Conceived and designed the experiments: NR UB. Performed the experiments: NR UB. Analyzed the data: NR UB. Contributed reagents/materials/analysis tools: NR UB. Wrote the paper: NR UB.
The authors have declared that no competing interests exist.
Just as complex electronic circuits are built from simple Boolean gates, diverse biological functions, including signal transduction, differentiation, and stress response, frequently use biochemical switches as a functional module. A relatively small number of such switches have been described in the literature, and these exhibit considerable diversity in chemical topology. We asked if biochemical switches are indeed rare and if there are common chemical motifs and family relationships among such switches. We performed a systematic exploration of chemical reaction space by generating all possible stoichiometrically valid chemical configurations up to 3 molecules and 6 reactions and up to 4 molecules and 3 reactions. We used Monte Carlo sampling of parameter space for each such configuration to generate specific models and checked each model for switching properties. We found nearly 4,500 reaction topologies, or about 10% of our tested configurations, that demonstrate switching behavior. Commonly accepted topological features such as feedback were poor predictors of bistability, and we identified new reaction motifs that were likely to be found in switches. Furthermore, the discovered switches were related in that most of the larger configurations were derived from smaller ones by addition of one or more reactions. To explore even larger configurations, we developed two tools: the “bistabilizer,” which converts almost-bistable systems into bistable ones, and frequent motif mining, which helps rank untested configurations. Both of these tools increased the coverage of our library of bistable systems. Thus, our systematic exploration of chemical reaction space has produced a valuable resource for investigating the key signaling motif of bistability.
How does a cell know what type of cell it is supposed to become? How do external chemical signals change the underlying “state” of the cell? How are response pathways triggered on the application of a stress? Such questions of differentiation, signal transduction, and stress response, while seemingly diverse, all pertain to the storage of state information, or “memory,” by biochemical switches. Just as a computer memory unit can store a bit of 0 or 1 through electrical signals, a biochemical switch can be in one of two states, where chemical signals are on or off. This lets the cell record the presence/absence of an environmental stimulus, the level of a signaling molecule, or the result of a cell fate decision. There are a small number of published ways by which a group of chemical reactions come together to realize a switch. We undertook an exhaustive computational exploration to see if chemical switches are indeed rare and found, surprisingly, that they are actually abundant, highly diverse, but related to one another. Our catalog of switches opens up new bioinformatics approaches to understanding cellular decision making and cellular memory.
Most chemical reaction systems have a single steady state, but a few interesting cases are known to oscillate
A few biochemical switches have been extensively analyzed, including complex enzyme mechanisms
A second observation about the known bistable switches is that they are quite different in their chemical topologies. While feedback loops are a recurring motif
While signaling models tend to result in rather complex reaction systems, a distinct approach to the study of chemical bistability is driven from theoretical analyses of enzyme kinetics and flux reaction systems
Necessary conditions for bistability, such as positive loops in the system Jacobian, have been well characterized
Here we systematically explore chemical reaction space to show that bistable chemical switches are remarkably common. We show that all small bistable systems are related, and that larger ones frequently share motifs that may be predictive of bistability.
In our first phase of analysis, we began with a basis set of 12 reactions (
(A) Basis set of reactions with signatures and examples. Reactions C, D, J, K, and L are enzymatic and the enzyme name is at the bend of the arrow. (B) Flowchart for finding bistability. (C) Example of steps 1–4 in the flowchart. (D) Number of possible configurations rises steeply with increasing number of molecules and initially also with the number of reactions. Shaded region indicates configurations fully sampled in this study, the remainder were subsampled. (E) Bistable configurations initially become more common with increasing numbers of reactions, and for 3 molecules the percentage declines for more than 6 reactions. (F) Bistability must persist over a wide parameter range to be detected. Fraction of parameter range = #parameters√(Propensity). Model class is expressed as m×n where m is the number of molecules and n is the number of reactions. (G) Frequency of occurrence of bistability as a function of propensity. Some configurations exhibit bistability over 30% of any parameter set in our selected range.
We observed a large number of bistable systems even with our very sparse sampling of reaction parameter space (
Admittedly, due to our sparse sampling of parameter space, there could be undetected bistables in the space of systems sampled here. While a single configuration is sufficient to prove that a network has the capability for exhibiting bistability, our analysis methods do not support an impossibility proof for bistability. The range in which a system exhibits bistability can depend intricately on how the phase space is structured in terms of the system parameters such as molecule concentrations and rate constants. Bifurcation analysis can shed insight into parameter ranges feasible for realizing bistability. Nevertheless, even with the possibility of false negatives, it is significant that nearly 10% of explored systems are bistable and this percentage can only improve with greater analysis and exploration.
The simplest bistable system (3×2M101) involved 3 molecules and 2 reactions (
(A) The simplest configuration, 3×2M101, i.e., having 3 molecules, 2 reactions, and being the 101st configuration in this class. (B) A simpler model that is bistable with Michaelis–Menten/Briggs–Haldane kinetics, but not with a mass action explicit representation of the enzyme-substrate complex. (C) Time-course of response of 3×2M101 to perturbations. (D) Stability diagram of 3×2M101, as a function of c0 (initial concentration of c). The bistable region is shaded. The specific model in panel A has c0 indicated by the dashed line. (E) All the 3×3 bistable configurations. (F) Two configurations with bistability propensity >70%. Model 3×3M445 in (E) is also over 70%. Note the model similarities and symmetry. (G) Three non-autocatalytic configurations with propensity 26%, 23%, and 16%, respectively. (H) Schematic of stability curve for an autocatalytic reaction (inset, thin arrow). There is a stable point at 0, and a saddle at 1. Addition of a rapidly saturating fast back-reaction (inset, thick arrow, and graph, crosses) converts this to a bistable model with the same configuration as (A). Now the stability curve (thick line) has a stable at 0, a saddle at ∼0.05, and another stable at ∼0.9.
Positive feedback loops, such as autocatalysis and catalytic loops, have been implicated as a common motif leading to bistability in signaling
(A) Number of nonautocatalytic configurations as a function of number of reactions. (B) Percentage of bistable configurations for 3, 4, and 5 molecules as a function of number of reactions. Same symbols as in (A). (C) Comparison of bistability percentage for entire dataset (
In addition to autocatalysis, we found several cases where bistability arose from more subtle chemical interactions (e.g.,
Are all discovered bistables distinct? Because isomorphisms were removed at the time of generating possible reaction signatures, we ensured that each discovered bistable mapped to a unique signature composed of the 12 basic reaction types. A remaining concern was that there might be equivalences in terms of the underlying dynamical system when the chemical systems were converted to mathematical models. We investigated this possibility by reducing all the composite reactions to approximate equivalences in the form of either a single reactant-single product reaction (type A) or a double reactant-single product reaction (type E) (see
Does bistability “run” in families of related reaction topologies? To test this hypothesis, we constructed a directed acyclic graph (DAG) of configurations where each bistable configuration was a node, and each addition/removal of a reaction between nodes was an edge. We found that almost all bistable configurations from the first phase (3,415/3,562 = 95.9%) formed a single, highly interconnected set, i.e., a giant component. Most of the 147 “orphans” occurred at the boundaries of our sampling (98 at 3×6 and 47 at 4×3). These may simply represent novel ‘roots’ that connect further up in the reaction hierarchy. In
(A,B) Representations of relationships between bistable configurations. Node color represents bistability propensity. Color scales for (A) and (B) are the same. (A) “Banyan tree” diagram showing multiple “root” bistable configurations that cannot be generated by addition of a single reaction to a smaller bistable configuration but are connected through larger configurations. Model classes are labeled on the left. Nodes are staggered vertically within bands for visualization. “Root” edges are in sky blue and deeper edges are in green. On the left are orphan models. (B) Minimum spanning tree rooted at 3×2M101. Inner nodes with smaller reaction sizes are drawn as larger circles. A few 3×3 bistables and the primarily low propensity systems they derive are not shown to minimize crowding. The “pie” denotes restriction of exploration of 4 molecule systems to only 3 reactions in this study. (C) The number of novel bistables drops sharply in larger reaction systems.
These graphs suggested that most bistable systems were derived from smaller ones. As reactions were added (
We tested two implications of the “bistables are related” observation. First, we asked if we could take a large published bistable system and remove one reaction at a time without losing bistability. If we could continue this process till we ended up at a bistable configuration present in our dataset, then we had a continuous trajectory from our known DAG of bistables to the published model. Second, we asked if the large bistable system was a superset of a known bistable configuration, without requiring that there were intermediate bistables between the two. We performed this analysis on several known bistable reaction systems from published work (
(A,B) Published bistables from
We therefore hypothesized that the DAG of bistables may be nearly complete for small systems, but the increasing degrees of freedom afforded by greater numbers of molecules and reactions helped realize bistability in new, unseen, ways. We developed two analysis tools that work in complementary ways to explore such larger configurations.
A suggestive observation from our first phase was that a large fraction of configurations (∼60%) contained saddle points and line solutions (
Our second tool used motif matching. We analyzed the configurations of smaller bistables plus the sparsely sampled larger bistables to find frequently occurring groups of reactions, and then searched for these motifs in unexplored configurations. We analyzed bistables in each configuration class (3 molecules, 4 molecules, 5 molecules) separately for frequent motifs. A motif must occur with sufficient frequency in the given class to be detected (see
The motifs were mostly independent and only one motif occurred in all three reaction classes. Coincidentally, this common, two-reaction, motif (composed of reactions DabX and Jbca) was identical to a bistable found both in our analysis and in previous work (
(A–C) Top 5 motifs mined for 3-molecule, 4-molecule, and 5-molecule bistable systems. (D) Motifs improve search for bistables by about 7-fold, for each of the reaction classes tested. (E) Venn diagram of motifs among different classes of reaction configurations. The one motif shared by all classes of configurations is the same as the smallest bistable, model M101.
Just as a motif occurred in multiple bistable systems, a given bistable system could exhibit many distinct motifs. We used this property to advantage to help rank untested configurations for their potential to exhibit bistability. In each configuration class (3-molecule, 4-molecule, or 5-molecule systems), we searched for motifs specific to that class, and ranked the (untested) configurations in terms of the number of motifs they exhibited. We evaluated ≥100 of the top configurations for each class, exploring 120 parameter sets for each. We found bistability in 96% of the 3 molecule systems (214/222), 49% of the 4 molecule systems (49/100), and 13% of the 5 molecule systems (82/641). These numbers significantly improved upon the random sampling results of the second phase of analysis (
Finally, we compared the motifs across the three classes (3, 4, and 5 molecule systems) to investigate whether there were any overlaps in mechanisms by which 3-molecule, 4-molecule, and 5-molecule systems exhibit bistability. On face value, there was little overlap between the motif sets taken pairwise (
3xn | 4xn | 5xn | |
3xn | – | 15% | 29% |
4xn | 21.5% | – | 1.6% |
5xn | 45% | 3.5% | – |
Rows: Motif set. Columns: Class of bistable systems.
Our study draws the first stability map of chemical reaction space. We find that bistables are common, especially in smaller reaction systems. They are also very robust, i.e., we find many configurations that are bistable over a very wide parameter range. Smaller bistables are all related to each other in a tree-like manner. While the overall configurations that support bistability are very diverse, there are frequently recurring motifs of reaction groupings in such configurations. These motifs serve to identify promising candidates in higher order systems.
Signaling motifs have been regarded as a good way to abstract out the chemical complexity of signaling
Bistable switches are important in biology in maintaining cellular history and decisions. Our study shows that there is a large repertoire of such switches for natural selection to draw upon, including many very simple switches. Furthermore, several of these switches are highly robust with respect to parameter variations. This has two implications for evolution. First, it is easy for evolving biochemical networks to stumble upon parameters that will give a switch. Second, such switches themselves will work effectively over a wide range of parameter conditions. The relatedness of the switches through addition or removal of individual reactions is also a good substrate for evolutionary modification. For example, a mutation that adds another enzyme regulator to a bistable switch is, by this argument, quite likely to retain the original bistability, along with the new regulatory properties. Overall, our survey of chemical topologies hints at an interconnected and rather well-populated terrain of bistability in a biologically biased region of chemical space.
We selected a set of 12 primary chemical reaction steps (
Our choice of primary chemical steps was biologically-inspired. In other words, we found a different proportion of bistables than we would see if we used, say, only the most elementary reactions such as type A and type E (
We constructed reaction architectures involving
We signed each reaction with a terse unique 4-character string that completely specified all reactants and products, so that the first character of a reaction signature denotes one of the 12 reaction types (A–L), and the remaining two or three characters denote the molecular species participating in various roles in the reaction. The signature for a reaction architecture was obtained by concatenating the signatures for the constitutent reactions. We checked for isomorphic signatures (see
Our set of configurations did not deal with two cases that have previously been analyzed for bistability: continuous flux and buffered systems
In this manner our reaction systems also accommodated steady state cases where continuous metabolic input was necessary to sustain stability. We stipulated that these ‘hidden’ molecules were stoichiometrically balanced within individual reactions, such as the enzymatic step above. We were able to approximate many cases of buffering simply by having a high concentration of the ‘buffered’ species.
In order to assess bistability we needed to work with specific models, with all parameters specified. We generated at least 100 models for each configuration we tested. Each model was generated from one of the configurations using Monte Carlo sampling to assign rate constants and concentrations. We chose concentrations using logarithmic sampling in the range 10 nM to 10 µM. This spans the concentration range of most biochemical reagents. We chose rate constants using logarithmic sampling in the range 0.01 µM−
Due to computational limitations we sampled only the smaller reaction sets completely for bistability. We completely sampled all configurations with 2 molecules, 3 molecules up to 6 reactions, and 4 molecules up to 3 reactions (
We found steady states for each model using two distinct methods: homotopy continuation
Supplementary Information about Methods and Algorithms To Accompany the Main Text of the Paper.
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Minimum Spanning Tree (A) and Banyan Tree Graph (B). (A) Minimum spanning tree derived from high propensity configurations (rooted at 3×3M445). Inner nodes with smaller reaction sizes are drawn as larger circles. Four levels of the giant component are shown that shrinks but remains connected as the propensity threshold is raised. Dark blue is the range 0<propensity<0.1, white is propensity ≥0.3. (B) “Banyan tree” graph of nonautocatalytic models, with a few orphan systems. Color scale indicates propensity of configurations.
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Isomorphic Mappings. (A) Table of Isomorphic Mappings from the Original 12-Reaction Set to a Minimal Set Consisting of Reactions A and E. Note that many mappings require the formation of intermediate molecular species and that enzyme reactions become bidirectional during the expansion. (B) An example of an isomorphic mapping where bistability is preserved, despite the change from unidirectional enzyme to two bidirectional conversion reactions. (C,D) Examples of isomorphic mappings that lose bistability. In both cases, the bistability is lost because the expanded form of an enzyme contains two bidirectional reactions, which can also be mapped to an enzyme with the reverse direction.
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We thank Sanjay Jain, Dennis Bray, Charles Wampler II, and Layne Watson for many ideas on the project. Vishal Surana and Evan Maxwell helped verify the results obtained from frequent motif mining. We acknowledge the use of the Anantham and System X clusters at Virginia Tech on which our simulations were conducted. We thank Harsha Rani for setting up the DOCSS web page and SBML conversions of models.