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Probabilistic Methods for Interpreting Electron-Density Maps

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### Probabilistic Methods for Interpreting Electron-Density Maps

ACMI Overview

Frank DiMaio

University of Wisconsin – Madison

Computer Sciences Department

High-Throughput Structure Determination

- Protein-structure determination important
- Understanding function of a protein
- Understanding mechanisms
- Targets for drug design
- Some proteins produce poor density maps
- Interpreting poor electron-density maps is very (human) laborious
- I aim to automatically interpret poor-quality electron-density maps

Density Map Resolution

1.0Å

2.0Å

3.0Å

4.0Å

Ioerger et al. (2002)

Terwilliger (2003)

Morris et al. (2003)

My focus

Thesis Contributions

- A probabilistic approach to protein-backbone tracingDiMaio et al., Intelligent Systems for Molecular Biology (2006)
- Improved template matching in electron-density mapsDiMaio et al., IEEE Conference on Bioinformatics and Biomedicine (2007)
- Creating all-atom protein models using particle filteringDiMaio et al. (under review)
- Pictorial structures for atom-level molecular modelingDiMaio et al., Advances in Neural Information Processing Systems (2004)
- Improving the efficiency of belief propagationDiMaio and Shavlik, IEEE International Conference on Data Mining (2006)
- Iterative phase improvement in ACMI

ACMI Overview

- Phase 1: Local pentapeptide search (ISMB 2006, BIBM 2007)
- Independent amino-acid search
- Templates model 5-mer conformational space
- Phase 2: Coarse backbone model(ISMB 2006, ICDM 2006)
- Protein structural constraints refine local search
- Markov field (MRF) models pairwise constraints
- Phase 3: Sample all-atom models
- Particle filtering samples high-prob. structures
- Probs. from MRF guide particle trajectories

ACMI Overview

- Phase 1: Local pentapeptide search (ISMB 2006, BIBM 2007)
- Independent amino-acid search
- Templates model 5-mer conformational space
- Phase 2: Coarse backbone model(ISMB 2006, ICDM 2006)
- Protein structural constraints refine local search
- Markov field (MRF) models pairwise constraints
- Phase 3: Sample all-atom models
- Particle filtering samples high-prob. structures
- Probs. from MRF guide particle trajectories

5-mer Lookup

…SAWCVKFEKPADKNGKTE…

- ACMI searches map for each template independently
- Spherical-harmonic decomposition allows rapid search of all template rotations

Protein

DB

map-regionsampled in

spherical shells

sampled region of

density in 5A sphere

template-densitysampled in

spherical shells

calculated (expected)

density in 5A sphere

5-mer Fast Rotation Searchelectron density map

pentapeptide fragment

from PDB (the “template”)

map-region spherical-harmonic coefficients

map-regionsampled in

spherical shells

correlationcoefficientas functionof rotation

template-densitysampled in

spherical shells

template spherical-harmonic coefficients

5-mer Fast Rotation Searchfast-rotation

function(Navaza 2006,

Risbo 1996)

correlation coefficients

over density mapti (ui)

probability distribution over density map

P(5-mer at ui|EDM)

Convert Scores to ProbabilitiesBayes’

rule

scan density map

for fragment

ACMI Overview

- Phase 1: Local pentapeptide search (ISMB 2006, BIBM 2007)
- Independent amino-acid search
- Templates model 5-mer conformational space
- Phase 2: Coarse backbone model(ISMB 2006, ICDM 2006)
- Protein structural constraints refine local search
- Markov field (MRF) models pairwise constraints
- Phase 3: Sample all-atom models
- Particle filtering samples high-prob. structures
- Probs. from MRF guide particle trajectories

Probabilistic Backbone Model

- Trace assigns a position and orientation ui={xi, qi} to each amino acid i
- The probability of a trace U={ui} is

- This full joint probability intractable to compute
- Approximate using pairwise Markov field

ALA

GLY

LYS

LEU

SER

Pairwise Markov-Field Model- Joint probabilities defined on a graph as product of vertex and edge potentials

ALA

GLY

LYS

LEU

SER

ACMI’s Backbone Model- Adjacency constraints ensure adjacent amino acids are ~3.8Å apart and in proper orientation
- Occupancy constraints ensure nonadjacent amino acids do not occupy same 3D space

Backbone Model Potential

Constraints between all other amino acid pairs

Backbone Model Potential

Observational (“template-matching”) probabilities

Inferring Backbone Locations

- Want to find backbone layout that maximizes

Inferring Backbone Locations

- Want to find backbone layout that maximizes

- Exact methods are intractable
- Use belief propagation (Pearl 1988) to approximate marginal distributions

Scaling BP to Proteins(DiMaio and Shavlik, ICDM 2006)

- Naïve implementation O(N2G2)
- N = the number of amino acids in the protein
- G = # of points in discretized density map
- O(G2) computation for each message passed
- O(G log G) as Fourier-space multiplication
- O(N2) messages computed & stored
- Approx (N-3) occupancy msgs with 1 message
- O(N) messages using a message accumulator
- Improved implementation O(NG log G)

Scaling BP to Proteins(DiMaio and Shavlik, ICDM 2006)

- Naïve implementation O(N2G2)
- N = the number of amino acids in the protein
- G = # of points in discretized density map
- O(G2) computation for each message passed
- O(G log G) as Fourier-space multiplication
- O(N2) messages computed & stored
- Approx (N-3) occupancy msgs with 1 message
- O(N) messages using a message accumulator
- Improved implementation O(NG log G)

Occupancy Message Approximation

- To pass a message

occupancy

edge potential

product of incoming msgs to iexcept from j

Occupancy Message Approximation

- To pass a message

- “Weak” potentials between nonadjacent amino acids lets us approximate

occupancy

edge potential

product of all

incoming msgs to i

Occupancy Message Approximation

ACC

1

2

4

5

3

6

Send outgoing occupancy message product to a central accumulator

Occupancy Message Approximation

ACC

1

2

4

5

3

6

Then, each node’s incoming message product is computed in constant time

BP Output

- After some number of iterations, BP gives probability distributions over Cα locations

…

…

ARG

LEU

PRO

ALA

VAL

…

…

…

…

…

ACMI’s Backbone Trace- Independently choose Cα locations that maximize approximate marginal distribution

Example: 1XRI

3.3Å resolution density map

39° mean phase error

prob(AA at location)

HIGH

0.9

0.1

LOW

0.9009Å RMSd

93% complete

Testset Density Maps (raw data)

75

60

Density-map mean phase error (deg.)

45

30

15

1.0

2.0

3.0

4.0

Density-map resolution (Å)

% backbone correctly placed

% amino acids correctly identified

Experimental Accuracy100

80

60

% Cα’s located within 2Å of some Cα / correct Cα

40

20

0

ACMI

ARP/wARP

Resolve

Textal

100

100

80

80

60

60

40

40

20

20

0

0

0

20

40

60

80

100

0

20

40

60

80

100

Experimental Accuracy on a Per-Protein Basis100

80

60

ACMI % Cα’s located

40

20

0

0

20

40

60

80

100

ARP/wARP % Cα’s located

Resolve % Cα’s located

Textal % Cα’s located

- Phase 1: Local pentapeptide search (ISMB 2006, BIBM 2007)
- Independent amino-acid search
- Templates model 5-mer conformational space
- Phase 2: Coarse backbone model(ISMB 2006, ICDM 2006)
- Protein structural constraints refine local search
- Markov field (MRF) models pairwise constraints
- Phase 3: Sample all-atom models
- Particle filtering samples high-prob. structures
- Probs. from MRF guide particle trajectories

Probability=0.4

Probability=0.35

Probability=0.25

Maximum-marginal structure

Problems with ACMI- Biologists want location of all atoms
- All Cα’s lie on a discrete grid
- Maximum-marginal backbone model may be physically unrealistic
- Ignoring a lot of information
- Multiple models may better represent conformational variation within crystal

ACMI with Particle Filtering(ACMI-PF)

Idea: Represent protein using a set of static 3D all-atom protein models

Particle Filtering Overview (Doucet et al. 2000)

- Given some Markov process x1:KXwith observations y1:K Y
- Particle Filtering approximates some posterior probability distribution over Xusing a set of N weighted point estimates

Particle Filtering Overview

- Markov process gives recursive formulation

- Use importance fn. q(x k |x 0:k-1 ,y k) to grow particles
- Recursive weight update,

Particle Filtering for Protein Structures

- Particle refers to one specific 3D layout of some subsequence of the protein
- At each iteration advance particle’s trajectory by placing an additional amino-acid’s atoms

Particle Filtering for Protein Structures

- Alternate extending chain left and right

Particle Filtering for Protein Structures

- Alternate extending chain left and right
- An iteration alternately places
- Cα positionbk+1 given bk
- All sidechain atomssk given bk-1:k+1

bk-1

bk

bk+1

sk

Particle Filtering for Protein Structures

- Key idea: Use the conditional distribution p(bk|bik-1,Map) to advance particle trajectories
- Construct this conditional distribution from BP’s marginal distributions

bk-1

bk

bk+1

sk

sk

bk+1

bk-1

Particle Filtering for Protein StructuresAlgorithm

place “seeds” bkifor each particle i=1…N

whileamino-acids remain

place bki+1 / bji-1 given bj:kifor each i=1…N

place ski given bki-1:k+1for each i=1…N

optionally resample N particles

end while

…

…

bk

1…L

b

k+1

b

b

k

k-1

Backbone Step (for particle i)place bki+1 given bkifor each i=1…N

(1) Sample Lbk+1’s from bk-1–bk–bk+1 pseudoangle distribution

1…L

b

k+1

2

L

1

pk+1(b )

pk+1(b )

pk+1(b )

k+1

k+1

k+1

b

b

k

k-1

Backbone Step (for particle i)place bki+1 given bkifor each i=1…N

…

(2) Weight each sample by its ACMI-computed approximate marginal

1…L

b

k+1

2

L

1

pk+1(b )

pk+1(b )

pk+1(b )

k+1

k+1

k+1

b

b

k

k-1

Backbone Step (for particle i)place bki+1 given bkifor each i=1…N

…

(3) Select bk+1 with probability proportional to sample weight

b

k+1

b

b

k-1

k

Backbone Step (for particle i)place bki+1 given bkifor each i=1…N

(4) Update particle weight as sum of sample weights

Sidechain Step (for particle i)

place ski given bki-1:k+1for each i=1…N

Protein

Data

Bank

(1) Sample sk from a database of sidechain conformations

1

2

3

pk(EDM |s )

pk(EDM |s )

pk(EDM | s )

k

k

k

Sidechain Step (for particle i)place ski given bki-1:k+1for each i=1…N

(2) For each sidechain conformation, compute probability of densitymap given the sidechain

1

3

2

pk(EDM |s )

pk(EDM | s )

pk(EDM |s )

k

k

k

Sidechain Step (for particle i)place ski given bki-1:k+1for each i=1…N

(3) Select sidechain conformation from this weighted distribution

Sidechain Step (for particle i)

place ski given bki-1:k+1for each i=1…N

(4) Update particle weight as sum of sample weights

Particle Resampling

wt = 0.4

wt = 0.4

wt = 0.2

wt = 0.3

wt = 0.3

wt = 0.2

wt = 0.1

wt = 0.1

wt = 0.2

wt = 0.1

wt = 0.1

wt = 0.2

wt = 0.1

wt = 0.1

wt = 0.2

Amino-Acid Sampling Order

- Begin at some amino acid k with probability

j

k

- At each step, move left to right with probability

Experimental Methodology

- Run ACMI-PF 10 times with 100 particles each
- Return highest-weight particle from each run
- Each run samples amino-acids in a different order
- Refine each structure for 10 iterations in Refmac5
- Compare 10-structure model to others using Rfree

ACMI-PF Versus ACMI-Naïve

Additionally, ACMI-PF’s models have …

- Fewer gaps (10 vs. 28)
- Lower sidechain RMS error (2.1Å vs. 2.3Å)

Refined Rfree

Number of ACMI-PF runs

0.65

0.65

0.65

0.55

0.55

0.55

0.45

0.45

0.45

0.35

0.35

0.35

0.25

0.25

0.25

0.25

0.35

0.45

0.55

0.65

0.25

0.35

0.45

0.55

0.65

0.25

0.35

0.45

0.55

0.65

ACMI-PF Versus OthersACMI-PF Rfree

ARP/wARP Rfree

Resolve Rfree

Textal Rfree

ACMI Overview

- Phase 1: Local pentapeptide search (ISMB 2006, BIBM 2007)
- Independent amino-acid search
- Templates model 5-mer conformational space
- Phase 2: Coarse backbone model(ISMB 2006, ICDM 2006)
- Protein structural constraints refine local search
- Markov field (MRF) models pairwise constraints
- Phase 3: Sample all-atom models
- Particle filtering samples high-prob. structures
- Probs. from MRF guide particle trajectories

- Phase 4: Iterative phase improvement
- Use particle-filtering models to improve density-map quality
- Rerun entire pipeline on improved density map
- Repeat until convergence

Phase Problem

Intensities

Measured by X-raycrystallography

Phases

Experimentally

estimated (e.g. MAD, MIR)

75

60

45

30

15

0

0

15

30

45

60

75

ACMI-PF’s Phase ImprovementError in ACMI-PF’s phases(deg. mean phase error)

Error in initial phases(deg. mean phase error)

Two-Iteration ACMI

100

90

% backbone locatedIteration 2

80

70

60

50

50

60

70

80

90

100

% backbone locatedIteration 1

60

50

20

40

30

15

20

10

10

0

5

0

1

2

3

4

0

1

2

3

4

5

?

?

Future Work: Many-iteration ACMIAverage % uninterpreted AAs

Average mean phase error

Number of ACMI iterations

Number of ACMI iterations

Conclusions

- ACMI’s three steps construct a set of all-atom protein models from a density map
- Novel message approximation allows inference on large, highly-connected models
- Resulting protein models are more accuratethan other methods

Ongoing and Future Work

- Incorporate additional structural biology background knowledge
- Incorporate more complex potential functions
- Further work on iterative phase improvement
- Generalize my algorithms to other 3D image data

Acknowledgements

- Advisor Jude Shavlik
- Committee
- George Phillips
- Charles Dyer
- David Page
- Mark Craven
- Collaborators
- Ameet Soni
- Dmitry Kondrashov
- Eduard Bitto
- Craig Bingman
- 6th floor MSCers

- Center for Eukaryotic Structural Genomics
- Funding
- UW-Madison Graduate School
- NLM 1T15 LM007359
- NLM 1R01 LM008796

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