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Automatic mechanistic inference from large
families of Boolean models generated by Monte
Carlo Tree Search
Bryan Glazer, Jonathan Lifferth, Carlos F. Lopez
February 2022
1 Abstract
Motivation: Many important processes in biology, such as signaling and gene
regulation, can be described using logic models. These logic models are typi-
cally built to behaviorally emulate experimentally observed phenotypes, which
are assumed to be steady states of a biological system. Most models are built
by hand and therefore researchers are only able to consider one or perhaps a
few potential mechanisms. We present a method to automatically synthesize
Boolean logic models with a specified set of steady states. Our method, called
MC-Boomer, is based on Monte Carlo Tree Search (MCTS), an efficient, paral-
lel search method using reinforcement learning. Our approach enables users to
constrain the model search space using prior knowledge or biochemical interac-
tion databases, thus leading to generation of biologically plausible mechanistic
hypotheses. Our approach can generate very large numbers of data-consistent
models. To help develop mechanistic insight from these models, we developed
analytical tools for multi-model inference and model selection. These tools re-
veal the key sets of interactions that govern the behavior of the models.
Results: We demonstrate that MC-Boomer works well at reconstructing
randomly generated models. Then, using single time point measurements and
reasonable biological constraints, our method generates hundreds of thousands
of candidate models that match experimentally validated in-vivo behaviors of
the Drosophila segment polarity network. Finally we outline how our multi-
model analysis procedures elucidate potentially novel biological mechanisms and
provide opportunities for model-driven experimental validation.
Availability: Code is available at: www.github.com/bglazer/mcboomer
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2 Introduction
Technological advances in high throughput sequencing have significantly in-
creased the amount of data available to biologists. However, the systems of
molecular interactions that generate many cellular phenotypes remain poorly
understood. This lack of understanding is a particularly pressing problem for
diseases such as cancer, in which small genetic perturbations can have dras-
tic clinical consequences. In order to understand and potentially intervene in
the mechanisms by which cellular systems become dysregulated, one must first
create a hypothesis of the system’s interactions.
2.1 Computational modeling of biology
Given the complexity and non-linearity of many biological systems, computa-
tional models are a key tool for hypothesis generation and testing, allowing
in-silico perturbation and experimentation. Much previous work has shown
the value of computational models of cellular systems for both understanding
mechanisms and predicting cellular response to perturbation (B´eal et al., 2019;
S´aez-Rodr´ıguez et al., 2007; Schlatter et al., 2009).
In this work, we focus on automatic synthesis of Boolean models. Boolean
models are a simple, two-state, dynamical system with Boolean logic update
rules introduced by Kauffman (1969). Boolean models do not have reaction
rate parameters; their behavior is entirely determined by the structure of the
update rules. Despite their simplicity, previous work has shown their accuracy
and utility in modeling a wide variety of biological systems.
2.2 Manually specifying models is difficult
However, manually creating these computational models can be time-consuming
and difficult for several reasons. First, selecting a set of interactions that lead
to the desired behavior is challenging due to the vast number of possible inter-
actions. Further, introducing a new interaction can create feedback loops that
change the model’s behavior in unintuitive ways. Finally, data is often limited,
only covering a limited set of conditions. Thus, many possible model config-
urations may have behavior that matches the (limited) data equally well. In
order to have a reasonable chance of finding a model that captures an accurate
representation of the biological system, including in conditions outside the given
data, one must create many models.
2.3 Our approach
Thus, automated model synthesis is desirable as it alleviates the difficulty of
manually constructing a wide variety of models that are consistent with data.
However, an efficient search algorithm is required to synthesize data-consistent
models from the vast space of possible Boolean models. Inspired by recent
work in reinforcement learning for games, which also have combinatorially large
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search spaces, we investigate Monte Carlo Tree Search (MCTS). Our method
uses MCTS to iteratively build Boolean models by adding molecular interactions
to the model’s update rules, similar to the way this algorithm is used to select
moves in the games of chess or Go (Gelly et al., 2006).
We show that MCTS works well for a wide variety of input data and model
structures by testing the algorithm’s ability to recover randomly generated
Boolean models. Further, we show that it works for a more biologically realistic
scenario: generating multi-cellular models of the Drosophila segment polarity
network. Our method generated hundreds of thousands of models of the segment
polarity network that are all consistent with experimental observations.
Having created a large collection of data-consistent models, one must derive
some insight into the key interactions or mechanisms which drive their behavior.
This is itself a challenging pattern recognition problem, which we address by
developing data driven methods to extract mechanisms from models. Specifi-
cally, we present methods for clustering models based on the structure of their
interactions. Using the structural clustering, our methods reveal the key inter-
actions that control model behavior. We employ this analysis to develop a novel
hypothesis for the mechanism of regulation of the wg gene by isoforms of CI in
Drosophila.
We call this pipeline of automated model generation and mechanism explo-
ration MC-Boomer, or Monte Carlo Boolean Modeler.
2.4 Previous Work
Approaches to inferring logical models with data-consistent behavior can be
divided into two categories: constraint solving and optimization. Constraint
solving based methods pose the problem as a series of logical constraints, e.g.
that the update functions must be consistent with steady states described in the
data. These constraints are typically encoded as Boolean logic equations or in
a more abstract formalism such as answer set programming (ASP) (Chevalier
et al., 2020, 2019) or satisfiability modulo theories (SMT) problems (Yordanov
et al., 2016; Fisher et al., 2015). Specialized solvers then find a set of models
which satisfy all the constraints specified by the data and the modeling assump-
tions.
Optimization methods use general purpose discrete optimization algorithms
to generate Boolean models, which are then scored according to a user-defined
objective function (incorporating e.g. similarity to data or model complexity).
The optimization algorithms then generate new models which are variations of
the best scoring models (Lim et al., 2016; Terfve et al., 2012).
2.5 Comparison to our method
Our method differs from previous approaches in several key ways. First, we
employ a heuristic optimization method, in contrast to linear programming or
satisfiability solver based approaches. This allows us to trivially encode more
complex model dynamics (e.g. multi-cellularity) and constraints on the form
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of update rules. Further, our optimization approach requires simulation of all
models, giving us a view into the state spaces of our models. This allows us
to characterize models according their behavior between initial conditions and
steady states, yielding greater insight into populations of models that all have
similar steady states. This comes at the cost of greater required computational
resources compared to methods based on specialized constraint solvers. How-
ever, our method is trivially parallelizable, which exploit to find large numbers
of data-consistent models in a reasonable time frame. Finally, our optimization
based approach immediately generates models that are partial matches to the
experimental data. In contrast, constraint solvers may neglect useful models
that do not perfectly satisfy constraints, even when those constraints are mis-
specified or based on noisy data. In the worst case, constraint solvers may yield
zero models after a lengthy search, while our approximation approach yields a
spectrum of models of varying complexity and goodness of fit to the data.
MC-Boomer is conceptually similar to optimization based approaches such
as BTR (Lim et al., 2016) and PRUNET (Rodriguez et al., 2015). One novel
aspect of our approach is that we employ Monte Carlo Tree Search (MCTS)
(Kocsis and Szepesv´ari, 2006). The efficiency of MCTS allows us to find large
numbers of models that fit the data well. Thus, we are able to make infer-
ences about possible mechanisms of biological systems that are based on families
of thousands of potential models. Some previous approaches (Saez-Rodriguez
et al., 2009) consider the relative probabilities of individual interactions, based
on the whole population of data-consistent models. However, we investigate
model structures with more sophisticated and fine-grained analyses, such as
structure-based clustering and cluster interpretability methods.
3 Methods
Here we describe the components of our framework for automated generation
and exploration of mechanistic hypotheses, which we call MC-Boomer (Monte
Carlo Boolean Modeler). As shown in Fig 1, our framework consists of three
steps: gathering data and prior knowledge (Fig 1, left), using Monte Carlo
Tree Search to generate and test model hypotheses (Fig 1, middle), and finally
analyzing the model collection using statistical and multi-model inference ap-
proaches (Fig 1, right). In this section, we primarily describe the second step,
the algorithmic components involved in generating models. We describe the
third step, analysis of the models generated by MCTS, in more detail in the
Results (Section 4), as part of our analysis of the segment polarity network.
We separate our discussion of model generation (Fig 1, middle) into three
sections: simulation, scoring, and search. We simulate our models with Boolean
update rules, introduce a novel edit distance metric for scoring, and employ
Monte Carlo Tree Search (MCTS) for search. Below we will describe each
component in more detail.
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Figure 1: MC-Boomer workflow. Broadly speaking, MC-Boomer consists of
three steps. The first is to gather data of the steady state expression levels
of the genes of interest in a system. Optionally, one can gather information
regarding known interactions between genes in the system. Potential sources for
this information include biochemical interaction databases, published literature,
or even prior experience and intuition. This prior knowledge can guide and
constrain the second step, model generation. We use Monte Carlo Tree Search
to generate models (Section 3.3 and Fig. 3). The objective of search is to find
models that have simulated attractor states that are similar to data, as measured
by an edit distance metric, described in Section 3.2. We test this algorithm’s
ability to recover random models in Section 3.4. With promising results on
random models, we apply the method to a more biologically realistic model: the
Drosophila segment polarity network, described in Section 4.1. This resulted
in >200khigh quality models, which we analyze further in Sections 4.1.2 to
extract structural features.
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3.1 Simulation
A Boolean model is composed of logic rules that determine the state of each
species in the system at the next step. We use synchronous updating, meaning
that at each step we apply the corresponding update equations to every species
in the model. Synchronous updating is deterministic and is guaranteed to reach
either a single stable attractor state or a sequence of periodically repeating
states, called a cyclic attractor. We detect both stable and cyclic attractors
by tracking previous states and halting simulation when the state matches a
previously simulated state.
We restrict the form of our Boolean update functions to only dominant
inhibition, having the form:
xt+1 =(actt
0or actt
1or ... actt
n) and not (inht
0or inht
1or ... inht
m)
Here, xis the species in the model that will be updated, actt
1...n and inht
1...m
are the states (at time t) of other species in the network that regulate the target
node. Both actiand inhican be a single species or composites of of two or more
species connected by an and clause, e.g. (aand b). A node is activated at t+ 1
only if one or more of its activators is active and no inhibitors are active at t.
A more comprehensive review of simulating biological systems with Boolean
networks can be found in Albert et al. (2008).
3.2 Scoring
We developed an edit distance metric that compares reference data to model
steady states (described in Figure 2). This distance is used to guide the MCTS
search algorithm towards models that generate steady states that are similar to
the data.
At each step of the distance calculation, we identify every attractor state
siin the simulation set where the occurrence count of the attractor is differ-
ent between simulation and data (ci=cj). We then calculate ”edits”, which is
changing one attractor state to another. The cost Cof an edit is the Manhattan
distance between the bit vectors representing the state of the individual species
in each attractor. We then find the edit with minimum cost that would maxi-
mally reduce ci−cj. We apply these edits by changing the occurrence count of
the edited simulation state, then repeat the process until all occurrence counts
are equal between simulation and data. By accumulating edit costs at each step
we obtain a total edit distance between simulated and measured attractor sets.
This is normalized between (0,1) by dividing by the maximum possible edit
distance |si| · Nc, where |si|is the number species in the model and Ncis the
sum of occurrence counts.
Dedit(Asim , Aobs) = PNe
k=1 Ck
|si|Nc
where Neis the number of edits required, and Ckis the cost of the edit at step
k. A detailed example of the edit distance calculation is shown in Fig 2.
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Figure 2: Example distance calculation for a system with five genes. Top row:
Example attractors (left) are generated by simulating a model. The expression
data (middle left) against which simulated attractors will be compared is also
shown. Note that the data has three unique attractor states denoted Diwhile
the simulation only has two, denoted Mi. To calculate the first entry in the dis-
tance matrix (right) attractor states M1and reference states D1are compared.
Differences are assigned a ”1” while matches are assigned a ”0”. As shown, the
distance between states M1and D1is ”3” because they differ at three genes
(C,D,E). Bottom boxes: Sequence of edits required to calculate the distance
between simulation attractors (M) and data attractors (D). In the first step
(1), we choose an edit by selecting the smallest valid distance from the distance
matrix. This edit changes one of the M1attractors to D2, but these are already
identical, so the cost is zero. In step two (2) we select the next smallest distance
(M2to D1, with distance two) and change two attractors for a total cost of
four. In step three (3) and four (4) we continue the same process. Note that
in step three we remove multiple edits involving M2from consideration, as all
of the available M2states have been edited already. In step four, the new state
exactly equals D, so we halt the process with a final edit distance of ten.
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3.3 Monte Carlo Tree Search
The core of MC-Boomer is Monte Carlo Tree Search (MCTS). In short, our
MCTS procedure iteratively adds interactions to a Boolean model’s update
rules, simulates the model, and then computes the similarity of simulated at-
tractors to reference data. Each unique combination of rules is represented by
a branch of the search tree. In Figure 3, each branch of the search tree is anno-
tated with the unique set of interactions that comprise the corresponding model.
On the left side of Figure 3 we show the Boolean models corresponding to the
grey labeled nodes (M1-M3). MCTS probabilistically chooses which branches to
continue expanding, based on a statistical upper bound on the similarity score
of models from each branch. The upper bound is called the UCT or Upper
Confidence bound for Trees. The upper bound is approximated by tracking the
number of times a branch has been explored (visit count) and the average sim-
ilarity scores of models on each branch of the search tree. These statistics and
an example upper bound are shown for each node in the search tree in Figure 3.
MCTS uses the upper bound to balance exploration of different rules versus
exploitation of rules that have already produced high scores. This is illustrated
in Figure 3. The leftmost branch is relatively unexplored but models on that
branch have high average similarity to the data. Thus, this branch has a high
upper bound and the MCTS algorithm will preferentially explore and expand it.
In contrast, the middle left branch has low average similarity scores so MCTS
has ”pruned” it from the search, leaving it unexplored. The middle right branch
has low similarity, but has been explored several times, yielding a very low upper
bound. Finally, the rightmost branch has high scores, but has been visited many
times, and so the upper bound is close to the average score.
We implemented several modifications to standard MCTS that have been
shown to improve the algorithm’s performance. RAVE is a simple modification
to the MCTS algorithm that shares value estimates of actions across all branches
of the search tree (Gelly and Silver, 2011). Nested search uses the actions from
the best random rollout to choose the next step, rather than selecting based
on upper confidence bound (Rosin, 2011). Branch retention keeps the upper
confidence bound from previous search iterations and reuses them for every
subsequent search step. These methods are described in more detail in the
supplementary methods.
3.4 Validation Experiments
We performed two experiments to demonstrate MC-Boomer for inferring Boolean
models. In sections 3.4.2 and 3.4.3, we randomly generated Boolean models of
various sizes, then tested MC-Boomer’s ability to recover the structure and be-
havior of the random models. Then, in section 4.1.1, we tested MC-Boomer’s
ability to recover the structure and behavior of the drosophila segment polarity
network, a complex multicellular model that accurately recapitulates one aspect
of drosophila embryo morphogenesis.
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Figure 3: Overview of MCTS. On the left are the Boolean models corresponding
to the branch of the search tree shown on the right, denoted M1, M2, M3. At
each node in the tree, we also show the average score of models on the branch
and the number of times the MCTS algorithm has visited the branch. These
statistics are used to calculate the upper bound. In the bottom right, we show
a conceptual overview of the functional form of the upper bound. In short,
MCTS will aggressively explore branches with high scores but low number of
visits. More exploration (i.e. a higher visit count) will progressively lower the
upper bound until MCTS chooses another branch to explore.
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3.4.1 Random Model Generation
Here we describe our validation experiments, showing that MC-Boomer can
find models with a wide variety of behaviors and structures. We tested this by
randomly generating models, simulating them, and then applying MC-Boomer
to generate models matching their steady states.
We randomly generated Boolean models with dominant inhibition update
rules by sampling uniformly from a list of all possible interactions between sets
of 8 or 16 species. Following this procedure, we generated 80 random networks.
Before testing MC-Boomer on the randomly generated models, we ensured
that the attractor states of the random models had realistic, diverse character-
istics. The attractors reached by the random models do not collapse to an all
active or inactive state, and instead have roughly one third active species (as
shown in supplementary table S2). We consider the characteristics of these at-
tractors to be biologically relevant, similar to data that might be obtained from
an experiment. Thus, good performance on these random models shows that
MC-Boomer can generalize to a realistic variety of input data distributions.
3.4.2 Steady State Behavioral Similarity
We applied MC-Boomer to attempt to recover these random models using only
their initial states and attractors as input data. Figure 4a shows that the
models generated by MC-Boomer at the beginning of the search process poorly
matched the behavior of the ground truth models. This is expected, as the
MCTS algorithm is effectively a random search process during the initial steps.
However, by the end of the search, MC-Boomer reliably found models that had
steady states with high similarity to the ground truth models. Across all model
sizes, MC-Boomer was able to find several exact behavioral matches, with a
majority having ¿95% similarity, as shown in Figure 4b.
3.4.3 Rule Set Similarity
In addition to the steady state behavior of the models, we are also concerned
with the content of the update rules generated by MC-Boomer. Many possible
rule sets can have the same steady state behavior. However, many of these
rule sets may be significantly different from each other and, most importantly,
different from the underlying biological system. Under novel perturbations or
conditions, these models may behave in radically different ways. Thus, we would
like MC-Boomer to find models that match both the steady state behavior
and the ”interaction topology” of the underlying system. To validate MC-
Boomer in this regard, we tested its ability to generate models with interactions
that are similar to the reference models. In our tests, we quantified similarity
by converting the update rules to sets of interactions for both the reference
(randomly generated) model and the model generated by MC-Boomer. We
then find the Jaccard index between the two interaction sets. This process is
illustrated in Figure 5. Higher Jaccard indexes indicate that the MC-Boomer
model matches the reference topology well.
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Figure 4: a) Orange histogram depicts distribution of similarities from the first
one thousand models sampled during the MCTS search. Blue histogram is the
distribution of similarities from the last thousand models. The blue distribution
is significantly shifted towards higher rewards, indicating that MCTS was sys-
tematically sampling good models. b) Distribution of highest reward obtained
by each independent search process. Most searches found models with ¿90%
similarity.
Figure 5: Example calculation of Jaccard similarity. We compare the structural
similarity between two Boolean models by computing the Jaccard similarity
between their sets of interactions. Here, shared interactions between the two
models are highlighted in green, while interactions that are unique to each model
are in black. In this example, the two models share three interactions in com-
mon, but have three more that are unique to each model. Thus they have a
Jaccard similarity of 3/(3 + 3) = 3/6 = 0.5
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Figure 6: Jaccard similarity between synthetic reference and generated models
with varying levels of prior knowledge. The violin plots show the distribution of
Jaccard similarities achieved by MC-Boomer for synthetic models. The horizon-
tal axis shows varying proportions of incorrect interactions randomly removed
from the list of actions that MC-Boomer can choose when generating models.
Removal of incorrect edges simulates the effect of prior knowledge, for example
using only interactions from a database of validated biochemical interactions.
As expected, higher levels of prior knowledge lead to higher Jaccard similarities,
as MC-Boomer has a higher probability of choosing correct interactions from a
smaller list.
With no restriction on the interactions selected by the model search process,
MC-Boomer was able to find models with behavior that exactly matched the
steady states of the reference models, but using rule sets that differed by as much
as 80%. This corresponds to the left-most column of Figure 6, with zero reduc-
tion in search space, indicating that MC-Boomer was generating models using
all possible interactions and no bias towards the true reference interactions.
We next investigated the effect of utilizing ”prior knowledge” on MC-Boomer’s
ability to recover correct rules. As noted above, model inference is an undercon-
strained problem with many possible models having data-consistent behavior,
and so ruling out infeasible interactions can reduce the number of spurious
models. We simulated varying levels of prior knowledge by randomly removing
incorrect interactions from MC-Boomer’s action list, while retaining all of the
correct interactions. We repeated the search five times, removing 10%, 25%,
50%, 75%, and then 90% of incorrect interactions from a set of 80 models. The
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aggregated Jaccard similarities for each percentage are shown in Figure 6. For
models with both 8 and 16 species, increasing prior knowledge increased the
Jaccard similarity to the reference data, as expected. Note that most protein-
protein interaction databases are much sparser than our highest tested level
of prior knowledge. For example, BioGRID (version 4.4.2021) has 26k genes
and 806k interactions, which corresponds to a 99.9% reduction from all possible
interactions (Oughtred et al., 2021). Thus, our tests simulate a very difficult
scenario, relying on much less prior knowledge than is available in biochemical
interaction databases.
4 Results
Here we show the result of applying MC-Boomer to the segment polarity network
(SPN). In sections 4.1.1 and 4.1.2, we describe the SPN and show MC-Boomer
can generate models that are structurally similar to it, automatically discovering
interactions that were previously manually selected by experts. Additionally we
describe the large collection of alternate mechanisms generated by MC-Boomer,
analyzing several in detail.
4.1 Segment Polarity Network (SPN)
As shown in the previous sections, MC-Boomer is able to generate models that
are behaviorally and structurally similar to a variety of synthetically gener-
ated reference systems. While this was useful for validation, we also applied
MC-Boomer to a more realistic setting to demonstrate the usefulness of the
proposed framework. To that end, we employed MC-Boomer to build models of
the Drosophila Segment Polarity Network (SPN), which is a gene circuit that
controls the formation of borders and directionality of body segments during
development of the drosophila embryo. As a reference, we have chosen a well-
studied model by Albert and Othmer (Albert and Othmer, 2003). Briefly, this
model comprises four cells, with several distinct components, including genes,
proteins, membranes, protein isoforms, and complexes. A diagram of the SPN
interactions is shown in Figure 7, and a complete listing of the reference rules are
shown in supplementary table S3. Albert and Othmer provided binarized ex-
pression levels for wild type conditions as well as three gene knockouts, shown in
supplementary table S4. We applied MC-Boomer with these expression profiles
to automatically generate models of the SPN.
4.1.1 Model Generation
First, we will describe how we initialized the model and performed the search.
We applied several constraints to the search process so that MC-Boomer
would only generate biologically plausible models. Membrane proteins (W G,
PTC,SMO,P H ,HH ) could interact with membrane proteins only on adjacent
cells. Internal proteins (EN ,S LP ,CI,CIR,CI A) could interact with other
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Figure 7: Reference Model for Segment Polarity Network. Diagram of the inter-
actions in Albert and Othmer’s model of the segment polarity network (Albert
and Othmer, 2003). Green edges indicate activating interactions. Red are in-
hibiting. Lower case ovals indicate genes and upper case indicate proteins. The
dotted border indicates the cell membrane, with membrane proteins straddling
the border. On the right is the adjacent cell, with several interactions spanning
between cells. Albert and Othmer’s model has four cells with the same inter-
actions inside each cell. Interactions between cells are symmetric, though only
one direction is shown in the diagram to maintain clarity.
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internal proteins, membrane proteins in the same cell, and genes in the same
cell. Genes (en,ci,ptc,hh) could only activate their corresponding protein,
and these gene-protein activating interactions were pre-specified in our search
process.
All interactions added during the search process were repeated across all
four cells. Multi-cellular membrane interactions were symmetric, added in both
directions between neighboring cells.
The reference SPN model specified initial and stable states for the wild type
network as well as initial and attractor states for knockouts of wg,hh, and en
(see Supplementary figure S4 for details).
We applied MC-Boomer to search for models that matched the behavior of
the reference SPN model across the wild-type and three knockout conditions. At
each search iteration, we simulated the model across all conditions, calculated
the edit distance between simulated and reference steady states, then averaged
all conditions’ similarity scores to get a final score for the iteration. We imple-
mented knockouts by removing all interactions to and from the hh, en and wg
genes across all cells.
We ran 1500 searches in batches of 30 in parallel on our institution’s com-
puting cluster. In each search step, MC-Boomer simulated 10k model variations
before adding the best interaction to the model and starting the next step. We
restricted the search to terminate after 30 steps, but not before completing 8
steps. Every search was run with RAVE, nested search, and branch retention
enabled with the same uniformly random sampled parameter distributions as in
the synthetic data experiments. The complete search process took 41 hours and
simulated 430 million unique models. Eleven of the 1500 search processes found
models with exactly the same steady states as the reference model for all four
conditions. Collectively, these eleven search processes generated ¿202kmodels
with perfect consistency to the attractor data.
4.1.2 Visualizing the Set of Data-Consistent Models
Given the our collection of models with consistent steady state behavior, we
were motivated to develop methods for visualization and exploration of large
numbers of models.
First, we applied dimensionality reduction and clustering methods to visu-
alize similarities between the models. We randomly sampled fifty thousand of
the 202kdata-consistent models and clustered them with the UMAP algorithm
(McInnes et al., 2018) using the interaction set Jaccard distance between mod-
els, as illustrated in Figure 5. Model sampling was necessary because UMAP
requires computation of a pairwise distance matrix that would have been infea-
sible for the full data set. Multiple different samples all gave similar results, thus
providing us with confidence that the sample analyzed here was representative
of the overall model population.
Applying UMAP with the Jaccard distance yielded the result shown in Fig-
ure 8 with eleven well separated clusters, corresponding to the eleven indepen-
dent searches that produced data-consistent models.
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Figure 8: Scatter plot depicting clustering of unique data-consistent segment
polarity models after UMAP projection to two dimensions. There are eleven well
separated clusters, corresponding to the eleven independent search processes
that found data-consistent models.
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Table 1: Structural Intersection with Reference Model
Intersection Cluster Size
Cluster Mean Median Max
0 4.80 5 8 5321
1 6.03 6 9 6566
2 6.32 6 9 1654
3 3.23 3 6 2057
4 6.87 7 9 5673
5 6.49 6 9 4805
6 3.30 3 6 6710
7 8.68 9 11 5493
8 5.22 5 8 2277
9 6.90 7 10 6413
10 8.27 8 11 3031
Table 2: For each cluster of models shown in Fig. 8 we computed the inter-
section between these common rules and the reference model. We show the
mean, median, and maximum intersection between each cluster’s models and
the reference. Cluster 7 has the highest intersection across all three statistics,
while cluster 3 shares the fewest interactions with the reference. We further
investigate the most common interactions in Cluster 7 in Fig. 10.
4.1.3 Structural Similarity between Clusters and Reference
We then compared the interactions in each MC-Boomer generated model with
the interactions in the reference model’s update rules to find the ”structural”
similarity.
The update rules of the reference model had 26 total interactions. We man-
ually pre-specified eleven of the interactions in the reference segment polarity
network. That is, all models generated by MC-Boomer included these interac-
tions as ”prior knowledge”. This included all the interactions in which a gene
activated its corresponding protein, as well as four interactions that did not fit
the dominant inhibition dynamics of the rest of the network (supplementary
figure S1). Our tests evaluated MC-Boomer’s ability to discover models that
included the remaining 15 interactions in the reference model.
Within each cluster of MC-Boomer models, we computed the mean, median,
and maximum size of the intersection between the cluster’s models’ interactions
and the reference model’s interactions, as shown in Table 2.
Comparison across clusters revealed a wide disparity in accuracy, with clus-
ter 3 having, on average, 3 rules in common with the reference SPN model.
We note that while the models in cluster 3 had low structural similarity to the
reference SPN model, all of the models in every cluster have the same steady
state attractors as the reference. Cluster 7 had the highest average intersection,
with several models in the cluster having 11 out of 15 rules in common with ref-
erence model. For cluster 7, we found the most common rules, i.e. those shared
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by ¿90% of the models in the cluster. Figure 10a shows these common rules
and Figure 10b/c shows ”false positive” and ”false negative” rules, respectively.
False positive were present in MC-Boomer models but not in the reference and
false negative rules were in the reference but not the MC-Boomer models. In
the following sections, we investigate two of these interactions, one of which was
not present in the reference model.
4.1.4 Investigating Common Mechanisms
The high-level clustering analysis shows that MC-Boomer generates models with
a wide variety of structures but identical steady state behavior. However, this
analysis is too broad to elucidate the precise nature of the mechanisms that
these models employ to generate this behavior. Accordingly, we more closely
investigated two key interactions that are present in every model generated
by MC-Boomer. Specifically, we consider ”EN inhibits ci” and ”CIR inhibits
CI A”, which are present in 100% of the data-consistent models.
First, we look at EN inhibiting CI , which is present in all of our models
and also present in the reference model. This indicates that this interaction is
a crucial link across the the very diverse mechanisms employed by the eleven
clusters of models and the reference model. Simulating a random sample of one
thousand models with this interaction knocked out resulted in a 28% average
absolute reduction in similarity to the reference steady state data. We observed
that knocking out the EN to ci interaction in the reference model also reduced
similarity to the reference data by 28%. Again, this indicates that the while the
models are structurally diverse, they share a similar reliance on this particular
interaction of EN and ci.
On the contrary, CIR inhibition of CIA is not present in the reference model.
This interaction is shared by more than two hundred thousand unique models
generated by MC-Boomer. The high frequency of the CIR inhibiting CIA in-
teraction motivated further investigation into CIR and CIA’s role in regulation
of the wg gene.
To provide necessary background for our discussion of wg regulation, we
briefly describe the key genes in this pathway. CIA is an activated, nuclear
transported form of the CI protein, while CIR is a proteolytically cleaved form
of CI which represses wg transcription. In the absence of HH, SMO forms a
complex with CIA and Cos2, a kinesin-like protein that binds and sequesters
CIA, preventing its nuclear translocation and permitting its cleavage into CIR.
In the presence of HH, SMO is activated and Cos2 releases CIA, which is then
transported to the nucleus, where it activates wg (Lum et al., 2003; Kalderon,
2004; Ranieri et al., 2012). The exact mechanisms and network dynamics behind
CI activation, cleavage, and nuclear translocation have long remained a point
of debate and uncertainty (Ruel et al., 2003).
In addition to CIR inhibiting CIA, MC-Boomer also suggests (in 34% of
models) an inhibitory interaction between CIR and SMO. The novel inhibition
of CIA and SMO by CIR can be interpreted in at least two ways.
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Figure 9: a) The reference model depicts the modification of CI as a forked
pathway, where the resulting product is determined by the activation state of
SMO. In the reference model, active SMO promotes CIA and inhibits CIR. b)
The MC-Boomer model, in contrast, includes two novel interactions where CIR
inhibits CIA and SMO. Figures partially based on Figure 3 from Hooper and
Scott (2005)
1. These interactions do not represent real signaling mechanisms. In accor-
dance with the reference model, the bi-directional inhibitory loop between
CIR and SMO may simply reflect the normal activation states of these
proteins. When SMO is active, CIR cannot be produced because SMO
destabilizes Cos2 and therefore all CI is available as CIA. Conversely, when
SMO is inactive, Cos2 binds CI and conversion to CIR occurs. There-
fore, the inhibition of CIA and SMO by CIR may not represent genuine
biochemical interactions, but may simply be artifacts of MC-Boomer’s
automated model generation process.
2. These interactions do represent real, redundant signaling mechanisms.
The novel inhibition of CIA and SMO by CIR may represent redundant
signals which prevent the possibility of competition at the target gene
binding site. This type of redundancy is a feature observed in other bio-
logical signaling networks (Albert 2011). CIR inhibition of CIA and SMO
in the cytosol ensures that CIR can bind and inhibit wg in the nucleus
without interference from CIA. In this interpretation, CIR is not just a
passive cleavage product, but also an active participant in a feedback loop
that inhibits the activity of CIA.
This second interpretation describes an instance of signaling redundancy. If
CIR inhibits SMO and CIA, this helps to ensure a full transition between on
and off network states and prevents any potential binding competition at the
target gene.
Overall, these observations show that the proposed method can both re-
produce the known biological features as well as provide novel insight into the
segment polarity network by generating new mechanistic hypotheses, which re-
quire further investigation through experiments.
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4.1.5 Identifying Unique Mechanisms in Model Clusters
We are able to analyze these two interactions in detail because they are shared
across all models and their limited scope eases their interpretation. However,
our clustering analysis showed that there are at least eleven groups of models
with widely differing structures. Accordingly, we also investigated the role of
interactions that are specific to individual clusters of models. We searched for
sets of up to 5 interactions that are present in a high proportion of models in
each cluster, while not being present in models in other clusters. We call these
distinguishing sets. We found between 25 and 21,570 distinguishing sets per
cluster.
Given the large number of distinguishing sets for some clusters, we needed
a measure of which sets are most important to the function of the models in
the cluster. We quantified this by simulating knock outs of each distinguishing
set in a sample of 100 models from their respective clusters and calculating the
reduction in similarity to the reference data caused by the knockouts. We refer
to distinguishing sets with the largest reduction in similarity as the maximally
disruptive sets. These maximally disruptive sets identify the unique mechanisms
that the models in each cluster most highly rely on to generate their behavior.
Comparing the interactions in the maximally disruptive sets revealed het-
erogeneity across the clusters. Most of the maximally disruptive sets shared
two or fewer interactions in common. For example, the maximally disrupt-
ing sets for cluster 7 (Figure 11a) and cluster 8 (Figure 11c) only share a single
interaction in common. Simulated knockouts of cluster 7 and 8’s maximally dis-
ruptive sets reduced similarity to reference data by 31% and 39%, respectively.
This indicates that models in these two clusters depend, to a similar degree,
on these distinct sets of interactions for generating correct behavior. Inspection
reveals that while the two mechanisms are not similar by a direct comparison,
they share functional similarity in primarily modulating the connectivity and
activity of EN . This corresponds with our previous analysis showing that EN
interactions are crucial for correct model behavior across our whole collection
of models. However, the actual mechanism by which EN activity is directed is
quite distinct. The interactions in cluster 7 (Figure 11a) give EN a mixed ac-
tivating/inhibiting role, while cluster 8 (Figure 11b) relies on several inhibitory
feedback loops centered on EN .
Similar to the case of CIR described in Section 4.1.4, many of the distin-
guishing sets do not have any effect on the behavior of the model; one such
example is illustrated by Figure 11b. One perspective is that these interactions
are redundant and only increase the complexity of the model. Accordingly,
several previous approaches (citations) penalize models with more interactions.
Another perspective is that these redundant connections may confer robustness,
i.e. an ability to recover from aberrant initial conditions or losses of function,
or as with CIR they could help ensure full response to inhibition or activation.
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Figure 10: a) Most common interactions in cluster 7. These interactions that
are found in ¿90% of models in cluster 7. Transparent lines represent interac-
tions that were pre-specified as prior knowledge, while more opaque interactions
were generated by MC-Boomer. b) Interactions highlighted in red and green
are common in cluster 7 but are not present in the reference model. Grey in-
teractions are shared between the reference model and cluster 7. Note the red
inhibiting interaction between CIR and CIA, which is investigated in more detail
in Section 4.1.2 c) The reference model is shown here, highlighting interactions
that are in not present in cluster 7. Again, note the red inhibiting interac-
tion between CIR and wg. Section 4.1.2 proposes an alternative mechanism for
inhibition of wg by CIR.
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Figure 11: a) Shown in red and green is the maximally disruptive set for cluster
7. These are interactions that are common in cluster 7, but are very uncom-
mon in other clusters. Additionally, knocking out these interactions reduces
similarity to the reference steady states more than other sets of common inter-
actions (shown in grey). b) Another distinguishing set of interactions, but these
are minimally disruptive. Knocking them out only reduces similarity to the
reference data by a negligible amount. c) Maximally disruptive set for cluster
8. These reduce similarity to reference data to a similar degree as the most
disruptive set in cluster 7, but these interactions utilize a different mechanism.
5 Discussion
Biology is inherently complex, yet our measurements capture only a limited slice
of the true activity within a cell. Current assay technology can only describe a
subset of biomolecules at low time resolution and with significant noise. From
this blurry view researchers must synthesize a model that can both describe the
phenomena under investigation and predict the system’s behavior in novel cir-
cumstances. Synthesizing a model can be made easier by choosing the simplicity
of the Boolean logic modeling formalism to represent the system. Nonetheless,
even for a small number of interacting species, the number of possible Boolean
models is vast. Consequently, a typical researcher, creating models through
trial and error, may only find one or perhaps a few models whose behavior is
consistent with the observed data. However, as we have shown in Section 4.1.1,
even in a small system with multiple measurements and reasonable prior as-
sumptions on model structure, there are hundreds of thousands of models that
are all consistent with the data.
This observation was made possible by using an efficient search technique,
Monte Carlo Tree Search, to build models. We demonstrate the power of MCTS
to synthesize models with the correct steady-state behavior and the correct
interactions in Section 3.4. While previous studies have shown that similar
optimization methods (e.g. tabu search in Aghamiri and Delaplace (2020)) are
effective for finding data-consistent models, they have focused on finding a single
model that is ”best” in terms of both complexity and fit to the data. In contrast,
we retain every model that fits the data well and in Sections 4.1.4 and 4.1.5 we
develop a set of techniques for making sense of this large collection of models.
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We approach this from a data-driven perspective, in the sense that our MCTS
algorithm generates data about the space of valid hypotheses. By clustering
models based on their structural features, we can find recurrent motifs across
the whole collection of models, as well as distinct motifs that discriminate the
structure of groups of models. Simulated knockouts of these motifs then reveal
that some are critical to the models’ correct behavior.
5.1 Using MC-Boomer to Design Experiments
As we describe in Section 4.1.4, analysis of the models generated by MC-Boomer
pointed us towards an alternate hypothesis for the mechanism by which CIR
and CI A regulate expression of the wingless gene (wg) in the segment polarity
network. An investigator using MC-Boomer to study this pathway may propose
that CIA activation of wg depends on both SMO (Smoothened) stabilization
and, as MC-Boomer suggests, the absence of CIR. The existence of these novel
inhibitory relationships could be experimentally validated by introducing CIR
into cells in which HH signaling has already activated SMO and CIA. Reduced
concentrations of active CIA or SMO would indicate that CIR does, in fact,
inhibit the activity of CIA and SMO.
5.2 Limitations and Future Work
Previous work (Faur´e et al., 2006) has suggested that the general asynchronous
updating scheme yields more biologically realistic results for Boolean network
simulations. While our current approach uses synchronous updating, extending
MC-Boomer to work with asynchronous updating would be straightforward.
The current approach is limited in its scalability to models with large num-
bers of interacting species by several key bottlenecks. First, this approach
requires simulation of every synthesized model, and simulation becomes pro-
hibitively expensive for large models. This could be alleviated through partial
or approximate simulations of the models. While this would yield an approxi-
mation of the model’s similarity to data, the UCT upper bound allows MCTS
to tolerate some noise in the search process. Second, the search space scales ex-
ponentially with the number of species in the model. We show that restricting
the search space through prior knowledge constraints on model structure is an
effective strategy for improving structural and behavioral accuracy of synthe-
sized models. The efficiency of the search algorithm could further be improved
by using deep learning to guide MCTS. This is similar to the approach used
by the AlphaZero algorithm (Silver et al., 2018), that proved to be exception-
ally effective at searching the combinatorially large space of moves in games
like chess and Go. We are currently exploring each of research directions as
potential optimizations of the MC-Boomer algorithm.
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