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Sequential Sampling Designs for Small-Scale Protein Interaction Experiments

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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

- 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

- E.g. Saccharomyces cerevisiae, various interaction types
- 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

- Modular organization into complexes/groups
- Measurement Error
- False positive/negative probabilities
- Chiang et al. (2007) Genome Biology, Chiang and Scholtens (2009) Nature Protocols

- False positive/negative probabilities
- Mostly large graphs
- 100s-1000s of nodes
- 1000s-10,000s of edges

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)

Three bait:prey pull-downs

from Gavin et al. (2002)

Apl5: Apl6, Apm3, Aps3, Ckb1

Apl6: Apl5, Apm3, Eno2

Apm3: Apl6, Apm3

One AP-MS

`pull-down’

bait

prey

Eno2

untested:

?

AP-MSdata capture bait-prey relationships:

a bait finds ‘interacting’ prey with

common membership in at least

one complex

Apl6

Apl6

Aps3

Apm3

Apl5

Apl5

Ckb1

tested:

absent

Maximal cliquesmap 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

B

C

Two protein complexes

with physical topologies

shown by black edges

D

A

A

F

E

If the AP-MS technology works perfectly (I.e. no false positives or false negatives)…

2 Baits: AB

3 Baits: ABC

6 Baits: ABCDEF

1 Bait: A

B

B

B

B

F

D

F

D

F

D

F

D

A

A

A

A

C

E

C

E

C

E

C

E

9 tested edges

7 present

2 absent

6 untested edges

12 tested edges

8 present

4 absent

3 untested edges

15 tested edges

9 present

6 absent

5 tested edges

5 present

0 absent

10 untested edges

- 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

- 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

- 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,

n

Pr(Sm= sm | Em,m) = π (pEmi)smi((1-pEmi))(1-smi)

i=1

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 do

not cover all

portions of the graph

that we would with

a full snowball

sample.

- 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-1

l-1

Test all 3 nodes/edges in 3 rounds:

p

p

p

1-p

p

p

p

Test 3 nodes/edges in 4 rounds:

- 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.

- 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)

2 Baits: AB

3 Baits: ABC

6 Baits: ABCDEF

1 Bait: A

B

B

B

B

F

D

F

D

F

D

F

D

A

A

A

A

C

E

C

E

C

E

C

E

9 tested edges

9/15 = 3/5 tested

12 tested edges

12/15 = 4/5 tested

15 tested edges

9 present

6 absent

5 tested edges

So 1/3 of possible

edges are tested

- 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

- 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)

B

C

Two protein complexes

with physical topologies

D

A

A

F

E

Affiliation matrix:

nodes to cliques

Incidence matrix

among nodes

Corresponding

AP-MS graph

B

A =

Y = AAT =

Boolean algebra:

1+1=1*1=1+0=1

0+0=0*0=0*1=0

F

D

A

C

E

AP-MS graph

B

Y =

F

D

A

C

E

- 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

AP-MS graph

B

A =

Y = AAT =

Boolean algebra:

1+1=1*1=1+0=1

0+0=0*0=0*1=0

F

D

A

C

E

Affiliation matrix:

nodes to strata

Affiliation matrix:

strata to cliques

BDF

X =

Q =

A

CE

- Note the following properties:
QQT is the incidence matrix

among strata

XQ = A

XQ(XQ)T = AAT = Y

- 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

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)

- 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

- 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

- 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

- 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).

- 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

of complexes

- 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

- 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

- 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