Constrained inference of protein interaction networks for invadopodium formation in cancer

Abstract

Integrating prior molecular network knowledge into interpretation of new experimental data is routine practice in biology research. However, a dilemma for deciphering interactome using Bayes’ rule is the demotion of novel interactions with low prior probabilities. Here the authors present constrained generalised logical network (CGLN) inference to predict novel interactions in dynamic networks, respecting previously known interactions and observed temporal coherence. It encodes prior interactions as probabilistic logic rules called local constraints, and forms global constraints using observed dynamic patterns. CGLN finds constraint‐satisfying trajectories by solving a k ‐stops problem in the state space of dynamic networks and then reconstructs candidate networks. They benchmarked CGLN on randomly generated networks, and CGLN outperformed its alternatives when 50% or more interactions in a network are given as local constraints. CGLN is then applied to infer dynamic protein interaction networks regulating invadopodium formation in motile cancer cells. CGLN predicted 134 novel protein interactions for their involvement in invadopodium formation. The most frequently predicted interactions centre around focal adhesion kinase and tyrosine kinase substrate TKS4, and 14 interactions are supported by the literature in molecular contexts related to invadopodium formation. As an alternative to the Bayesian paradigm, the CGLN method offers constrained network inference without requiring prior probabilities and thus can promote novel interactions, consistent with the discovery process of scientific facts that are not yet in common beliefs.

1 Introduction

Data from a single biological experiment are often inadequate for reliable network inference [1 ]. Researchers thus routinely apply prior knowledge in the form of sparse network topology into deciphering messages hidden in new experimental data. With the rise of comprehensive biological network databases [2 ], automation of the underlying reasoning process becomes feasible and is expected to expedite the complete characterisation of interactomes in systems biology [3 ].

Discrete dynamic network models have indeed been widely used to delineate molecular systems. They include probabilistic Boolean networks [4 ], dynamic Bayesian networks [5 ], language‐generating automata [6 ], and generalised logical networks [7 ], among others. Given a probabilistic Boolean network and a desirable state, computational approaches have been developed to search for optimal intervention [8, 9 ]. Although discrete dynamic models are flexible for representation of highly non‐linear dynamics, their inference has been a challenge to date, because entirely data‐driven approaches are limited by the small sample size relative to the high dimensionality of typical molecular biology experimental data.

Bayes’ rule underlies most attempts to integrate prior knowledge into network inference [10 –12 ]. By this rule, a posterior probability determines the overall strength of an interaction, in proportion to the product of a prior probability and the likelihood of an interaction given observed data. A limitation of the Bayesian paradigm is that a novel interaction may have a zero or low prior probability such that even strong evidence (likelihood from the observed data) may still be insufficient to promote the interaction. Recent diffusion approaches [13 ] propagate prior belief based on local connectivity to estimate prior probabilities of novel interactions, applicable only when a path is known beforehand to connect the two nodes in a potentially novel interaction.

Therefore, we sought to provide an alternative framework to Bayes’ rule to reduce the impact of prior probability on new interactions. In this framework, we formulate the network inference problem under two types of constraints. Local constraints are topological and probabilistic logic rules describing prior qualitative knowledge about interactions in a network. Global constraints have observed temporal patterns of a system. Dynamic patterns need to be neither complete for every node in the system nor tightly sampled in time. The goal is to find dynamic networks to satisfy both types of constraints. Although previous work has extensively addressed global constraints [14, 15 ], integration of local constraints is under‐investigated other than using Bayes’ rule.

Here, we report a constrained generalised logical network (CGLN) inference method. Generalised logical networks (GLNs) are synchronous and deterministic discrete networks where each node can take two or more finite values and has a generalised truth table (gtt) [7, 16 ]. CGLN interpolates new states into the global constraints based on the local constraints, and then reconstructs one or more constraint‐satisfying GLNs. Interpolation is an enabling step to discover novel interactions that may resolve inconsistency between local and global constraints. We heuristically solve an NP‐complete k ‐stops problem of finding a simple path in the state space of a GLN. Each stop is a dynamic pattern in the global constraint which must be visited in the order observed in experiments. Several methods were developed to find simple paths with some must‐pass nodes without a required order [17 ]. An approximate algorithm by Vardhan et al. [18 ] finds a simple path with must‐pass nodes in mandatory order, but not practical for the scale of our problem.

We then applied CGLN to predict novel protein interaction networks driving invadopodium formation. Invadopodia are F‐actin based membrane protrusions in cancer cells that degrade the extracellular matrix (ECM). Invadopodium formation goes through the stages of induction, initiation, assembly, and maturation [19 ]. Each stage is marked by a signature dynamic pattern of protein activities. In the early stages, a cell activates actin polymerisation and secretes matrix metalloproteinase (MMPs). Actin polymerisation forms a protrusion on the ventral cell surface that extends vertically from the ventral cell membrane into the ECM [20 ]; MMPs’ proteolytic activity degrades the ECM. The combination of physical and chemical stresses creates a channel for the cancer cell to migrate and invade. After migration, invadopodia are disassembled, although some structures, such as the actin‐rich core, can persist for hours [19 ]. Due to this association of invadopodia with cell invasion, their presence is an indicator of a cancer cell's capacity to metastasize into surrounding tissues [21 ]. By applying CGLN on a protein network for invadopodium formation with local and global constraints, we identified a total of 134 novel protein interactions with varying strengths of evidence for their involvement. Fourteen protein interactions have been reported in various molecular contexts, some directly related to invadopodium formation. The molecular roles of these novel protein interactions are of great interest as they may lead to the identification of potential therapeutic targets to block cancer metastasis.

2 Methods

2.1 Network inference by constrained path finding in the state space

We represent molecular interaction networks by GLNs [7, 16 ]. Each node in the GLN corresponds to a molecule, a cell, or a species. As a discrete‐time and discrete‐value dynamical system model, a GLN is represented by a directed graph with a gtt attached to each node in the graph. The gtt allows a discrete variable to take more than two possible values and to reflect more subtle changes than Boolean logic. Thus, a gtt can encode biological mechanisms by which molecules use to interact with each other. The state of a GLN is a unique assignment of values to each node in the GLN. The number of unique states is the product of the number of levels for all nodes. The transition graph of a GLN is directed and represents every state as a vertex. An edge in the graph indicates a transition from one state to its successor state. Only one outgoing edge is allowed for each vertex in the transition graph because the GLN is deterministic. The complete dynamics of a GLN can be visualised by a transition graph in its state space. A trajectory is a path in the transition graph, capturing dynamics of the GLN from an initial state.

We present a method called constrained generalised logical network inference by finding GLNs to satisfy given local and global constraints. We solve the CGLN problem by searching for a constraint‐satisfying trajectory in the state space of the GLN and reconstruct a GLN from the trajectory. The CGLN method proceeds in three steps outlined in Fig. 1. The first step is to construct a super transition graph in the state space of a GLN. In the step two, a novel stochastic and iterative deepening depth‐first search (SIDDFS) algorithm is applied to traverse the graph, searching for segments or simple paths between each pair of consecutive dynamic patterns in the global constraint. In this step, segments are concatenated to form a simple long path with detouring if necessary. In the step three, a GLN will be reconstructed from this simple long path (Supplementary Alg. S6 and Supplementary Alg. S7). Steps two and three will be repeated until no new distinct GLNs can be found. CGLN will finally return a list of previously unknown interactions ranked by frequency among all identified constraint‐satisfying GLNs. The list serves as novel hypotheses for further biological validation.

Fig 1.

Overview of CGLN inference

The three main steps are shown in the ovals. Local constraints are both topological and dynamical. Global constraints are dynamic patterns from experimental data and constitute k stops. k is three in this example and the ‘stops’ or dynamic patterns are listed in the global constraint box, where the dots indicate unmeasured values. The global‐constraint satisfying stops are also marked in three framed boxes in the super transition graph. CGLN operates in the super transition graph and finds a constraint‐satisfying trajectory (solid path) in the graph. The output is generalised logical networks reconstructed from the trajectory that satisfy both local and global constraints and also contain newly inferred interactions (dark edges)

The constrained network inference problem is computationally hard. With the above formulation of the CGLN problem, the k ‐stops problem is equivalent to finding an ordered simple path with mandatory nodes in a directed graph. The latter has been proven to have a computational complexity that is NP‐complete [22 ]. The proof was based on the theoretical computer science result on the k vertex‐disjoint path problem, previously proven to be NP‐complete [23 ].

2.2 Local constraints to generalised logic rules

We represent local constraints by a probabilistic logic rule to describe known qualitative relationships between nodes in a network. Let {U _t :t ∈ ℕ} and {V _t : t ∈ ℕ} be two discrete‐time stochastic processes in a dynamical system. We define a local constraint for the interaction between parent U _{t −τ} and child V _t as follows

$$lc(U_{t-\tau },V_t)=\left\{\begin{array}{cc}+,& \rho (U_{t-\tau },V_t)>0(\mathrm{positive}\mathrm{local}\mathrm{constraint})\\ 0,& \rho (U_{t-\tau },V_t)=0(\mathrm{undefined}\mathrm{local}\mathrm{constraint})\\ -,& \rho (U_{t-\tau },V_t)<0(\mathrm{negative}\mathrm{local}\mathrm{constraint})\end{array}\right.$$ (1)

where ρ is the correlation coefficient of two random variables and τ is a time delay for the interaction. We also call U _t is the parent and V _t is the child in this relationship. Local constraint satisfaction is verified by plugging the observed data for the two variables into the definition. We use τ = 1 in this paper.

This representation of local constraints does not require strict logic rules or probability distributions, so as to be robust to biological variation and measurement errors. The usage of signs is consistent with how mainstream biology literature presents results and allows convenient integration of research findings into constrained network inference.

2.3 Dynamic patterns as global constraints

We denote global constraints by a temporal sequence of k dynamic patterns gc [0], …, gc [k − 1] associated with concerned system events. Signature dynamic patterns can be summarised from either the literature or experimental data. In a molecular network, an event could be signified by a chorus of signature molecules, or a subset of nodes. Once the chorus reaches a state, the event will take place. This subset will be assigned a dynamic pattern associated with the event. Other unconstrained variables are marked as wildcards so that more than one state of the dynamic network can satisfy a given dynamic pattern. For example, a global constraint including k = 4 dynamic patterns for a 7‐node GLN could be

$$\begin{array}{rl}gc[0]& =(0,2,\ast ,2,\ast ,1,\ast )\\ gc[1]& =(1,1,\ast ,\ast ,\ast ,0,\ast )\\ gc[2]& =(1,0,\ast ,2,\ast ,1,\ast )\\ gc[3]& =(1,\ast ,1,2,1,0,1)\end{array}$$

where * represents an unconstrained value in a dynamic pattern. When there are multiple observations at the same time point from biological replicates under the same experimental condition, one can assign the replicate median to each variable to capture the overall trend qualitatively. A trajectory satisfies a global constraint when all its dynamic patterns are matched in the right order.

2.4 Super transition graph in the state space

The super transition graph is defined by the union of edges in all transition graphs of feasible GLNs for the given local constraints. Potential successors of a state in the super transition graph are determined by local constraints. In a local constraint, a child variable is active if and only if its positive parents are all active and negative parents are all inactive; and it is inactive if and only if its positive parents are all inactive and negative parents are all active; otherwise, it is undetermined and can take any values only limited by its radix. For a state with undetermined variables, all possible successor states are included in the super transition graph. Algorithm make‐super‐transition‐graph (lc, variables ) (Supplementary Alg. S1) generates a super transition graph from given local constraints.

2.5 Stochastic and iterative deepening depth‐first search

To find a simple path between two consecutive stops – vertices in the super transition graph, we developed the SIDDFS algorithm (Supplementary Alg. S2). It is based on iterative deepening depth‐first search (IDDFS) – a depth‐limited depth‐first search (DFS) that runs with incremental search depth until either a solution is found or a search depth limit is reached [24 ]. The iterative‐deepening strategy combines the advantages and avoids the disadvantages of DFS and breadth‐first search (BFS). BFS can be memory intensive, because it records all vertices in the adjacency lists to be visited and a super transition graph can be very large. On the other hand, DFS may return a long path increasing the risk for constraint violation. Algorithm SIDDFS is a randomised version of IDDFS with constraints on vertices to avoid. SIDDFS repeatedly calls stochastic‐DFS‐visit (Supplementary Alg. S3) to traverse the graph only up to an increasing maximum depth. The steps added to IDDFS in SIDDFS finish linearly to the length of adjacency lists. Thus the increase in runtime is slower than the order of O (b ^d ). Therefore, the time complexity of SIDDFS is not worse than IDDFS [24 ]. The user is provided an option to specify the search depth limit based on available CPU cycles.

2.6 Resolving conflicting segmental paths by detouring

Given a super transition graph G and a global constraint gc, we first use SIDDFS to find a long initial path which may not be simple (Supplementary Alg. S4). The long path is composed of segments, i.e. simple paths between each pair of consecutive dynamic patterns gc [i ] and gc [i + 1] in the global constraint. If the initial long path is not simple, here we attempt to resolve any vertices used by more than one segment by detouring. We call a segmental path pure if and only if all its vertices are marked by its own segmental ID; otherwise, the segmental path is impure. Detouring will first attempt at an alternative path without using vertices already used or forbidden. If a new path is found, the conflict is resolved and detouring will move on to next impure segment until no impure segment is left. If no alternatives could be found, it will then try to find a path considering vertices already used. If an alternative is found, this segment will be locked to be unchangeable and detouring will move to next impure segment until no impure segment is left. If still no path can be found, it will stop and report the conflict cannot be resolved. Segments with smaller IDs will be routed first, since in initial path finding, paths for segments with smaller IDs were searched first. Supplementary Alg. S5 DETOUR(G, gc, LongPath ) implements the detour strategy. Let k − 1 be the number of segments in the global constraints with k must‐stop states. For a given segment in the long path, if it is conflicting with other k − 2 segments and detouring is made for it every time without locking it, SIDDFS could be applied on this segment k − 2 times. Thus, the worst‐case runtime for this detour algorithm will be O (k ² b ^d ), where O (b ^d ) is the runtime of one run of the iterative‐deepening depth‐first search algorithm.

3 Results

3.1 Benchmarking network inference effectiveness of CGLN by simulation studies

We generated 100 artificial dynamic Boolean networks to evaluate the performance of CGLN, in comparison with other unconstrained or partially constrained network inference methods. Ground‐truth networks were randomly generated GLNs [7 ]. Each GLN contains eight Boolean nodes with from 0 to 8 parents. Each GLN trajectory, simulated using a Markov order of one, contains 256 time points.

Local and global constraints were generated from the ground‐truth GLNs. For each GLN, three sets of local constraints were randomly selected from the edges at increasing percentages of 30, 50, and 70%; the polarities of local constraints were determined by the logic rules in the GLNs. Each global constraint contains ten dynamic patterns randomly sampled in temporal order from GLN trajectories. In SIDDFS, the traversing order of an adjacency list was set to be random. The search depth limit of SIDDFS was set to five.

The CGLN method was compared with two other methods: unconstrained and topology‐constrained network reconstructions. The unconstrained network reconstruction does not utilise any prior knowledge regarding interactions and is thus free of local constraints; it builds GLNs entirely based on the observed dynamics contained in the global constraints without temporal interpolation. It represents the most typical data‐driven network inference methods [7 ]. Topology‐constrained network reconstruction infers GLNs from observed dynamic trajectories contained in global constraints without temporal interpolation; in addition, it gives priority to the topology encoded in local constraints by selecting the edge corresponding to a previously known interaction against another previously unknown one if they both satisfy the temporal logic underlying the observed dynamics during network reconstruction.

Performance of CGLN, unconstrained network reconstruction, and topology‐constrained network reconstruction were evaluated by precision–recall (PR) and receiver operating characteristic (ROC) curves against the 100 ground‐truth networks. Comparisons between them were visualised as area under the curves (AUCs) as a function of network inference runs. The results are shown in Fig. 2 where 50% of ground‐truth interactions are given as local constraints. CGLN achieved the greatest numerical values in AUC for both ROC and PR curves (Figs. 2 a and b ). Both constrained methods achieved statistically significantly better AUCs under the ROC curves than the unconstrained method (Fig. 2 c ). The PR AUC of CGLN is statistically significantly better than the other two methods (Fig. 2 c ). Additional results are shown in Supplementary Fig. S1 and S2 for using 30 and 70% interactions as local constraints. CGLN performed numerically worse than, but statistically indistinguishable from the other two methods when 30% interactions were known (Supplementary Fig. S1); but CGLN performed statistically significantly the best when 70% interactions are known among all three methods (Supplementary Fig. S2). The most remarkable advantage of CGLN is demonstrated when at least half of interactions are included in the local constraints (Fig. 2 ).

Fig 2.

CGLN considerably outperformed less constrained methods for network inference with 50% local constraints given

Each of 45 randomly generated networks contains eight nodes and ten observed dynamic patterns constituting the global constraint. CGLN is set to random traversing in path finding; the other two methods are topology‐constrained and unconstrained network reconstruction.

a Boxplots of areas under PR (left) and ROC (right) curves

b Average areas under PR (left) and ROC (right) curves and their corresponding standard deviations

c t ‐tests on the performance of each pair of methods in terms of AUC for PR (left) and ROC (right) curves. The null hypothesis is the left AUC to > is not greater than that of the right AUC

The transition graphs of two networks where CGLN performed best and worst are visualised in Fig. 3 with 50% of edges in the ground‐truth networks included in the local constraints. Examining the best and worst cases suggests that GLNs with more attractors, i.e. steady states or cycles, were harder to be inferred correctly by CGLN. Similar observations are true for the 30% (Supplementary Figs. S3 and S4) and 70% (Supplementary Figs. S5 and S6) results.

Fig 3.

State transition graphs of two dynamic networks that CGLN inferred most and least accurately

Each box represents a state indexed by the number inside from an artificially generated 8‐node GLN. States marked with a star are attractor states and all other states are transient states. Fifty percent of interactions in the network were used as local constraints and the global constraint includes a sequence of ten dynamic patterns.

a CGLN achieved the best performance or the greatest AUC in this network with a single attractor cycle of three states

b CGLN performed poorly with the least AUC in this network containing 25 attractors

3.2 Protein interaction networks for invadopodium formation in cancer cells

3.2.1 Local and global constraints to protein networks of invadopodium formation

The molecular process of invadopodium formation in cancer cells has been recently reviewed [19, 21 ], and goes through induction, initiation, assembly, and maturation stages. All known protein interactions directly involved in invadopodium formation and their polarities are represented as local constraints in an 18‐node protein network (Fig. 4 ).

Fig 4.

Local constraints are known molecular interactions regulating invadopodium formation

The 17 ovals represent protein molecules or protein complexes. The diamond represents the ECM. Edges represent known physical interactions among proteins that are documented specifically for invadopodium formation [19, 21 ]. The 27 interactions and their polarity (marked along each edge) constitute the local constraints for protein network inference

In the initiation stage of invadopodium formation, invadopodia and focal adhesion have a reciprocal relationship at the site of ECM contact [19 ] and the dissolution of focal adhesion is required for the formation of invadopodia [25 ]. In this process, focal adhesion kinase (FAK) acts as a negative regulator of invadopodia through the control of active SRC proto‐oncogene, non‐receptor tyrosine kinase (SRC) [25 ]: active FAK will attract SRC to the focal adhesions, inhibiting the formation of invadopodia; lack of FAK will release SRC back to the invadopodia, enhancing invadopodium formation. Also in the initiation stage, phosphatidylinositol‐4,5‐bisphosphate 3‐kinase, catalytic subunit alpha (PI3K) creates PtdIns(3,4)P2, which recruits SH3 and PX domains 2A (TKS5) to the invadopodia [26 ]. TKS5 then co‐localises with cortactin at the sites of ECM [27 ]. The phosphorylation of cortactin will then release active cofilin, initiating actin polymerisation [27 ].

In the assembly phase, TKS5 and cortactin bind with other regulators to form the protruding structure of the invadopodium [28 ]. Also, cofilin released from cortactin severs actin filaments and create free barbed ends for actin elongation [29 ]. Actin related protein 2/3 (Arp2/3) also binds to these newly created ends, increasing actin polymerisation and enlarging the protrusions [21 ]. The microtubule network is closely linked with the actin cytoskeleton [30 ] and delivers components to the protruding tip to help the elongation of invadopodia. After the protrusion is assembled, cortactin is dephosphorylated, stopping the action of cofilin, marking the completion of assembly stage and preparing invadopodia for maturation [27 ]. Knockdown of TKS5 will inhibit invadopodium formation [31 ] and MMPs are required for matured invadopodia [32 ]. Cortactin and SH3 and PX domains 2B (TKS4) regulate the secretion and localisation of MMPs respectively [33 ]; the localised MMPs are then secreted to degrade the ECM.

Lastly, serine phosphorylation of actin filament associated protein 1 (AFAP110) and tyrosine phosphorylation of cortactin mark the dissolution of invadopodia, also called the turnover phase [27, 34 ]. During turnover, the receptor, scaffold proteins and other components surrounding invadopodia are dissembled and/or inhibited [35 ]. The actin‐rich core is left and can remain for hours, and is thus not considered part of the turnover phase.

Growth factors, including platelet‐derived growth factor (PDGF), transforming growth factor, beta 1 (TGFβ ), and epidermal growth factor (EGF), induce invadopodium formation by stimulating downstream molecules including integrins, SRC, PKC, reactive oxygen species (ROS), and microRNAs. Integrins are receptors that transmit extracellular signals into the cell and vice versa, which in turn modulate the formation of invadopodia [36 ]. ROS increases the expression of MMPs, activates SRC and PKC and inhibits protein tyrosine phosphatases and some lipid phosphatases [37 ]. Activated SRC has many functions: first, it coordinates with TKS5, AFAP110 (a.k.a AFAP1), and breast cancer anti‐estrogen resistance 1 (p130CAS) [38, 39 ] to positively control invadopodium formation; second, SRC accounts for cortactin phosphorylation, possibly through control of the Abl‐related gene Arg [21 ]; third, SRC connects integrin signals to actin cytoskeleton rearrangement [40 ]; and fourth, SRC functions with PKC, which induces AFAP110, and together the proteins regulate invadopodium formation [41 ].

In the invadopodium protein network (Fig. 4 ), the local constraints consist of 27 known direct interactions. For example, as ROS is known to increase the expression of MMPs during invadopodium formation, we thus have a local constraint lc (ROS, MMPs)=+.

The global constraint of invadopodium formation includes four dynamic patterns representing induction, initiation, assembly, and maturation, respectively [19 ]. They are summarised in those entries marked in Fig. 5 a. For example, when invadopodia mature, the levels of cell division cycle 42 (Cdc42), cortactin, and certain other genes are high, but cofilin and SRC are under‐expressed. In addition, some spatial constraints are captured in the global constraint as well. For example, cortactin controls the secretion of MMPs and TKS4 relocalises MMPs to ECM [19 ]. This implies that both cortactin and TKS4 are essential for MMPs’ function. However, it is improper to encode the relationship between TKS4 and MMPs as a positive local constraint from TKS4 to MMPs. Indeed, whether or not TKS4 is present does not affect the existence of MMPs. Therefore, such requirement of TKS4 for MMPs is captured in the dynamic pattern representing maturation, rather than in a local constraint. It could be interpreted that for invadopodia to mature, TKS4 must be present as well as MMPs. Finally, we use a pseudo node to represent the existence of ECM: a value of 0 for this node denotes that the ECM has been degraded by MMPs; a value of 1 indicates that the ECM is still present and has not been degraded yet.

Fig 5.

Constraint‐satisfying trajectory and an inferred network during invadopodium formation

a Inferred and given dynamic protein patterns during four stages of invadopodium formation. The rows represent proteins in invadopodium formation network. The columns are the time course marked by the four developmental stages. Dark entries are dynamic patterns in the mandatory global constraint; the light entries in the columns of Induction, Initiation, Assembly, and Maturation, not specified in the literature, were inferred by CGLN; and other entries in the unnamed columns are interpolated states by CGLN

b Constraint‐satisfying network inferred by CGLN includes novel protein interactions. The frequency of each novel interaction among the 100 inferred GLNs is marked next to its corresponding edge. The dotted edges are given known protein interactions as local constraints. Four novel interactions (solid edges) are found in the literature, and eight dashed edges are novel protein interactions not previously known

3.2.2 Inferred protein interactions during invadopodium formation

The input to CGLN includes both the local and global constraints in protein networks for invadopodium formation described above. The parameters of CGLN were the same as used in the simulation study: traverse order of adjacency list was set to be random and the search depth limit was set to 5.

CGLN returned 100 GLNs and one of them is visualised in Fig. 5. The dynamics of this GLN are shown in Fig. 5 a with highlighted entries covering the dynamic patterns required for induction, initiation, assembly, and maturation respectively. In addition, eight states were interpolated between the four stages in order for the inferred GLN to satisfy all constraints. This resulting network (Fig. 5 b ) includes 12 novel protein interactions with four of them (FAK $\to$ TKS4, FAK $\to$ cortactin, NCK adaptor protein 1 (Nck1) $\to$ FAK, and PtdIns(3,4)P2 $\to$ TKS4) being known protein interactions in biological processes related to invadopodium formation.

A total of 134 distinct novel interactions (not already in Fig. 4 ) in the resulting 100 constraint‐satisfying GLNs are listed in Table 1. The table also gives the frequencies of each interaction among the 100 GLNs. We found literature support for 14 predicted protein interactions regarding their biological function in contexts relevant to invadopodium formation. Their potential molecular mechanisms and sources of publications are summarised in Table 2. The remaining 120 interactions are unknown in the literature and constitute potential candidates for follow‐up studies.

Table 1.

CGLN predicted 134 protein interactions as ranked by frequency in the 100 constraint‐satisfying networks

Rank	Parent $\to$ Child	Frequency, %
1	Cofilin $\to$ TKS4	33
2	Cortactin $\to$ FAK	32
3	PI3K $\to$ FAK	31
4	TKS5 $\to$ FAK	29
5	Cortactin $\to$ TKS4	28
***6	ROS $\to$ FAK	28
7	PtdIns(3,4)P2 $\to$ FAK	28
8	MMPs $\to$ TKS4	26
9	AFAP110 $\to$ TKS4	26
***10	PtdIns(3,4)P2 $\to$ TKS4	25
11	MMPs $\to$ FAK	25
12	TKS4 $\to$ TKS4	25
13	PKC‐alpha $\to$ TKS4	25
14	ROS $\to$ TKS4	24
***15	Cofilin $\to$ Cortactin	24
16	FAK $\to$ FAK	23
*17	TKS4 $\to$ Cortactin	22
18	Cofilin $\to$ FAK	22
19	TKS4 $\to$ FAK	22
20	AFAP110 $\to$ FAK	22
21	PI3K $\to$ TKS4	21
22	N‐WASP/WIP $\to$ FAK	21
*23	FAK $\to$ TKS4	20
24	ROS $\to$ Cortactin	20
***25	N‐WASP/WIP $\to$ TKS4	19
26	PKC‐alpha $\to$ PKC‐alpha	19
27	Cortactin $\to$ Cortactin	19
***28	SRC $\to$ TKS4	18
29	Cdc42 $\to$ FAK	18
30	FAK $\to$ MMPs	18
***31	FAK $\to$ Cortactin	18
***32	Nck1 $\to$ FAK	17
33	PtdIns(3,4)P2 $\to$ Cortactin	17
34	Nck1 $\to$ PKC‐alpha	17
35	MMPs $\to$ Cortactin	17
***36	TKS5 $\to$ TKS4	17
*37	Integrin $\to$ FAK	17
38	PI3K $\to$ ROS	16
*39	PKC‐alpha $\to$ FAK	16
40	GrowthFactor $\to$ TKS4	16
***41	Cortactin $\to$ AFAP110	16
42	MMPs $\to$ PKC‐alpha	16
43	Nck1 $\to$ TKS4	15
44	GrowthFactor $\to$ Cortactin	15
45	SRC $\to$ PKC‐alpha	15
46	Integrin $\to$ Cortactin	15
47	Integrin $\to$ TKS4	15
48	Cofilin $\to$ TKS5	14
49	PI3K $\to$ Cortactin	14
50	Cortactin $\to$ ROS	14
51	TKS5 $\to$ MMPs	14
52	Cdc42 $\to$ TKS4	13
53	Cdc42 $\to$ Cortactin	13
54	Nck1 $\to$ Cortactin	13
55	PI3K $\to$ MMPs	13
56	N‐WASP/WIP $\to$ PKC‐alpha	13
57	PtdIns(3,4)P2 $\to$ MMPs	13
58	TKS4 $\to$ TKS5	13
59	Cortactin $\to$ PKC‐alpha	13
60	ROS $\to$ AFAP110	13
61	AFAP1 $\to$ PKC‐alpha	12
62	PI3K $\to$ SRC	12
63	AFAP110 $\to$ TKS5	12
64	Cofilin $\to$ SRC	12
65	MMPs $\to$ TKS5	12
66	Cofilin $\to$ PKC‐alpha	12
67	GrowthFactor $\to$ FAK	11
68	TKS4 $\to$ MMPs	11
69	Cdc42 $\to$ PKC‐alpha	11
70	Nck1 $\to$ SRC	11
71	ROS $\to$ TKS5	11
72	FAK $\to$ AFAP110	11
73	FAK $\to$ ROS	11
74	GrowthFactor $\to$ MMPs	11
***75	SRC $\to$ FAK	10
76	PKC‐alpha $\to$ SRC	10
77	PtdIns(3,4)P2 $\to$ ROS	10
78	TKS4 $\to$ PKC‐alpha	10
79	AFAP110 $\to$ Cortactin	10
80	Cofilin $\to$ ROS	10
81	Integrin $\to$ TKS5	10
82	Integrin $\to$ PKC‐alpha	10
83	FAK $\to$ PKC‐alpha	10
84	Integrin $\to$ AFAP110	10
85	MMPs $\to$ SRC	10
86	N‐WASP/WIP $\to$ Cortactin	10
87	AFAP110 $\to$ MMPs	10
88	N‐WASP $\to$ SRC	9
89	AFAP110 $\to$ ROS	9
90	Cofilin $\to$ AFAP110	9
91	Cofilin $\to$ MMPs	9
92	N‐WASP/WIP $\to$ MMPs	9
93	FAK $\to$ TKS5	9
94	PKC‐alpha $\to$ MMPs	9
95	Cortactin $\to$ TKS5	9
96	TKS4 $\to$ AFAP110	8
97	PtdIns(3,4)P2 $\to$ SRC	8
98	TKS4 $\to$ SRC	8
99	MMPs $\to$ AFAP110	8
100	PKC‐alpha $\to$ TKS5	8
101	PKC‐alpha $\to$ ROS	8
102	N‐WASP/WIP $\to$ ROS	8
103	TKS5 $\to$ AFAP110	8
104	PKC‐alpha $\to$ Cortactin	8
105	PI3K $\to$ PKC‐alpha	7
106	Cortactin $\to$ SRC	7
107	TKS5 $\to$ PKC‐alpha	7
108	Integrin $\to$ ROS	7
109	Nck1 $\to$ TKS5	7
110	PI3K $\to$ AFAP110	7
111	GrowthFactor $\to$ TKS5	6
112	Cdc42 $\to$ SRC	6
113	MMPs $\to$ ROS	5
114	SRC $\to$ ROS	5
115	Nck1 $\to$ AFAP110	5
116	MMPs $\to$ MMPs	5
117	Integrin $\to$ SRC	5
118	TKS5 $\to$ SRC	5
119	AFAP110 $\to$ AFAP110	4
120	TKS5 $\to$ TKS5	4
121	ROS $\to$ ROS	4
122	PtdIns(3,4)P2 $\to$ PKC‐alpha	4
123	PI3K $\to$ TKS5	4
124	N‐WASP/WIP $\to$ TKS5	4
125	GrowthFactor $\to$ AFAP110	4
126	Cdc42 $\to$ TKS5	4
127	Cdc42 $\to$ ROS	4
128	PtdIns(3,4)P2 $\to$ AFAP110	3
129	Nck1 $\to$ ROS	3
130	Cdc42 $\to$ AFAP110	3
131	SRC $\to$ SRC	2
132	Nck1 $\to$ MMPs	2
133	Integrin $\to$ MMPs	2
134	N‐WASP/WIP $\to$ AFAP110	1

The rank (first column) is determined by the frequency (third column: frequency) at which each interaction (second column: Parent $\to$ Child) shows up in the 100 constraint‐satisfying networks. These 134 protein interactions include ten known direct directional protein–protein interactions such as phosphorylation, or direct, but directionless protein–protein interactions such as protein complex formation, all marked by ‘***’. There are also four pairs of proteins which interact at least indirectly, as marked by ‘*’. Most of these 14 pairs of protein interactions, though reported in the literature for various contexts, are not yet validated for their role in invadopodium formation. No literature is available for the other 120 interactions.

Table 2.

Novel protein interactions found by CGLN confirmed by the literature and reported in various contexts relevant to invadopodium formation

Interaction (frequency)	Mechanism and supporting publications
Direct (***) interactions
ROS $\to$ FAK (28%)	ROS induces tyrosine phosphorylation of FAK in cancer [42 ]
PtdIns(3,4)P2 $\to$ TKS4 (25%)	PX domain of TKS4 binds PtdIns(3,4)P2 [43 ]
Cofilin $\to$ Cortactin (24%)	Cortactin directly binds cofilin and inhibits its severing activity during invadopodium formation [27 ]
N‐WASP/WIP $\to$ TKS4 (19%)	Four SH3 domains of TKS4 bound weakly to N‐WASP in HEK293 cells [44 ]
SRC $\to$ TKS4 (18%)	SRC phosphorylates TKS4 for invadopodium formation in human colon cancer cells [45 ]
FAK $\to$ Cortactin (18%)	FAK phosphorylates cortactin and a FAK‐cortactin signalling complex is formed in fibroblasts [46 ]
Nck1 $\to$ FAK (17%)	Nck is an adapter in integrin‐mediated signalling that binds to many proteins to recruit them together. Nck forms a complex with FAK in lymphocytes, which use podosomes to move around [47 ]
TKS5 $\to$ TKS4 (17%)	SH3 domains of TKS4 bind the scaffolding protein TKS5, another member of the TKS protein family [44 ]
Cortactin $\to$ AFAP110 (16%)	AFAP110 (AFAP1) interacts with cortactin in podosome formation [48 ]
SRC $\to$ FAK (10%)	SRC phosphorylates FAK. SRC is generally associated with cancer [49 ]
Indirect (*) interactions
TKS4 $\to$ Cortactin (22%)	TKS5 recruits several proteins, including vinculin and cortactin. TKS4 then stimulates this complex to start actin polymerisation [43 ]. If not directly, TKS4 probably interacts with a protein bound onto cortactin
FAK $\to$ TKS4 (20%)	FAK modulates localisation of TKS5 and phosphorylation of TKS5 particularly increased in FAK‐deficient cells [50 ] and TKS4 can complex with TKS5 [44 ]
Integrin $\to$ FAK (17%)	FAK interacts with talin or other cytoskeletal proteins that in turn associate with the cytoplasmic domain of integrin beta subunits in normal cells [51 ]
PKC‐alpha $\to$ FAK (16%)	PKC‐alpha is linked to FAK activation by a molecular adaptor mda‐9/syntenin in a pathway of fibronectin adhesion by human breast cancer and melanoma cells [52 ]

Frequency is the percentage of times an interaction was reported in the 100 constraint‐satisfying networks for invadopodium formation.

The top predicted protein interactions (Table 2 and Fig. 5 b ) centre around the focal adhesion kinase FAK and the tyrosine kinase substrate TKS4, an adaptor protein. They are the targets of 17 and 16 protein interactions, respectively. They are the only two proteins lacking upstream protein connections in the published invadopodium formation network (Fig. 4 ). Indeed, only few interactions involving FAK and TKS4 have been reported in the literature (Table 2 ). Regarding FAK, both ROS [42 ] and SRC [49 ] are involved in the phosphorylation of FAK in cancer and Nck1 forms a complex with FAK in podosomes [47 ]. Regarding TKS4, four SH3 domains of TKS4 bind weakly to Wiskott‐Aldrich syndrome‐like (N‐WASP) [44 ] and also TKS5, another member of the TKS family [44 ], and SRC phosphorylates TKS4 at the PX domain [45 ]. A third protein with heavily predicted novel interactions is cortactin, which promotes actin polymerisation. It binds cofilin to inhibit its severing activity during invadopodium formation [27 ] and was predicted in 24% of the constraint‐satisfying networks. Encouragingly, supported gene interactions relevant to invadopodium formation are densely located in the top ranking interactions (Table 1 ). These results together suggest that CGLN can capture highly relevant interactions by exploiting local and global constraints.

To show how constrained network inference differs from existing approaches, we also applied locally unconstrained and topology constrained approaches to derive protein networks underlying invadopodium formation. The two methods are the same as defined in the simulation study in Section 3.1. Each method generated 100 inferred networks each from a different initial state. The interactions as edges ranked by their frequencies in the inferred networks are shown in Supplementary Tables S1 and S2. Two example inferred networks one from each method are shown as Supplementary Figs. S7 and S8. Both methods generated substantially different results from CGLN. Due to the enforcement of polarity in local constraints and temporal interpolation in the global constraints by CGLN inference, the highest ranked interactions by CGLN (Table 1 ) show more consistency in the inferred networks as indicated by their higher frequencies than those by the other two methods.

4 Discussion

Constrained network inference discovers novel interactions missed by unconstrained approaches given the same observed data. A novel interaction can also be missed by Bayesian methods if its prior probability is low. Therefore, our CGLN method moves beyond the paradigm of Bayes’ rule. In proof‐of‐principle analyses CGLN identified novel interactions involved in invadopodium formation. The qualitative representations for both local and global constraints for CGLN are robust to complex dynamics and biological variations. Local constraints as probabilistic logic rules capture negative or positive molecular interactions. The qualitative dynamic patterns used in global constraints alleviate stringent requirements for normalisation. We showed in our simulation study that when moderate local constraints are available, the CGLN exhibits an extraordinary power in recognising novel interactions.

Representation of dynamic networks by state‐transition graphs requires memory exponential to the number of vertices in the graph [53 ]. In the 18‐node network for invadopodium formation (Fig. 4 ), there are 2¹⁸ = 262,144 vertices in the state‐transition graph. If every vertex has 5000 adjacent vertices, it needs about 10 GB memory to store all adjacency lists. One possible solution is to generate adjacency lists on‐the‐fly: build the adjacency list of each vertex as needed and destroy it after use. In our implementation, when the number of adjacent vertices is greater than 100 000 and the search depth limit is greater than 10, the SIDDFS algorithm automatically switches to the on‐the‐fly strategy. Since SIDDFS is a variant of iterative‐deepening DFS, saving in space comes at the expense of possible multiple visits to the same vertex.

The novel protein interactions predicted as being necessary for invadopodium formation suggests the incompleteness of previously reported protein interaction networks. However, the constraint‐satisfying networks are divergent: not a single interaction is predicted to be required for all networks. To further reduce the search space of the novel interactions, one can test those top ranked interactions and thus eliminate a large number of candidate networks. Using CGLN iteratively, one is equipped to efficiently zoom into the most likely network until all constraints are fully resolved with confirmed edges.

Several simplifications were made in reconstructing protein networks underlying invadopodium formation. Since the detailed control from growth factors (PDGF, TGFβ, and EGF) to downstream molecules are yet to be fully documented in the literature, we decided to use one node ‘Growth Factor’ in the network to collectively represent all growth factor stimuli. Similarly, the node ‘Integrin’ is used to represent a set of integrins found in different types of cells. Furthermore, microRNAs were not included in the modelling as their dynamics or interactions were not described at the protein level for invadopodium formation in the literature. In addition, MMPs (e.g. MT1MMP) could be delivered to invadopodia in three different ways [19 ], but in our model, we assume that the presence of MMPs will contribute directly to invadopodium assembly. All these simplifications were made due to unavailability of data and could have important consequence on the inferred protein network models. As the data become more available, the constrained network inference can be re‐run to incorporate new information to further approach the true underlying molecular mechanisms for invadopodium formation.

5 Conclusions

As it is yet feasible to build a complete biological network from data collected in one experiment, we explored the CGLN method that searches for expanded networks to satisfy both local and global constraints to a dynamic network. CGLN inferred dynamic networks that aligned best with the existing literature, and still encouraged novel interactions because no prior probabilities were used. Predicted novel interactions among proteins mediating invadopodium formation constitute new components in the underlying molecular mechanisms for cancer cell migration. Such knowledge can potentially expedite identification of target molecules in cancer cells to block invadopodium formation, which is key to tumour invasion and metastasis. We anticipate that constrained network inference will provide new opportunities to progressively expand our understanding of novel molecular mechanisms in complex biological systems.

6 Acknowledgments

The reported work was supported in part by a New Mexico Cowboys for Cancer Research grant to AWN, LGH, and MS and grants to MS under US NIH Grant 2P20GM103451‐14 New Mexico IDeA Networks of Biomedical Research Excellence, US NIH Mountain West Clinical Translational Research Grant 1U54GM104944‐2, and US NSF MRI Grant CNS‐1337884.

Supporting information

Sup 1: Supplementary Data

Sup 2: Supplementary Data