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 ILDepartment of Statistics and Institute for Policy Research

Large Scale Protein Interaction Graphs

Fig. 4, Gavin et al. (2002) Nature

Top panelNodes: protein complex estimatesEdges: common members

Bottom panelNodes: proteinsEdges: 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

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, Ckb1Apl6: Apl5, Apm3, Eno2Apm3: Apl6, Apm3

Apl6

Apl5

untested: ?

tested:absent

Apl6

Apm3 Apl5

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

Inference using a portion of possible baits

B

DF

A

C

AE

Two protein complexes with physical topologiesshown 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

C E

1 Bait: A

15 tested edges 9 present 6 absent

5 tested edges 5 present 0 absent 10 untested edges

B

D

A

F

C E

2 Baits: AB

9 tested edges 7 present 2 absent 6 untested edges

B

D

A

F

C E

3 Baits: ABC

12 tested edges 8 present 4 absent 3 untested edges

B

D

A

F

C E

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

Link tracing designs(or snowball sampling)

Start with a set of nodes as starting baits (S0) Identify interacting partners Use interacting partners as new set of baits, excluding those already used as baits Identify their interacting partners Etc….

S0 S1 S2 S3

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

€

n number of nodes

Yij ,characteristic of a pair of nodes here a binary variable

Yij =1 if the edge between nodesi andj exists and0 otherwise

Y n×n matrix ofYij

Sm n×1 binary vector withi 1 th entry equal to if nodei is in them ( ) th wave of sampled nodes here bait nodes

0 . and otherwise LetS0 be the initial sample and in our case

.let it be defined a priori

Cm Cumulative set of sampled nodes afterm ,waves . . i e Stt=0

m∑

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

€

Dm n×n (matrix of tested edges in the graph withi, j) th element equal

1 to if thei / th and orj th element ofCm 1. equals Note:

Dm =Cm1T +1CmT−CmCm

T for an undirected graph

Dm =Cm1T for a directed graph

In traditional link- ,tracing designs we defineSm as follows

Sm = YSm−1 ⋅(1 −Cm−1 ) > 0[ ]

wherex[ ] 1 is ifx 0 .is true and otherwise

So all nodes with an edge to a node inSm−1 are included inSm , .unless they have been used in prior waves

Link tracing notation

€

,If we are satisfied with estimates of cliques with some untested edges .then define the following

Em n×1 binary vector for the set of nodes eligible to be

sampled on wavem (m=1,...,k)ψm n×1 vector of selection probabilities for units at wavem

,For the first wave we can write

E1 =[YS0 ⋅(1 −S0 ) > 0] and

(Pr S1 =s1 |E1 ,ψ1 ) = (ψ1ii=1

n∏ E1i )

s1 i ((1−ψ1i )E1i +(1−E1i ))

(1−s1 i)

fors1 ∈ {0,1}n.

Link tracing notation

€

,More generally for wavem (m=1,...,k)

Em =[YC m-1 ⋅(1 −Cm-1 ) > 0] and

(Pr Sm =sm |Em,ψm) = (ψmii=1

n∏ Emi)

smi ((1−ψmi)Emi+(1−Emi))

(1−smi)

forsm ∈ {0,1}n.

A simple scheme

Let ψm 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(Sm = sm | Em,ψm) = π (pEmi)smi((1-pEmi))(1-s

mi)

i=1

n

Sampling 1/4 of all eligible baits…

S0 = {n1,n2,n3}E1 = {n4,n6,n12,n13,n14,n15,n16,n17}S1 = {n4,n12}

E2 = {n6,n13,n14,n15,n16,n17,n34,n35, n36,n37,n38,n59,n97,n98,n99n100,n194}S2 = {n15,n59,n97,n98,n99}

Etc…

Note that we donot cover all portions of the graphthat we would witha full snowballsample.

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) = ( )pl(1-p)m-l

m=l,l+1,…

m-1l-1

p p p

p p p

1-p

Test all 3 nodes/edges in 3 rounds:

Test 3 nodes/edges in 4 rounds:

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.

B

D

A

F

C E

6 Baits: ABCDEF

B

D

A

F

C E

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

C E

2 Baits: AB

9 tested edges 9/15 = 3/5 tested

B

D

A

F

C E

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)

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

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 baitsbq 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)

A modification: capturing dependency among nodes

B

DF

A

C

AE

Two protein complexes with physical topologies

B

D

A

F

C E

CorrespondingAP-MS graph

c1

c2

A 1 1

B 1 0

C 0 1

D 1 0

E 0 1

F 1 0

A B C D E F

A 1 1 1 1 1 1

B 1 1 0 1 0 1

C 1 0 1 0 1 0

D 1 1 0 1 0 1

E 1 0 1 0 1 0

F 1 1 0 1 0 1

A = Y = AAT =

Boolean algebra:1+1=1*1=1+0=10+0=0*0=0*1=0

Affiliation matrix:nodes to cliques

Incidence matrixamong nodes

Strata:Nodes with identical adjacency

B

D

A

F

C E

AP-MS graph A B C D E F

A 1 1 1 1 1 1

B 1 1 0 1 0 1

C 1 0 1 0 1 0

D 1 1 0 1 0 1

E 1 0 1 0 1 0

F 1 1 0 1 0 1

Y =

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

B

D

A

F

C E

AP-MS graph

BDF

A

CE

c1

c2

A 1 1

B 1 0

C 0 1

D 1 0

E 0 1

F 1 0

A B C D E F

A 1 1 1 1 1 1

B 1 1 0 1 0 1

C 1 0 1 0 1 0

D 1 1 0 1 0 1

E 1 0 1 0 1 0

F 1 1 0 1 0 1

A = Y = AAT =

Boolean algebra:1+1=1*1=1+0=10+0=0*0=0*1=0

g1

g2

g3

A 1 0 0

B 0 1 0

C 0 0 1

D 0 1 0

E 0 0 1

F 0 1 0

X =

Affiliation matrix:nodes to strata

c1 c2

g1 1 1

g2 1 0

g3 0 1

Q =

Affiliation matrix:strata to cliques

Note the following properties:

QQT is the incidence matrix among strata

XQ = A

XQ(XQ)T = AAT = Y

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

Stratified sampling

Testing at least half of the edges within a stratum with 10 member nodes:

At least 3 baits are required

Have 1 baitChoose 2 more baits

Have 2 baitsChoose 1 more bait

Have 4 baitsDon’t sample from this stratum(or do so with small probability)

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

Stratified sampling Algorithm:

Specify starting baits S0 and form E1

Impute edges among prey-only nodes with at least one common bait

Estimate strata according to matching adjacency in Y1 to form X1

Calculate fraction of tested edges for each stratum determined by X1

Determine number of additional baits required for each stratum and sample accordingly to form S1

Repeat

At each step k, we can also estimate Qk, Yk and/or Ak

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

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).

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

Mean number correctly identified high-quality complexes

Standard errors on number of correctly identified complexes

Standard error / number identified

Cumulative number of baits

mean numberof complexes

Number of baits per complex

Number of complexes vs. number of baits

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

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

Extensions

Measurement error can be easily included in specification of Em, 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