely, similar to the phasetransition result for the Boolean satisfiability problem^{18}. For Boolean processes with and and ending with an attractor, take the case as an example (see section “Methods” for precise definition of a). We generated ten million processes with and the process with a node which has never been activated or inhibited is excluded (this criterion is also used in the following sampling or enumeration). For each process, we tested whether or not it satisfies Eq. (1). The feasibility f for is then calculated as the fraction of processes that have solutions, which has the value 0.0141. We also examined the case in the same manner (The processes with mean activity a < 11/121 are unfeasible). The results are presented in Fig. 2A. The same studies were performed for the case with robustness constraints (Eqs (2, 3, 4, 5, 6, 7)), with the results presented in Fig. 2B. Moreover, we can estimate the overall feasibility f_{OA}, the ratio of the total amount of feasible Boolean processes to the total amount of arbitrary Boolean processes. In the original model, f_{OA} = 2.95 × 10^{−7}; in the new model, f_{OA} = 1.72 × 10^{−14}. Both values are very small and indicate that feasible processes are rare in the process space.
The cellcycle processes are highly designable
An organism undergoing evolution undergoes frequent changes at the evolutionary time scale, with adaptations that involve forming new biological functions (processes). The formation of new functions entails rewiring networks of molecular interactions. However, if a process is fragile so that any small change to the network results in dysfunction (not carrying out the original process), it is not likely to survive. Instead, the trick is to maintain the process (function) while simultaneously making small evolutionary (wiring) changes for adaptation. A process with high designability makes it more likely that a small change will merely result in another network solution that satisfies the process.
Therefore, one wonders whether known biological processes (budding yeast and fission yeast, in our examples) have a large designability as compared with other feasible processes with the same size. Here the results of budding yeast are presented, while those of fission yeast are included in Supplementary information. We randomly sampled one million processes with and (the size of budding yeast cellcycle process) in the space of all feasible processes and computed designability of each process; this is shown in Fig. 3A with the designability plotted against activity, where the size of each dot corresponds to the number of processes for that combination of activity and designability. One sees that the budding yeast cellcycle process (the red triangle) has the largest designability among all the processes with that we generated.
One can also analyze the minimal individual designability . We produce Fig. 3B, which is the same as Fig. 3A except that the yaxis now represents . One sees that the budding yeast cellcycle process has higher than almost all the sampled processes with . Similar results were obtained for the fission yeast (see Supplementary Fig. 2).
The cellcycle processes can be highefficiency designed
Beyond designability, it is of interest to ask whether a process can be realized with an efficient network, that is, a network with few edges. In comparing two possible networks for the same process, one might reason that the network with more edges is harder for evolution to design. To see whether the budding yeast cellcycle process is efficient in this sense, we compared it with one million randomly generated feasible processes of the same size and . For every process, we computed the number of edges in the minimal network needed for that process (the minimality m). As Fig. 4 shows, the budding yeast cellcycle process (the red triangle) has much lower minimality (with among all sampled processes. A similar result was obtained for the fission yeast model (see Supplementary Fig. 3), indicating that these two biological processes can be designed with highefficiency.
Robustness constraints make better processes
Recall that we imposed two robustness criteria and thus, one might ask whether there is a cost to robustness, for example, whether robust processes (here “robust process” means that a process is feasible under Eqs (2, 3, 4, 5, 6, 7), the same below) are less designable or need more edges (have higher minimality). For Fig. 5A–C we computed the designabilty D and minimality m for randomly generated processes that are robust (blue curve), and compared these metrics for random processes that are just merely feasible (red curve). Interestingly, robust processes tend to have higher designability and lower minimality.
Next, we ask the question: are the two cellcycle processes robust? That is, do they satisfy every robustness equation among Eqs (2, 3, 4, 5, 6, 7)? To address this question, we examined each molecule in each step of these processes, and computed the fraction (percentage) of matrix entries that satisfy the conditions. The percentage of molecular states that satisfy these criteria is 99%: only one molecule in one step of the budding yeast does not satisfy the second robustness criterion (the first molecule is a trigger molecule and we do not check it). A similar result is obtained in the fission yeast and also only one molecule in one step does not satisfy the second robustness criterion. That means that both two cellcycle processes are robust and might indicate that these two robustness criteria might occur widely in biological systems.
Biological processes are cascadelike and sequentially activated
We now turn to the question at the beginning about whether biological processes are likely to be cascadelike. First, for any cascade, we compute the cascadeness distance c that measures how different a process is from a cascade (see section “Methods” for a precise definition). This cascadeness distance is shown along with designability and minimality in Fig. 5A–C. Thus, robust processes have a better (lower) cascadeness distance. Next, for further investigation, we enumerated all processes of size 8 × 9, and identified all the 112,877,531 robust processes among them. For each robust process, we calculated minimality m, designability D, and minimal individual designability . Inspiring by the earlier results on designability and minimality, we identified 89,733 processes in the top 1% with regard to m, D and as the candidates for the biological systems. After rearranging the columns using a smartexhaustive algorithm (see details in Supplementary information), we found that all of them are cascadelike (see the distribution of the cascadeness distance c in Supplementary Fig. 4). Furthermore, we clustered these 89,733 processes using the kmeans clustering algorithm with . For each cluster, an “average” process is calculated and shown in Fig. 6A–C, respectively, which clearly demonstrate their cascadeness.
One can also assess sequentiality by examining networks. Some networks have a layer like structure in which some molecules activate others, which activate yet others, and so on, in a feedforward fashion. Figure 1D shows how the budding cellcycle network can be redrawn to emphasize this treelike structure. We examined all minimal networks of each of the 89,733 processes. Among them, we find that there are 47,078 processes all of whose minimal networks have this layerlike structure, and an additional 20,399 processes have more than half of their minimal networks with this type of structure. Because this type of network structure inherently implies sequentiality, this result explains why biological processes have cascadelike behavior.
Are biological processes cascadelike? For the naturally occurring biological processes, we examined the budding yeast cellcycle process^{8} and the fission yeast cellcycle process^{9}. After canonical rearrangement of columns, they are both found to be cascadelike (see Fig. 1A and Supplementary Fig. 1A).
Cascadelike processes are not necessarily biological
We have shown that biologically robust processes must be, in a statistical sense, cascadelike. The reverse question is also relevant: is an arbitrary cascadelike process always biological, that is, does a cascadelike process automatically satisfy all requirements from a biological system? We examine processes that are very cascadelike, including the strict cascade, which is when we get a simple diagonal of 1’s in the matrix (Fig. 7A). We use the set of cascadelike processes with and as an example to characterize cascadelike processes. For every process, the network states at and are the same, implying a steady state has been reached in the end.
Actually the strict cascade turns out to be infeasible under our robustness criteria. To ensure feasibility and avoid the process ending with a allzero state, the minimal activity A is equal to and totally there are N difference processes (Fig. 7B–L). Certainly, all of them are cascadelike. We call these processes the pure cascades. We determined their minimality m, designability D, and minimal individual designability with our two robustness constraints and the results are shown in Fig. 7B–L. By these measures, the pure cascades appear to be similar to the cellcycle processes. Unfortunately, a closer examination reveals that the dynamic properties of the corresponding networks are not suitable, as we now explain. For example, consider the last pure cascade (Fig. 7L) as an example. We studied the dynamics of each of the 1,024 minimal networks for this process (since it is impossible to simulate every one of the solutions for this process). The basin size B (range from 1 to of each network was calculated: among 1,024 networks, 384 networks with , 192 networks with , 192 networks with and only 256 networks with B < 200. At first glance, this appears to be a positive result, suggesting that the pure cascades have sufficiently large basin sizes for their main attractor state. However, these networks exhibit poor dynamics when one examines the flow of states through the process, as described in^{8}. We use the term flux for a state to indicate how many paths from other states pass through the given state. A state with small flux indicates that it is only a small branch of the tree. If the flux of all nodes is small (which is corresponding to a low average flux), it means that lots of states have independent paths to the attractor, an indication of a “brittle” process. Instead, what is desired is high flux for the states in the process so that most perturbations will return back to the process^{8}. For the pure cascade shown in Fig. 7L, consider the flux of the 9th time step, among all 1,024 networks. The maximum flux is only 107, indicating that most of states go through different trajectories. We also tested the other pure cascades and obtained similar results, suggesting that at least these pure cascades are not suitable as biological processes. When combined with the observations that biological processes are cascadelike, this might mean that biological processes are concentrated at the “edge” of cascadeness, where the few additional nonpure cascade “1” elements in the process confer desired dynamical properties.
Discussion
In this paper, we investigated the common perception that events in biomolecular networks occur in a cascadelike manner. For example, both budding and fission yeast cellcycle show the cascadeness in processview (see Fig. 1A and Supplementary Fig. 1A) and the sequentiality in networkview (see Fig. 1D and Supplementary Fig. 1D). To analyze the phenomenon, we used Boolean methods to model biological systems. When Boolean models are augmented to include designability and robustness to meet the biological requirements, our results show that the qualified Boolean processes are statistically very likely to be cascadelike, providing a theoretical explanation to this common phenomenon.
Now, let us examine our two additional dynamical rules again: the first rule is to make every transition of the node’s status reliable and the second rule is to make the dynamics trajectory tree convergent. The two rules might seem arbitrary, but in this section we provide some arguments for their generality. First, note that though the two rules have different meanings, they have similar mathematical form (see Eqs (2, 3, 4) and Eqs (5, 6, 7)) and both of them improve the beginning model which is described in Eq. (1) via adding the additional control of the middle state “” (see definition in the section “Methods”). If a small perturbation occurs on S_{t} and leading to a middle state “”, these rules can guarantee that this perturbation will not diverge and can converge to the same steady state. In particular:
The first rule requires the following: if , then for each , in the next step , the status of i must be equal to . This requirement also implies that the quick flips of the node, such as “” or “”, are forbidden. The reason is that: S_{t+1} can be also seen as a special middle state , so if , as we describe above, S_{t+1} as a member of , in the next step t + 1, the status of i must be equal to .
The second rule requires the following: if , that is , then for each , in the next step , it must go to , it also means that the status of i must be equal to .
These interpretations show that both rules are not limited to the Boolean model under dominant inhibition rule and can be easily extend to an arbitrary Boolean model. We tested them with the majority rule model of the budding yeast cellcycle process^{8} and obtained a similar result: only one molecule in one step does not satisfy the second robustness criterion. Majority rule is another threshold rule and the difference between the models under dominant inhibition rule and majority rule is: in the former, the inhibition is dominant; but in the latter, the importance of the inhibition and the activation is equivalent. These observations indicate that both rules might be not specfic model dependent and they are applicable universally in biological systems.
Furthermore, let us turn our attention back to the middle state . If we treat as the product of an asynchronous update, obviously, our two rules enhance the ability of robustness to this timing variations. As it turns out, this general notion has been considered by others in different contexts. For example, Peixoto and Drossel proposed the concept of “reliable dynamics”: a fully reliable process is the process that at each time step only one node’s status is changed, so it is independent of the order in which the nodes are updated^{19}. It is easy to see that fully reliable process is a special case in consistent with our second robustness criterion. Mangla et al. proposed a similar concept, entitled “timing robustness”, to describe the ability of a system to maintain its function(process) in the presence of timing perturbations. Next, they found that both budding and fission yeast cellcycles were highly timingrobust and the reason was due to evolutionary pressue^{20}. Our work give a comprehensive understanding of why two cellcycles systems are highly timingrobust and may have implications for the design of the system with timing robustness.
Methods
A Boolean model for biological processes
Boolean networks were first proposed by^{3}, and then used as a theoretical model for studying inherent properties of gene regulation networks^{19,20,21,22,23,24,25}. Variants of the original random Boolean network have been widely adopted by recent works^{9,26,27,28} to capture biological phenomena, especially for those largescaled systems^{29,30,31} not handled by traditional methods such as ordinary differential equations.
The Boolean model in this paper is based on our previous work^{13,32}. In the model, molecules update their statuses simultaneously in a deterministic manner and can be described succinctly as the following Boolean equation (Attn: it’s not an algebraic equation):
Here or 1 represents the status of node j (molecule j) at time t; it is the entry in row t and column j in the Boolean process (in the matrix or tabular form); is a Boolean variable that represents a putative inhibitory (red) edge from node j to node i; similarly, g_{ji} represents a putative stimulatory (green) edge from node j to node i; addition represents the Boolean operator OR; multiplication represents AND; the bar on a variable represents NOT. The model reflects the basic rules of biomolecular interaction: an active molecule can stimulate (via a green edge) another molecule into being active, but only if no other active molecule has a red edge to the target molecule. Further descriptions are included in Supplementary information and more details are given in^{13,32}.
To study the impact of robustness, we need to add to the model the two robustness criteria described earlier:
• Reliable transition requirement. First note that the activation of node i at time t (the transition) depends on an activation edge from some node j and the condition that node j is active at time t . This explains the appearance of in Eq. (1). On the other hand, the transition can be blocked by any inhibition, which explains the term in Eq. (1). However, although for some j can make the transition occur, the transition can be halted if . Therefore, it is desirable to have a persistently active node to enforce the activation. This stable activation can be represented as . Similarly, for some j will try to revert this transition. That means when designing a robust transition: we not only allow to perform the inhibition, but also . That is, captures the above requirement of inhibition for the transition. By combining these requirements, one obtains Eq. (2). Similar considerations for the transition lead to Eq. (3). For the cases and , the corresponding equations (Eq. (4)) remain the same as the original ones. In summary, one has the following modified model:
for and
for and
for .
• Convergence requirement. A biological process is a trajectory of state transitions. For robustness, the model ought to allow some perturbations of individual states while maintaining the trajectory. As we discuss before in the “Introduction” section, let us consider the middle state between the state S_{t} and the state S_{t+1}. Here let’s give a precise definition of . For molecule i, all possible statuses of i in is termed as and we require that it should satisfy the following properties: if for any , then ; if , then can be arbitrary. For example, if and , then = 1 * 0 * *, where * represents arbitrary 0 or 1. Thus, there are middle states , which include S_{t} and S_{t+1}. For the convergence purpose, we expect goes to in the next step and this expectation really occurs in the budding yeast cellcycle (see Fig. 8). If , then one must have . That is, no matter what is, it must transit to . To guarantee this transition, one must have Eq. (5). If , then one must have . That is, no matter what is, it must transit to . To guarantee this transition, one must have Eq. (6). For the cases , the corresponding equations (Eq. (7)) remain the same as the original ones. In summary, one has the following modified model:
for
for
for .
We point out that sometimes, as is the case with Cln3 for the budding yeast cellcycle, a molecule’s purpose is to trigger a process, in which case it appears just once in the first row and never again; in this case, we do not include it in the above analysis because it is not part of the core network.
Taken together, the above additions to the model impose some robustness: any solution of these equations results in a network that can tolerate perturbation errors as defined above. And, more importantly, for a given process, since all status variables are known, it’s easy to see that Eqs (2, 3, 4, 5, 6, 7) have the same mathematical function form as the Eq. (1), that means that we don’t need to pay any additional computational cost to acquire these two robustness criteria. We can use the same analytical algorithm of the original model directly^{13}.
Phenotypical features of Boolean processes
In this paper, we are interested in examining the space of Boolean processes and asking questions about what types of processes are biologicallike, and what the characteristics of biological processes are (regardless of what network actually generates the process). To this extent, we define the following characteristics or phenotypes of a Boolean process:
Feasibility f. A feasible process refers to a Boolean process with nonzero designability. In other words, for a feasible process, we can find at least one network that can explain it under Eq. (1) or Eqs (2–7). As we mention before, we term the process which is feasible under Eqs (2–7) as robust process. Moreover, we can also define the feasibility for a given set of many Boolean processes, which is the percentage of feasible processes.
Activity A/Mean Activity a. These are a coarse indication of the level of activity in a process. The term activity refers to the number of molecule activations (the number of “1” in the matrix) in the Boolean process and mean activity is the percentage with respect to the size of the matrix. Notably, in this paper, we only focus on the process which is ended with an attractor, so the last repeated row is excluded from the calculation. For example, in the budding yeast cellcycle process (see Fig. 1A), A = 41 and .
Designability D/Minimal Individual Designability d_{min}. The term designability is, as mentioned earlier, the number of networks that can realize a given Boolean process. Our prior work^{13} explains how this number can be computed efficiently. Note that D can be written as , the product of the individual node designabilities^{12,13}, it is possible that D is large but is very small for some i. Then, small would be a bottleneck that limits mutation stability. To reflect this case, we also calculated the minimal individual designability , that is the smallest , for i = 1, 2, …, N.
Minimality m. A feasible process can be often realized by many networks. Some networks have more edges (corresponding to molecular interactions) and some have fewer. Minimal networks refer to the networks with the smallest number of edges. The number of edges in a minimal network is called the minimality of the process.
Cascadeness distance c is used to quantify “cascadeness”, the degree to which a Boolean process is cascadelike. It is defined to be
where is the state of node j at time step t. Intuitively, it measures the “distance” to the diagonal and thus, the lower the value of c the more the process resembles a cascade.
Note that a given Boolean process can be made to look noncascade like by shuffling the columns (the indices of the nodes). Thus, to compute the cascadeness distance, we first need a canonical transformation (permutation of columns) to enable meaningful comparisons. We do this through a smartexhaustive algorithm that rearranges the columns to make a process as cascadelike as possible. The idea behind this algorithm is to swap columns heuristically to decrease the cascadeness distance, and to repeatedly do so until all possible combinations are enumerated. The detailed description of the algorithm is included in Supplementary information.
Additional Information
How to cite this article: Chen, H. et al. Boolean Models of Biological Processes Explain CascadeLike Behavior. Sci. Rep. 6, 20067; doi: 10.1038/srep20067 (2016).
Change history
30 June 2017
A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has been fixed in the paper.
References
 1.
Morris, M. K., SaezRodriguez, J., Sorger, P. K. & Lauffenburger, D. A. Logicbased models for the analysis of cell signaling networks. Biochemistry 49, 3216–3224 (2010).
 2.
Wang, R.S., Saadatpour, A. & Albert, R. Boolean modeling in systems biology: an overview of methodology and applications. Phys Biol 9, 055001 (2012).
 3.
Kauffman, S. A. Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of theoretical biology 22, 437–467 (1969).
 4.
Rohlf, T. & Bornholdt, S. Criticality in random threshold networks: annealed approximation and beyond. Physica A: Statistical Mechanics and its Applications 310, 245–259 (2002).
 5.
Szejka, A., Mihaljev, T. & Drossel, B. The phase diagram of random threshold networks. New Journal of Physics 10, 063009 (2008).
 6.
Zanudo, J. G., Aldana, M. & MartnezMekler, G. Boolean threshold networks: Virtues and limitations for biological modeling. Information Processing and Biological Systems Springer: Berlin Heidelberg, 11, 113–151 (2011).
 7.
Rybarsch, M. & Bornholdt, S. Binary threshold networks as a natural null model for biological networks. Physical Review E 86, 026114 (2012).
 8.
Li, F., Long, T., Lu, Y., Ouyang, Q. & Tang, C. The yeast cellcycle network is robustly designed. Proc Natl Acad Sci USA 101, 4781–4786 (2004).
 9.
Davidich, M. I. & Bornholdt, S. Boolean network model predicts cell cycle sequence of fission yeast. PLoS ONE 3, e1672 (2008).
 10.
Davidich, M. I. & Bornholdt, S. Boolean network model predicts knockout mutant phenotypes of fission yeast. PLoS ONE 8, e71786 (2013).
 11.
Li, H., Helling, R., Tang, C. & Wingreen, N. Emergence of preferred structures in a simple model of protein folding. Science 273, 666–669 (1996).
 12.
Nochomovitz, Y. D. & Li, H. Highly designable phenotypes and mutational buffers emerge from a systematic mapping between network topology and dynamic output. Proc Natl Acad Sci USA 103, 4180–4185 (2006).
 13.
Wang, G. et al. Processbased network decomposition reveals backbone motif structure. Proc Natl Acad Sci USA 107, 10478–10483 (2010).
 14.
Okazaki, N., Asano, R., Kinoshita, T. & Chuman, H. Simple computational models of type i/type ii cells in fas signalinginduced apoptosis. J Theor Biol 250, 621–633 (2008).
 15.
Raychaudhuri, S. A minimal model of signaling network elucidates celltocell stochastic variability in apoptosis. PLoS ONE 5, e11930 (2010).
 16.
Perkins, T. J., Wilds, R. & Glass, L. Robust dynamics in minimal hybrid models of genetic networks. Phil Trans R Soc A 368, 4961–4975 (2010).
 17.
Finlayson, M. R., HelferHungerbühler, A. K. & Philippsen, P. Regulation of exit from mitosis in multinucleate ashbya gossypii cells relies on a minimal network of genes. Mol Biol Cell 22, 3081–3093 (2011).
 18.
Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B. & Troyansky, L. Determining computational complexity from characteristic ‘phase transitions’. Nature 400, 133–137 (1999).
 19.
Peixoto, T. P. & Drossel, B. Boolean networks with reliable dynamics. Phys. Rev. E 80, 056102 (2009).
 20.
Mangla, K., Dill, D. L. & Horowitz, M. A. Timing robustness in the budding and fission yeast cell cycles. PLoS ONE 5, e8906 (2010).
 21.
Kauffman, S., Peterson, C., Samuelsson, B. & Troein, C. Genetic networks with canalyzing boolean rules are always stable. Proc Natl Acad Sci USA 101, 17102–17107 (2004).
 22.
Shmulevich, I. & Kauffman, S. A. Activities and sensitivities in boolean network models. Phys. Rev. Lett. 93, 048701 (2004).
 23.
Ciliberti, S., Martin, O. C. & Wagner, A. Innovation and robustness in complex regulatory gene networks. Proc Natl Acad Sci USA 104, 13591–13596 (2007).
 24.
Drossel, B. Number of attractors in random boolean networks. Phys. Rev. E 72, 016110 (2005).
 25.
Payne, J. L. & Wagner, A. Constraint and contingency in multifunctional gene regulatory circuits. PLoS Comput Biol 9, e1003071 (2013).
 26.
AlvarezBuylla, E. R. et al. Floral morphogenesis: Stochastic explorations of a gene network epigenetic landscape. PLoS ONE 3, e3626 (2008).
 27.
Krumsiek, J., Marr, C., Schroeder, T. & Theis, F. J. Hierarchical differentiation of myeloid progenitors is encoded in the transcription factor network. PLoS ONE 6, e22649 (2011).
 28.
Choi, M., Shi, J., Jung, S. H., Chen, X. & Cho, K.H. Attractor landscape analysis reveals feedback loops in the p53 network that control the cellular response to DNA damage. Sci. Signal. 5, ra83 (2012).
 29.
SaezRodriguez, J. et al. A logical model provides insights into t cell receptor signaling. PLoS Comput Biol 3, e163 (2007).
 30.
SaezRodriguez, J. et al. Discrete logic modelling as a means to link protein signalling networks with functional analysis of mammalian signal transduction. Mol Syst Biol 5 (2009).
 31.
Kazemzadeh, L., Cvijovic, M. & Petranovic, D. Boolean model of yeast apoptosis as a tool to study yeast and human apoptotic regulations. Front. Physio. 3, 446 (2012).
 32.
Wang, G. et al. Processdriven inference of biological network structure: Feasibility, minimality, and multiplicity. PLoS ONE 7, e40330 (2012).
Acknowledgements
The work is supported by National Science Foundation (NSF) Grant CDI0941228 (CZ, RS, GW).
Author information
Affiliations
Department of Physics, The George Washington University, Washington, DC 20052, USA
 Hao Chen
 , Guanyu Wang
 , Chenghang Du
 & Chen Zeng
Department of Chemistry, Wuhan Polytechnic University, Wuhan 430023, China
 Hao Chen
Department of Biology, South University of Science and Technology of China, Shenzhen 518055, China
 Guanyu Wang
Department of Computer Science, The George Washington University, Washington, DC 20052, USA
 Rahul Simha
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
 Chen Zeng
Authors
Search for Hao Chen in:
Search for Guanyu Wang in:
Search for Rahul Simha in:
Search for Chenghang Du in:
Search for Chen Zeng in:
Contributions
H.C. and C.Z. conceived the project, H.C. conducted the experiments and analyzed the results. All authors discussed and contributed to the results, and cowrote the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Chen Zeng.
Supplementary information
PDF files
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Further reading

Handling variability and incompleteness of biological data by flexible nets: a case study for Wilson disease
npj Systems Biology and Applications (2018)

The combination of the functionalities of feedback circuits is determinant for the attractors’ number and size in pathwaylike Boolean networks
Scientific Reports (2017)

Observability of Boolean multiplex control networks
Scientific Reports (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.