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Sequential Sampling Designs for Small-Scale Protein Interaction Experiments Denise Scholtens, Ph.D. Associate Professor, Northwestern University, Chicago IL Department of Preventive Medicine, Division of Biostatistics Joint work with Bruce Spencer, Ph.D. Professor, Northwestern University, Evanston IL Department of Statistics and Institute for Policy Research

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Large Scale Protein Interaction Graphs Fig. 4, Gavin et al. (2002) Nature Top panel Nodes: protein complex estimates Edges: common members Bottom panel Nodes: proteins Edges: complex co-membership (often called indirect interaction) Often steady-state organisms E.g. Saccharomyces cerevisiae, various interaction types Gavin et al. (2002, 2006) Nature, Ho et al. (2002) Nature, Krogan et al. (2006) Nature, Ito et al. (1998) PNAS, Uetz et al. (2000) Nature, Tong et al. (2006) Science, Pan et al. (2006) Cell Topology Modular organization into complexes/groups Bader et al. (2003) BMC Bioinformatics, Scholtens et al. (2005) Bioinformatics, Zhang et al. (2008) Bioinformatics, Qi et al. (2008) Bioinformatics Global characterization as small-world, scale-free, hierarchical, etc. Watts and Strogatz (1998) Nature, Barabási and Albert (1999) Science, Sales-Pardo et al. (2007) PNAS Measurement Error False positive/negative probabilities Chiang et al. (2007) Genome Biology, Chiang and Scholtens (2009) Nature Protocols Mostly large graphs 100s-1000s of nodes 1000s-10,000s of edges

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Sampled data AP-MS data capture bait-prey relationships: a bait finds ‘interacting’ prey with common membership in at least one complex bait prey One AP-MS `pull-down’ Three bait:prey pull-downs from Gavin et al. (2002) Apl5: Apl6, Apm3, Aps3, Ckb1 Apl6: Apl5, Apm3, Eno2 Apm3: Apl6, Apm3 Apl6 Apl5 untested: ? tested: absent Apl6 Apm3Apl5 Eno2 Aps3 Ckb1 Maximal cliques map to protein complexes: when all proteins are used as baits, all nodes have edges to all other nodes in the clique, and the clique is not contained in any other clique NOTE: Failure to test all edges means we typically cannot identify maximal cliques

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Inference using a portion of possible baits B D F A C A E Two protein complexes with physical topologies shown by black edges 6 Baits: ABCDEF If the AP-MS technology works perfectly (I.e. no false positives or false negatives)… B D A F CE 1 Bait: A 15 tested edges 9 present 6 absent 5 tested edges 5 present 0 absent 10 untested edges B D A F CE 2 Baits: AB 9 tested edges 7 present 2 absent 6 untested edges B D A F CE 3 Baits: ABC 12 tested edges 8 present 4 absent 3 untested edges B D A F CE

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Smaller-scale studies What if we are interested only in a portion of the graph? Cataloguing complexes/ describing the local neighborhood for a pre- specified set of starting baits Comparing local neighborhoods for different sample types disease vs. normal treated vs. untreated Starting bait of interest Interesting neighbor Less interesting neighbor Uninteresting neighbor

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Link tracing designs (or snowball sampling) Start with a set of nodes as starting baits (S 0 ) Identify interacting partners Use interacting partners as new set of baits, excluding those already used as baits Identify their interacting partners Etc…. S0S0 S1S1 S2S2 S3S3

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Link tracing notation Adapted from Handcock and Gile (2010) Annals of Applied Statistics

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Link tracing notation

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A simple scheme Let m remain constant over all sampling waves, e.g. choose a fixed proportion p of all eligible baits at each wave. This leads to a simplification in the probability of observing a specific sample. In particular, Pr(S m = s m | E m, m ) = π (pE mi ) s mi ((1-pE mi )) (1-s mi ) i=1 n

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Sampling 1/4 of all eligible baits… S 0 = {n1,n2,n3} E 1 = {n4,n6,n12,n13,n14,n15,n16,n17} S 1 = {n4,n12} E 2 = {n6,n13,n14,n15,n16,n17,n34,n35, n36,n37,n38,n59,n97,n98,n99n100,n194} S 2 = {n15,n59,n97,n98,n99} Etc… Note that we do not cover all portions of the graph that we would with a full snowball sample.

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Negative binomial In this setting, a path of length l extending from one of the starting baits follows a negative binomial distribution for being tested (and therefore observed) in m rounds of sampling (0 < l ≤ m). Pr(observing a path of length l in m rounds) = ( ) p l (1-p) m-l m=l,l+1,… m-1 l-1 ppp ppp 1-p Test all 3 nodes/edges in 3 rounds: Test 3 nodes/edges in 4 rounds:

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Cumulative probabilities The cumulative probability for observing paths with nodes that are sampled early on is higher than those that enter later. When nodes are tightly grouped in cliques, this can lead to over- sampling in regions of the graph with high-confidence clique estimates. Ie, we may be ‘satisfied’ with a clique estimate that has a certain proportion of tested edges, but if the involved nodes are identified early in the process, chances are they will eventually enter the sample…so how can we move on and sample other areas? There is also great dependency among joint probabilities of testing any pair (or larger collection) of paths, especially among nodes with common paths extending from the starting baits.

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B D A F CE 6 Baits: ABCDEF B D A F CE 1 Bait: A 15 tested edges 9 present 6 absent 5 tested edges So 1/3 of possible edges are tested B D A F CE 2 Baits: AB 9 tested edges 9/15 = 3/5 tested B D A F CE 3 Baits: ABC 12 tested edges 12/15 = 4/5 tested Tested fraction of edges In addition, we are interested in complexes with a certain proportion of tested edges out of those that are possible, not necessarily a proportion of tested baits (although they are related)

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Edge imputation Assume a simple edge imputation scheme in which untested edges are assumed to exist if the involved prey share at least one common bait. This is consistent with high clustering coefficients observed for these types of graphs as well as existing clique estimation algorithms on partially observed graphs. A complex (or clique) estimate may be considered ‘high quality’ if more than half of the involved edges are tested and observed. High Quality: 9/15=0.6 edges observed Low Quality: 13/28=0.46 edges observed

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Tested fraction of edges In a collection of nodes involving b baits and q prey-only nodes with no measurement error for edge observations, we have: b(b-1)/2 tested edges among baits bq tested edges among bait-prey pairs (b+q)(b+q-1)/2 possible edges among all nodes So then the proportion of observed edges is b(b-1) + 2bq (b+q)(b+q-1)

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A modification: capturing dependency among nodes B D F A C A E Two protein complexes with physical topologies B D A F CE Corresponding AP-MS graph c1c1 c2c2 A11 B10 C01 D10 E01 F10 ABCDEF A111111 B110101 C101010 D110101 E101010 F110101 A = Y = AA T = Boolean algebra: 1+1=1*1=1+0=1 0+0=0*0=0*1=0 Affiliation matrix: nodes to cliques Incidence matrix among nodes

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Strata: Nodes with identical adjacency B D A F CE AP-MS graph ABCDEF A111111 B110101 C101010 D110101 E101010 F110101 Y=Y=

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All nodes with matching colors on the previous slide are connected to each other, and have matching sets of adjacent nodes In some sense, they contain ‘redundant’ information And in a measurement error setting, extremely highly correlated information If we know the strata, and we know the set of adjacent nodes for one member node, then we know the set of adjacent nodes for all other strata constituents For sampling purposes, it seems reasonable to represent these subpopulations by design

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B D A F CE AP-MS graph BDF A CE c1c1 c2c2 A11 B10 C01 D10 E01 F10 ABCDEF A111111 B110101 C101010 D110101 E101010 F110101 A = Y = AA T = Boolean algebra: 1+1=1*1=1+0=1 0+0=0*0=0*1=0 g1g2g3 A100 B010 C001 D010 E001 F010 X = Affiliation matrix: nodes to strata c1c2 g1 11 g2 10 g3 01 Q = Affiliation matrix: strata to cliques

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Note the following properties: QQ T is the incidence matrix among strata XQ = A XQ(XQ) T = AA T = Y

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Stratified sampling The idea: use estimated strata to inform sampling Maintain a constant fraction of tested edges within each estimated strata This will help identify strata and summarize their connectivity to other strata It will also help focus our resources in areas that require more observations as opposed to those that have been adequately sampled according to some desired threshold for the fraction of tested edges

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Stratified sampling Testing at least half of the edges within a stratum with 10 member nodes: At least 3 baits are required Have 1 bait Choose 2 more baits Have 2 baits Choose 1 more bait Have 4 baits Don’t sample from this stratum (or do so with small probability)

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Stratified sampling While the strata and the fraction of tested edges within them determine the number of additional baits to include, the samples do also include observations of edges connecting pairs nodes in different strata Tested edge within strata Tested edge between strata

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Stratified sampling Algorithm: Specify starting baits S 0 and form E 1 Impute edges among prey-only nodes with at least one common bait Estimate strata according to matching adjacency in Y 1 to form X 1 Calculate fraction of tested edges for each stratum determined by X 1 Determine number of additional baits required for each stratum and sample accordingly to form S 1 Repeat At each step k, we can also estimate Q k, Y k and/or A k

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A comparison: Threshold sampling Similar to the simple random sampling scheme introduced earlier Rather than specifying a set proportion of baits to test, sample the appropriate number to test a certain fraction of all possible edges in the graph given the identified nodes

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Simulation: In silico Interactome We used the ScISI Bioconductor package to create an ‘in silico interactome’ containing protein complex data reported in the Cellular Component Gene Ontology and at MIPS for Saccharomyces cerevisiae. The largest connected component of the resultant graph contains 1404 nodes and 86609 edges. 197 protein complexes are represented with a range of sizes from 2 to 308 (median 18).

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Simulation Study Compared stratified(str) and threshold (thresh) sampling schemes Specified tested fractions of 1/10 and 1/20 of all possible edges Called a complex ‘high quality’ if at least 1/2 of the edges were tested For each iteration, randomly chose 3 nodes with close proximity as starting baits 250 rounds for each scheme

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Mean number correctly identified high- quality complexes

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Standard errors on number of correctly identified complexes

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Standard error / number identified

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Cumulative number of baits mean number of complexes

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Number of baits per complex

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Number of complexes vs. number of baits

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Discussion Large-scale protein interaction experiments are very costly and may not be of interest in smaller lab settings or for investigations of particular cellular functions As long as we are comfortable with some estimation of untested edges, sampling identified prey to create the next bait set may yield considerable savings

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Discussion Using estimated sampling strata seems to provide a greater balance of resource allocation across the graph Work still in progress suggests that this is due to a reduction in cumulative sampling variability across the graph As long as the per-bait cost is less than the per-sampling-round cost, stratified sampling appears to be a better approach

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Extensions Measurement error can be easily included in specification of E m, and adaptations of clique identification (e.g. the penalized likelihood method in Bioconductor’s apComplex ) can be used instead of straightforward imputation This would also be a natural starting point for adaptively designing experiments to compare different sample types

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