1 protein interaction networks :...

Biological Networks, CSCI 3352Lecture 7

Prof. Aaron ClausetSpring 2020

1 Protein Interaction Networks : Structure

1.1 What are proteins and what do they do?

Proteins are large molecules, created by cells, that carry out specific cellular functions. They arethe machinery of the cell. Every protein is encoded in an organism’s genome by a specific gene.Each proteins is built from some specific string of amino acids, which then folds up into a 3d shape.The folding process itself can be simple or complicated (sometimes requiring the presence of addi-tional proteins or other molecules to fold properly). Once folded, we often say that proteins arecomposed of different “domains.”1 Some of these domains create structures like a pocket or otherbinding-ready shape, and it is through these structures that proteins mainly interact.

There are two main types of binding: (i) stable binding, in which the two proteins stay stuck to-gether after the binding event, and (ii) transient binding, in which they do not. Transient bindingcan be short, or long, and is the type of binding used in information relays as in a signaling pathway.

When we visualize proteins, there are three ways to shown them (see Fig. 1 below)2 : (i) cartoons,(ii) helices and sheets (also called “secondary structure”),3 and (iii) atomic structures (visualizingindividual atoms). For this class, the “cartoon” visualization is sufficient, and we will generally notcare about secondary structures or protein domains. The cartoon is sufficient to capture the ideathat a protein is a thing that binds to something else.

Figure 1: Visualizing proteins as (a) cartoons and (b) helices and sheets, and an atomic structure.

1See https://proteinstructures.com/Structure/Structure/protein-domains.html2Source: https://bit.ly/2v6xoPp3See https://proteinstructures.com/Structure/Structure/secondary-sructure.html

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1.2 Protein interaction databases

There are many databases that list protein interaction data. The STRING database is one thatincludes both known interactions (meaning: those with evidence from experiments) and predictedinteractions (meaning: predicted based on protein domain structure, co-expression data, etc.). Inthe “v11” version, STRING covers nearly 10,000,000 proteins across more than 2000 organisms,making it one of the largest available.

Here is the basic search box looks for interfacing with STRING at https://string-db.org/, alongwith one result obtained via the “Random entry” option:

In this case, STRING picked the protein ANI_1_2124 (proteins generally have names like this;sometimes they’re meaningful, but usually they’re just a unique string), and then shows us its localneighborhood in the PPIN. Initially, this view into the network shows just immediate neighbors,but we can use the +More button to show nodes +1 hop away.4 The red node in the middle isANI_1_2124.

Below this visualization is a Legend that illustrates the rich kinds of network data that STRINGstores, including different types of edges, edge weights, edge signs, node metadata, etc. It is aremarkable amount of information. And yet, overall, we know remarkably little about the set ofproteins, their properties, and their interactions. Such is the scale of the task of just describing allthe pieces of a PPIN.

4Source https://version-11-0.string-db.org/cgi/network.pl?networkId=pNkiZSN3VyXo

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1.3 Signaling pathways

If we zoom in on certain parts of the full PPIN, we can find what are called “signaling pathways,”which are a set of nodes and edges (usually directed) that represent a kind of cellular sensor. Someof these are extremely well studied because when they break, they lead to diseases or other issuesof particular interest. For instance, the MAPK pathway is used by cells to detect the presence ofa protein called EGF (epidermal growth factor) outside their cell membrane, and, if it is, changewhat genes are expressed in the nucleus (contributing to cell proliferation). Changes in the MAPKpathway are implicated in diseases including cancer, inflammatory diseases, obesity, and diabetes.5

extracellular nucleus

EGFR

GRB2

SOS

RAS

RAF

MEKEGF

MYC

CREBMNK

MAPKS6RSK transcription

P

P

P

P

P

P

P

Pcytoplasm

Figure 2: The MAPK pathway, which detects the presence of the EGF protein outside a cell.

But, the MAPK pathway is itself embedded within a larger signaling network, composed of manymore proteins. The visualization below shows this larger network, which also contains the WNT

5See Lawrence et al., “The roles of MAPKs in disease.” Cell Research 18, 436–442 (2008).https://www.nature.com/articles/cr200837

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pathway, another well-studied pathway implicated in a variety of diseases,6 and a variety of intra-cellular proteins that interact with these pathways. Within this larger network, we may observethat there are two different kinds of protein interactions shown: activation edges (an arrow) andinhibition edges (a line with a blunt end). For instance, the structure around the JNK node (in themiddle) indicates it has five input interactions: two activation inputs and three inhibition inputs.Hence, JNK works like an elaborate conditional function in a programming language: only if all itsinputs are satisfied, will it be able to activate itself and continue the information cascade towardthe nucleus.

Figure 3: The MAPK and WNT signaling pathways, along with a variety of intracellular proteinsthat interact with them. EGF is located in the upper-left corner.

6Source: https://en.wikipedia.org/wiki/File:MAPKpathway.png

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1.4 Where do the data come from?

Data in a PPIN come from one of four main sources:

1. yeast two-hybrid (Y2H)Y2H is an edge sampling experiment, which tests for binding between a specific pair of proteinsX (the “bait”) and Y (the “prey”). Y2H does this by inserting the potential interactionbetween the binding and activation domains (BD and AD below) of the Gal4 transcriptionfactor so that the downstream gene is only expressed if X and Y successfully bind.

upstream activating sequence reporter gene

Gal4 AD

Gal4 BD

bait X

Y

High-throughput versions of Y2H allow testing for many such pairings in parallel, but it alsohas many limitations. In particular, the Y2H method works best for nuclear proteins, andit can have a high false positive rate (activated gene is expressed even when X and Y don’tbind) and a high false negative rate (no expression even when X and Y do bind). Variousimprovements to Y2H have aimed to mitigate some of these issues.

2. affinity purification and mass spectrometry (AP-MS)AP-MS is a node sampling method, which takes a protein of interest X and make an arrayof them on a plate. We then wash the plate with a mix of other proteins and use mass spec.to identify what stuck to the Xs.

candidateproteins

plate of X plate of X +bound proteins

non-boundproteins

AP-MS can also have a high false positive and a high false negative rate, but for differentreasons than Y2H. The false positives come in part from the fact that we mass spec. everythingon the plate, meaning we cannot tell the difference between all proteins {Y, Z,A} binding toX from a “chain” of bound sequentially bound proteins, e.g., the edges (X,Y ), (Y,Z), (Z,A),of which only Y binds directly with X. False negatives from from weak binding events wherenone of the protein Y remains bound to X after washing the plate. Various improvements toAP-MS have aimed to mitigate some of these issues, as well.

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3. text miningWe can also use natural language processing methods to trawl the scientific literature toidentify interactions that were reported via various (typically low-throughput) experimentalapproaches. This approach also has a non-trivial false positive and false negative rate, becauseit can be tricky to correctly identify an interaction from a paper’s language alone.

4. prediction from protein domain analysis, etc.Based on the secondary and tertiary structures of two proteins, e.g., the shape and locationof their binding domains, we may also be able to predict that proteins X and Y interact byidentifying that X has a domain that could bind to one of Y ’s domains.

It is important that each of the above methods has non-trivial false positive and false negativerates. Because real-world networks are typically sparse, and PPINs are no different, there will beΘ(n2) non-edges, each of which could be misclassified as an edge (a false positive), and Θ(n) trueinteractions, each of which could be misclassified as a non-edge (a false negative). It is likely thatin the real PPIN data we have, a great many of the observed interactions (derived using the abovemethods) are in fact false positives, and a great many of the true interactions are missing (falsenegatives). Basically, PPIN data is messy, and it is important to bear that in mind as we learnabout what we think we know about their structure.

In addition, PPIN data typically omits other types of biologically useful information about proteins:

• binary interactions only, meaning we don’t have edge weights (binding affinity) or edge sign(activation or inhibition);

• no data on where in the cell the protein occurs, or when in the cell cycle the protein isexpressed, meaning two proteins that bind in an experiment may never interact in the cell;

• no data on protein concentrations in the cell, i.e., is it a rare or abundant protein; and

• many proteins are missing functional labels, meaning we know they exist, but we do not havemuch sense, or a complete sense, or what cellular functions they are involved in.

1.5 What do people do with PPINs?

1. fill in the detailsThere is value in just having as complete a description of the PPIN as possible, having acomplete list of proteins, and a complete list of their interactions, and a complete list ofannotations on the edges and nodes. On the computational side of things, this often means• predict missing node attributes (functional labels, etc.)• predict missing (and spurious) links

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2. find modules (“the building blocks of complexity”)• these may be small, as in “complexes”• these may be large, as in communities (the way we’ve defined them in this class)

3. understand diseaseThe PPIN is different from social networks in that it is functional, meaning it carries outspecific functions and its correct functionality is essential to life. When it breaks, it cancreate disease.• understand the pathways and partners of proteins associated with specific diseases, e.g.,Alzheimers, Parkinson’s, Schziophrenia (see below), various cancers, etc.• understand what specific proteins regulate, and what regulates them, both of which mightbe targets for therapies• what modules and pathways are associated with specific diseases, which ones are sharedacross many diseases• modeling the interaction dynamics of these pathways to understand how they lead to disease• do specific (local) network structures induce greater or lesser resilience to disease states?

4. systems biology• what can the architecture of the entire PPIN tell us about how cells work?• are there general principles that structure it?• the PPIN may look like spaghetti, but it is evidently both highly functional and highlyrobust. How is that possible?

The tools researchers use in these efforts are familiar:

• motifs (small subgraphs) and their frequencies relative to null models. Motifs are believedto act like little logical circuits, and hence hooking them together in a network should (weimagine) create a kind of dynamic computational program. Finding the motifs that are moreabundant than we would expect (under the null) may offer hints about how it works.

• modules, meaning communities, which provide a coarse graining of the network that allowsus to think about it as a lower-dimensional object

• pathways, especially in dynamic models, where we model the abundance of individual pro-teins and their sequential activation

• hubs (“high” degree nodes), for example, whether they interact with many other proteinsat the same time (“party hubs”), or over time (“date hubs”). There is often great emphasisplaced on proteins with “high” degrees (aka, hubs), but it’s not clear how much of theirapparent importance is driven entirely by the fact that they simply have more connections(and thus more chances for being connected to other things).

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1.6 From structure to dynamics: computing with protein interaction networks

What makes protein interaction networks interesting is that they are functional, meaning theirdynamics produces useful phenotypic behavior for the cell. There are three general approaches tomodeling how the structure of a PPIN produces biological function:

1. rate (differential) equation models, which aim to capture the temporal dynamics ofprotein concentrations and rely on detailed knowledge of the biochemical kinematics of theassociated interactions,

2. linear models, which approximate the rate equations with simple linear functions, and

3. boolean network (logical) models, which approximate the linear models with simplybinary logic.

These models are typically deterministic, but more realistic versions can incorporate stochasticity.In all three models, the crucial beginning assumption is that the network structure G is known.For simplicity, we start with linear models.

1.6.1 A simple, linear circuit

In signaling, protein interactions can operate like little circuits. For instance, consider a pair ofmembrane-bound proteins (the pacmans below) that activate molecules X1 and X2, respectively,when they detect their respective preferred molecule outside the cell membrane (green band). Bothof these molecules can then bind to Y , but do so with different weights w1 and w2, respectively.

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In a linear model of this system, we ignore the underlying rate equations that govern how thedynamics actually work. Let X1 and X2 denote concentrations ranging over [0, 1], and let w1 andw2 denote each molecule’s “binding strength” with Y . We then say that Y becomes activated ifthe linear combination of its inputs exceeds a threshold:

Y =

{1 if X1w1 + X2w2 > 10 otherwise.

The right-hand plot above maps out the response of Y to variations in X1 and X2: the two shadedregions represent the joint concentrations X1 and X2 that activate Y , for two different choices of

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weights w1, w2. (The values of w1, w2 determine the slope and intercept of the line that divides theinput space into Y = 1 and Y = 0. Do you see why?) In this sense, we can say that this motif,i.e., this subgraph of the larger PPIN, computes this function.

1.6.2 Combining linear circuits

Now consider putting two of the above circuits in parallel with each other, so that X1 and X2 bothfeed into molecules Y1 and Y2, and these two in turn feed into a new molecule Z. Now we have sixfree parameters, and three corresponding “activation plots,” one for the activation of each of Y1,Y2, and Z.

But, the interesting part comes in the way the two input plots interact to produce the plot for Z:if the weights w1 and w2, which feed into Z are both very strong, then activating either Y1 or Y2 issufficient to activate Z. But if they are individually too weak to activate Z, but are strong enoughtogether, then we must activate both Y1 and Y2 in order to activate Z. That is, depending on howwe set the weights (w1, w2), this little circuit is either an OR gate or an AND gate for X1 and X2.(In this example, all the weights are positive, representing excitatory interactions, but negativeweights are possible, which represent inhibitory interactions.)



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w1 > 1, w2 > 1<latexit sha1_base64="KGqMOW8PH8p3M4aVYl+IIb1KpHM=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBg5RNLeipFLx4rGA/oF2WbJq2odnskmQtZenf8OJBEa/+GW/+G9N2D9r6YODx3gwz84JYcG1c99vJbWxube/kdwt7+weHR8Xjk5aOEkVZk0YiUp2AaCa4ZE3DjWCdWDESBoK1g/Hd3G8/MaV5JB/NNGZeSIaSDzglxkq9iY9r+ApN/EoN+8WSW3YXQOsEZ6QEGRp+8avXj2gSMmmoIFp3sRsbLyXKcCrYrNBLNIsJHZMh61oqSci0ly5unqELq/TRIFK2pEEL9fdESkKtp2FgO0NiRnrVm4v/ed3EDG69lMs4MUzS5aJBIpCJ0DwA1OeKUSOmlhCquL0V0RFRhBobU8GGgFdfXietShlflysP1VK9msWRhzM4h0vAcAN1uIcGNIFCDM/wCm9O 4rw4787HsjXnZDOn8AfO5w+5f5Ah</latexit>

w1 < 1, w2 < 1<latexit sha1_base64="Z/H/U5EOPNVNDyj47OsSAbAF1lA=">AAAB83icbVA9SwNBEJ2LXzF+RS1tFoNgIeE2BrRIEbCxjGA+IDmOvc0mWbK3d+zuGcKRv2FjoYitf8bOf+MmuUITHww83pthZl4QC66N6347uY3Nre2d/G5hb//g8Kh4fNLSUaIoa9JIRKoTEM0El6xpuBGsEytGwkCwdjC+m/vtJ6Y0j+SjmcbMC8lQ8gGnxFipN/FxDV+hiV+pYb9YcsvuAmid4IyUIEPDL371+hFNQiYNFUTrLnZj46VEGU4FmxV6iWYxoWMyZF1LJQmZ9tLFzTN0YZU+GkTKljRoof6eSEmo9TQMbGdIzEivenPxP6+bmMGtl3IZJ4ZJulw0SAQyEZoHgPpcMWrE1BJCFbe3IjoiilBjYyrYEPDqy+ukVSnj63LloVqqV7M48nAG53AJGG6gDvfQgCZQiOEZXuHNSZwX5935WLbmnGzmFP7A+fwBs12QHQ==</latexit>

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Hence, simple circuits can be combined to compute more complicated functions over the inputspace, and this is what we imagine that signaling pathways do; the more complicated the pathway,the more complicated a function the circuit can compute.

However, there are several large caveats for applying these ideas more broadly to analyzing a PPIN:

1. Generally, we only know the weights (binding strengths, etc.) for a small subset of very wellstudied protein interactions, even in crucial pathways like MAPK. Hence, we cannot perform

9



large-scale analysis of the kind of functions that the PPIN computes, because changing theweights necessarily changes the function (as the example above shows).

2. Drawing small circuits is easy in isolation, but real PPINs, and even just real signalingpathways, are a mess of cross-talking, interacting, overlapping circuits, and it’s a big stretchto go from understanding how to build an OR or AND gate to understanding the architectureof an entire computer (or cell).

For these reasons, developing a realistic linear model of how PPINs compute often still remainsoutside our reach. These constraints also tend to constrain the use of rate equation models tovery simple circuits, like the well-studied Lac operon in bacteria that controls the expression of thelactase enzyme.

1.7 Boolean circuits

In a boolean network model, we let each node have a binary state si ∈ {0, 1}. The particularassignment of state values s ∈ {0, 1}n defines the system’s state, and the following simple updaterules define the way the network traverses this space:

si(t + 1) =

{1 if

∑nj=1Aijsj(t) + c > 0

0 otherwise,

where A is the adjacency matrix,c is a threshold, as in the linear model case, and we update allstates from si(t) to si(t + 1) synchronously. Letting the network be a signed graph, in whichAij ∈ {−1, 0, 1} allows this model to capture both activation (Aij = 1) and inhibition (Aij = −1)interactions. (Do you see how this model is a simplification of the linear model in Section 1.6.1?)

1.7.1 A simple circuit

To see how a boolean network works, consider two very simple circuits: a feedback loop (FBL)with inhibition, and a feedback loop with activation. Let’s see how these systems evolve when weinitialize them to a few different states, with c = 0 in the update equation.

s1

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s2

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s3

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+1

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�1

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+1


s1

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s2

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s3

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+1


+1


feedback loop (inhibition) feedback loop (activation)

+1


10



• In both cases, if all nodes are inactive, then s(0) = 000. Applying our update equation, wefind that s(1) = s(0), meaning this state is an “attractor,” and specifically a “fixed point,”meaning that once the system reaches this state, it states in it forever.

• For the inhibition FBL, if we initialize the system as s(0) = 100, we then visit s(1) = 010,s(2) = 001, and finally s(3) = 000, as before. Qualitatively, this circuit runs once, and thenturns off. If there were an active input to s1, and an output at s3, this circuit would producea “pulse” signal that emerges from s3 exactly once every three time steps, but only so longas the input were active.

• For the activation FBL, if we initialize the system as s(0) = 100, we then visit s(1) = 010,s(2) = 001, and then back to s(3) = s(0), and so on forever. This sequence of states is a“limit cycle,” the remains active forever, once turned on. If there were an output attachedto s3, this circuit would generate the same “pulse” signal as the other FBL, but would do soeven after s1 was deactivated.

1.7.2 More complicated circuits

Boolean networks can be used to model more complicated systems. For example, they have beenused to illustrate the robustness of the regulatory network structure that governs the cell cycle inthe budding yeast Saccharomyces cerevisiae, a single-cell model eukaryotic organism.7

The yeast cell cycle consists of four phases: G1 → S → G2 → M → G1, which correspond to cellgrowth, DNA synthesis, a “gap” phase, and finally the division phase, after which each daughtercell returns to the G1 state and awaits a signal to move to initiate another cycle by moving toS. Although more than 800 genes are involved in the overall cell cycle, a simple model of 11 keyproteins, plus one “check-point” state, provides a minimal representation of the regulatory network(Fig. 4).

With 11 nodes, there are 211 = 2048 possible states for this system, which is small enough toenumerate every state transition s(t) → s(t + 1), for all possible choices of s(t). (The exponentialsize of the state space means this approach quickly becomes computationally intractable for evenmodest-sized boolean networks.) These pairs are themselves edges in a graph where nodes areentire states of the system, and analyzing the structure of that graph lets us understand the kindof dynamics the boolean network produces. Because each state “points” to a single next state, thisgraph is a forest, and each component is a “basin of attraction,” in which starting the system atany node in that tree will lead it to converge on a fixed point or a limit cycle. The larger the tree,the more states in the basin.

7Li et al., “The yeast cell-cycle network is robustly designed.” Proc. Natl. Acad. Sci. USA 101(14), 4781–4786(2004). https://www.pnas.org/content/101/14/4781

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In this model of the yeast cell-cycle network, there are 1764 states (86%) are connected in a singletree, and a line of 13 of those states maps precise onto the four phases of the cell cycle: a STARTstate, three G1 states, an S state, five M states, and then two G1 states (Fig. 4). The remaining14% of states group into smaller basins, composed of 151, 109, 9, 7, 7, and 1 state.

The large size of the largest component in this “state space” graph means that yeast’s cell-cyclenetwork is very robust. That is, for a very large number of perturbations to the system’s statevector, i.e., changes to s, the network will “relax” back to the main sequence of states representingthe four cell-cycle phases.

where aij ! ag for a green arrow from protein j to protein i andaij ! ar for a red arrow from j to i. We first focus on the casewhere all of the other checkpoints are always off, except thatof the cell size. That is, the cell size checkpoint will act as a startsignal, whereas other checkpoints will always let the ‘‘traffic’’pass when it come to it. We therefore arrive at a slightlysimplified network shown in Fig. 1B, with 11 nodes (plus asignal node). We have also added ‘‘self-degradation’’ (yellowloops) to those nodes that are not negatively regulated byothers. [This is a simplification for the actual degradationprocesses. See supporting information for details.] The deg-radation is modeled as a time-delayed interaction: if a proteinwith a self yellow arrow is active at time t (Si(t) ! 1) and if itstotal input is zero from t " 1 to t ! t " td, it will be degradedat t ! t " td, i.e., Si(t " td) ! 0. The results presented belowwere obtained with ag ! #ar ! 1 and td ! 1. As will bediscussed later, the overall dynamic properties of the networkare not very sensitive to the choice of these parameters.

Fixed Points. We use the dynamic model described above to studythe time evolution of the protein states. First, we study theattractors of the network dynamics by starting from each of the211 ! 2,048 initial states in the 11-node network of Fig. 1B. Wefind that all of the initial states eventually f low into one of theseven stationary states (fixed points) shown in Table 1. Amongthe seven fixed points, there is one big fixed point attracting 1,764or $86% protein states. Remarkably, this super stable state isthe biological G1 stationary state. The advantage for a cell’sstationary state to be a big attractor of the network is obvious:the stability of the cell state is guaranteed. Under normal

conditions, the cell will be sitting at this fixed point, waiting forthe signal for another round of division.

Biological Pathway. Next, we start the cell-cycle process by ‘‘ex-citing’’ the G1 stationary state with the cell size signal, andobserve that the system goes back to the G1 stationary state. Thetemporal evolution of the protein states, presented in Table 2,indeed follows the cell-cycle sequence, going from the excited G1state (the START) to the S phase, the G2 phase, the M phase,and finally to the stationary G1 state. This is the biologicaltrajectory or pathway of the cell-cycle network.

To investigate the dynamical stability of this biological path-way, we study the dynamic trajectories of all 1,764 protein statesthat will f low to the G1 fixed point. In Fig. 2, each of these proteinstates is represented by a dot, with the arrows between themindicating dynamic flows from one state to another. The bio-logical pathway is colored in blue and so is the node representingthe G1 stationary state. We see that the dynamic flow of theprotein states is convergent onto the biological pathway, makingthe pathway an attracting trajectory of the dynamics. With sucha topological structure of the phase diagram of protein states, thecell-cycle pathway is a very stable trajectory; it is very unlikelyfor a sequence of events, starting at the beginning (or at anyother point) of the cell-cycle process, to deviate from thecell-cycle pathway. Interestingly, the topology of the convergingtrajectories shown in Fig. 2 is reminiscent of the convergingkinetic pathways in protein folding where a protein sequence isfacing the challenge of finding the unique native state among ahuge number of conformations (10–12).

Comparison with Random Networks. To investigate how likely a bigfixed point and a converging pathway can arise by chance, we

Fig. 1. (A) The cell-cycle network of the budding yeast. (B) Simplified cell-cycle network with only one checkpoint ‘‘cell size.’’

Table 1. The fixed points of the cell-cycle network

Basin size Cln3 MBF SBF Cln1,2 Cdh1 Swi5 Cdc20 Clb5,6 Sic1 Clb1,2 Mcm1

1,764 0 0 0 0 1 0 0 0 1 0 0151 0 0 1 1 0 0 0 0 0 0 0109 0 1 0 0 1 0 0 0 1 0 0

9 0 0 0 0 0 0 0 0 1 0 07 0 1 0 0 0 0 0 0 1 0 07 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0 0 0

Each fixed point is represented in a row. The first column is the size of the basin of attraction for the fixed point; the other 11 columnsshow the protein states of the fixed point. The protein states of the biggest fixed point correspond to that of the G1 stationary state.

4782 ! www.pnas.org"cgi"doi"10.1073"pnas.0305937101 Li et al.

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study an ensemble of random networks (13, 14) that have thesame numbers of nodes and links in each color as in the cell-cyclenetwork. We find that random networks typically have more

attractors (fixed points and limit cycles), with the averagenumber being 14.28. The sizes of the basins of attraction in therandom networks have a power-law distribution, as shown in Fig.

Table 2. Temporal evolution of protein states for the simplified cell-cycle network of Fig. 1B

Time Cln3 MBF SBF Cln1,2 Cdh1 Swi5Cdc20 and

Cdc14 Clb5,6 Sic1 Clb1,2 Mcm1!SFF Phase

1 1 0 0 0 1 0 0 0 1 0 0 START2 0 1 1 0 1 0 0 0 1 0 0 G1

3 0 1 1 1 1 0 0 0 1 0 0 G1

4 0 1 1 1 0 0 0 0 0 0 0 G1

5 0 1 1 1 0 0 0 1 0 0 0 S6 0 1 1 1 0 0 0 1 0 1 1 G2

7 0 0 0 1 0 0 1 1 0 1 1 M8 0 0 0 0 0 1 1 0 0 1 1 M9 0 0 0 0 0 1 1 0 1 1 1 M

10 0 0 0 0 0 1 1 0 1 0 1 M11 0 0 0 0 1 1 1 0 1 0 0 M12 0 0 0 0 1 1 0 0 1 0 0 G1

13 0 0 0 0 1 0 0 0 1 0 0 Stationary G1

The right column indicates the cell-cycle phases. Note that the number of time steps in each phase do not reflect its actual duration.

Fig. 2. Dynamical trajectories of the 1,764 protein states (green nodes) flowing to the G1 fixed point (blue node). Arrows between states indicate the directionof dynamic flow from one state to another. The cell-cycle sequence is colored blue. The size of a node and the thickness of an arrow are proportional to thelogarithm of the traffic flow passing through them.

Li et al. PNAS " April 6, 2004 " vol. 101 " no. 14 " 4783

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Figure 4: (left) A simplified model of 11 key factors that regulate cell division in yeast, whichspan four classes: 4 cyclins (Cln1,2; Cln3; Clb1,2; Clb5,6, which bind to the kinase Cdc28); 3inhibitors, degraders, and competitors of the cyclin/Cdc28 complexes (Sic1, Cdh1, Cdc20&Cdc14);4 transcription factors (SBF, MBF, Mcm1/SFF, Swi5); and 1 check-point (Cell Size). Greenarrows represent activation, and red inhibition. The cell cycle begins by activating Cell Size, whichactivates Cln3. (right) The largest component of state transitions, covering 1764 (86%) statestotal, with the main sequence of states corresponding to the four cell-cycle phases shown in blue.(Reproduced from Li et al. (2004).)

1.8 The structural compromise: Counting motifs

As a compromise to the computational complexity of modeling biological circuit dynamics, re-searchers often resort to a structural approach: identify and count subgraphs that are computa-tional building blocks. When doing this, we typically draw a distinction between a subgraph (a

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specific, and usually small set of nodes and edges) and a motif:

A motif is a subgraph that appears more often than expected.

The latter part of this definition implies a choice of a null model, i.e., some Pr(G). We have alreadyseen several such possibilities (see Lectures 3 and 5). Usually, researchers choose the configurationmodel as Pr(G), meaning they are measuring the abundance of a given motif of interest relative toa random graph with the same degree sequence.

Before we can count motifs, however, we must choose what counts as a new subgraph. There aretwo choices:

1. non-identical subgraphs, meaning subgraphs that different in at least 1 node or edge

2. edge-disjoint subgraphs, meaning two subgraphs that do not share any edges, i.e., we cannotreuse a node or edge in a second subgraph, once it’s been included in another.

To illustrate the distinction, consider the following example. On the left is a graph G, and on theright are two possible motifs. Under the edge disjoint definition, there’s 1 subgraph of type (a) or1 subgraph of type (b). (Do you see why?) In contrast, under the non-identical definition, there’sstill only 1 subgraph of type (a), but now we can find 6 subgraphs of type (b). Which of these isthe right way to count? The answer is: it depends. It depends on the type of scientific question wewant to answer.

(a) (b)

The maximal motif Gs is defined as a subgraph of G that is not contained in any other subgraph ofG that is a motif. That is, it’s the largest subgraph that is also a motif. If Gs is contained withinanother motif, then it’s not the maximal one.

Suppose we have an algorithm that can count how many of a given motif Gk are within a graphG, which we can denote f(Gk). To determine whether or not that count f(Gk) is more than weexpect, we often calculate a z-score under the given null model:

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z-score =f(Gk)− E[f(Gk)]null√

V[f(Gk)]null(1)

Basically, this calculation imagines that f(Gk) is drawn from a Normal distribution, and convertsit into units of the standard deviation of that distribution. There are basically three possibilities:

1. If z = 0 (or close to it), then f(Gk) is about what we would expect under the null.

2. If z > 0 (by some margin, often 2), then f(Gk) is higher than we expected under the null.

3. If z < 0 (by some margin, often 2), then f(Gk) is lower than we expected under the null.

Nearly any null model of graphs that we might use, e.g., the configuration model or the Chung-Lu model, will generate a distribution of motif frequencies Pr(Gk) that is not Normal, meaning az-score should only be interpreted as a (rough) descriptive statistic, rather than as a test statistic.8

1.8.1 Algorithms for counting

There are basically two kinds of algorithms for counting motifs. The key difficulty that all ofthese algorithm face is accounting for what’s called subgraph isomorphism. Two subgraphs areisomorphic if we could permute the names of the nodes, and still have the same motif. In the aboveexample, motif (a) is a triangle, and there’s one of them in the left-hand graph. But, we couldwrite down the names of the three nodes that compose it in 3! = 6 different ways, and hence anyalgorithm would need to avoid over counting these isomorphisms. (The larger the subgraph, themore isomorphisms it will have.)

1. exhaustive search, which is usually very slow, and works mainly for k < 5. These algorithmsoften enumerate all possible listings of the k vertices, and then checks if they have the targetset of edges among them. These algorithms run in time O(nk). They can be made somewhatmore efficient if we are smart about recognizing how motifs can be nested, e.g., the motif (b)above contains (a) as a subgraph, and so if there are no (a) motifs in a graph, there can beno (b) motifs either, and we don’t need to check for them.

2. sampling, which is faster than exhaustive search, but still not scalable; it works mainly fork < 20. There are various strategies for sampling, one of which is simply to sample the ex-haustive search approach. A different approach is to find some subgraph in G, repeatedly, andto track which ones are found. In this approach, a problem arises, which is that some motifsare easier to find (sample) than others, depending on the method of grabbing a subgraph outof the larger graph G. Fixing this problem requires re-weighting the estimated frequencies inorder to produce an unbiased estimate.

8A better statistic can be found in Picard et al., “Assessing the Exceptionality of Network Motifs.” Journal ofComputational Biology 15(1), 1–20 (2008). http://doi.org/10.1089/cmb.2007.0137

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Supplemental readings

Protein-protein interaction networks:

1. Szklarczyk et al., “STRING v10: proteinprotein interaction networks, integrated over thetree of life.” Nucleic Acids Research 43, D447D452 (2015).https://www.ncbi.nlm.nih.gov/pubmed/25352553

2. Giot et al., “A Protein Interaction Map of Drosophila melanogaster.” Science 302, 1727(2003).https://www.ncbi.nlm.nih.gov/pubmed/14605208

3. Chautard et al., “Interaction Networks: From Protein Functions To Drug Discovery. Areview.” Pathologie. Biologie. 57, 324-333 (2009).https://www.ncbi.nlm.nih.gov/pubmed/19070972

Boolean network models:

1. Bornholdt, “Boolean network models of cellular regulation: prospects and limitations.” J. R. Soc.Interface 5, S85–S94 (2008).https://dx.doi.org/10.1098/rsif.2008.0132.focus

2. Grieco et al., “Integrative Modelling of the Influence of MAPK Network on Cancer Cell FateDecision.” PLoS Comput. Biol. 9(10), e1003286 (2013).https://dx.doi.org/10.1371/journal.pcbi.1003286

Motif discovery:

1. Ciriello and Guerra, “A review on models and algorithms for motif discovery in protein-protein interaction networks.” Briefings in Functional Genomics and Proteomics 7, 147–156(2008).https://www.ncbi.nlm.nih.gov/pubmed/18443014

2. Ahmed al., “Efficient Graphlet Counting for Large Networks.” IEEE International Confer-ence on Data Mining (ICDM) (2015).http://nesreenahmed.com/publications/ahmed-et-al-icdm2015.pdf

3. Grochow and Kellis, “Network Motif Discovery Using Subgraph Enumeration and Symmetry-Breaking.” RECOMB (2007).http://compbio.mit.edu/publications/C04_Grochow_RECOMB_07.pdf

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