Delta: a Toolset for the Structural Analysis of Biological Sequences - - PowerPoint PPT Presentation

Motivation The Delta Toolset Applications Summary

Delta: a Toolset for the Structural Analysis

• f Biological Sequences
• n a 3D Triangular Lattice

Minghui Jiang Martin Mayne Joel Gillespie

Department of Computer Science, Utah State University

International Symposium on Bioinformatics Research and Applications, Atlanta, Georgia, May 7–10, 2007

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

Protein folding in HP model

20 types of amino acids: hydrophobic (H) or hydrophilic (P). A sequence folds on a lattice:

each element occupies a unique lattice point; consecutive elements in the sequence occupy adjacent lattice points.

Optimization: to maximize the number of H-H contacts. Extensively studied: folding kinetics simulation, computational complexities, approximation algorithms, alternative lattices.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

An example on 2D hexagonal lattice

Minghui Jiang and Binhai Zhu. Protein folding on the hexagonal lattice in the HP model. Journal of Bioinformatics and Computational Biology, 3(1):19–34, 2005.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

RNA folding

Previous work Dynamic programming. Secondary structure (base pairs) first, tertiary structure next.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

RNA folding

Previous work Dynamic programming. Secondary structure (base pairs) first, tertiary structure next. Our approach: RNA folding on a 3D triangular lattice. Simulate tertiary structure directly, then derive secondary structure.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

RNA folding

Previous work Dynamic programming. Secondary structure (base pairs) first, tertiary structure next. Our approach: RNA folding on a 3D triangular lattice. Simulate tertiary structure directly, then derive secondary structure. Two advantages: The predicted structure is always realizable in 3D. Arbitrary pseudoknots.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

RNA folding

Previous work Dynamic programming. Secondary structure (base pairs) first, tertiary structure next. Our approach: RNA folding on a 3D triangular lattice. Simulate tertiary structure directly, then derive secondary structure. Two advantages: The predicted structure is always realizable in 3D. Arbitrary pseudoknots. 3D triangular lattice is non-trivial: the Delta toolset.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

3D triangular lattice

x y z √ 2 1 1 x y z z = 0 z = 1

A closest cubic packing of spheres. 12 neighbors on three square lattices in parallel planes. Better than cubic lattice: denser; no parity constraint.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Representation of a structure

3 primary axes and 3 auxiliary axes:

• x
• y
• z
• u =

x + z

• v =

y + z

• w =

x + y + z External: a turn sequence over the alphabet {U, V, W, X, Y, Z, u, v, w, x, y, z} Internal: an array of (x, y, z) coordinates and a hashtable (collision detection in expected constant time).

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Balls and sticks

Zooming, rotation, interactive manipulation, . . .

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Manipulation of structures: flip moves

Flip moves are local: constant time per move

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Manipulation of structures: pivot moves

Pivot moves are global: Ω(n) time per move on average.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Manipulation of structures: pull moves

Introduced by Lesh et al. for protein folding on square lattice.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

We chose pull moves

Almost-local: each random move relocates only a small number of elements on average

sequence length 32 64 128 256 512 average # of relocated elements 2.4 3.8 4.1 3.4 3.5 standard deviation 1.3 2.7 3.2 2.0 2.4

Reversible: A → B ⇐ ⇒ B → A Complete: A B

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Simulated annealing

At step i, Choose a random pull move, which transforms the configuration from vi−1 to vi. If score(vi) ≥ score(vi−1), commit the move and continue to the next step. Otherwise, commit the move with a probability of pi = 2(score(vi)−score(vi−1))/Ti, where Ti = c/ log2(1 + i/n).

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Pair score s(i, j)

For two adjacent bases i and j, |i − j| ≥ 4 (the hairpin loop distance), G-C: 9 A-U: 6 G-U: 1 Other pairs: −1. Justification: A G-C pair forms 3 hydrogen bonds; An A-U pair forms 2 hydrogen bonds; The wobble pair G-U and other pairs are rare.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Normalized score s′(i, j)

sum(i) =

• (i,j) a pair

|s(i, j)|. s′(i, j) = s(i, j) sum(i) · s(i, j) sum(j) · s(i, j), The two ratios

s(i,j) sum(i) and s(i,j) sum(j) represent the levels of

“commitment” of i and j in forming the pair (i, j). O(

• |V||E|) = O(n1.5) time maximum matching is avoided;

rare secondary structures such as base triples are still possible.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Structure Representation Visualization and Manipulation Folding Simulation

Stacking score s′′(i, i + 1, j − 1, j)

For two pairs (i, j) and (i + 1, j − 1) such that s′(i, j) > 0 and s′(i + 1, j − 1) > 0, s′′(i, i + 1, j − 1, j) = s′(i, j) + s′(i + 1, j − 1). The score of a configuration is its total stacking score. A subtlety in our scoring function implicitly discourages the unnecessary clustering of bases. Each anneal step takes only constant time on average.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Local RNA sequences from PseudoBase

Test set 229 sequences (20 ≤ ℓ ≤ 131) 150 sequences (20 ≤ ℓ ≤ 40) # of rounds k 5 10 5 10 Accuracy A 69.2% 73.4% 83.4% 87.2% Accuracy A′ 60.3% 64.2% 83.4% 87.5%

Simulated annealing options are set to

• a 104ℓ 103ℓ k -z 104ℓ,

for each sequence of length ℓ. For each sequence si in a test set of N sequences,

ni is the number of real base pairs, mi is the number of base pairs predicted correctly.

Define A = (

i mi/ni)/N and A′ = ( i mi)/( i ni).

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Both accurate and fast

Eddy noted that “current RNA folding programs get about 50–70% of base pairs correct, on average.” 1 Our prediction accuracy is around 70% on average and almost 90% for short sequences. For a sequence of length 40, a 20-round run typically takes about two minutes. 2 Easy trade-off between running time and prediction accuracy: more steps/rounds = ⇒ more accurate

• 1S. R. Eddy. How do RNA folding algorithms work? Nature Biotechnology,

22(11):1457–1458, 2004.

2On an Apple iMac computer: 2GHz PowerPC G5 processor; 2GByte

DDR SDRAM memory; MacOS 10.4.8; GCC 4.0.0.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

A few screenshots of predicted structures

CCCCUUUACUUGAGGGAAAUCAAGC :(((::::[[[[[))):::]]]]]: XwwuuyvvYYWZUWWuvvxzwyyW

BSBV1

CCCCCAUCCGGAGGGUUAUCCGGC :(((:::[[[[[))):::]]]]]: WWVxZwyWWXzwwwUWUYZxxwy

SBWMV2

UAGGGGCUUACCGAAAUAAGCC :(((:[[[[[))):::]]]]]: wyVxYUWyyzvZWZxuYYwuy

CcTMV UPD-PK2

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Predicted structures may not be realizable

The hierarchical approach has the drawback that the predicted secondary structures (say, by mfold) may not be realizable in 3D as tertiary structures due to steric constraints.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Predicted structures may not be realizable

The hierarchical approach has the drawback that the predicted secondary structures (say, by mfold) may not be realizable in 3D as tertiary structures due to steric constraints. To model the hairpin loop: two bases forming a base pair must be separated by at least 3 other bases in the sequences.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Predicted structures may not be realizable

The hierarchical approach has the drawback that the predicted secondary structures (say, by mfold) may not be realizable in 3D as tertiary structures due to steric constraints. To model the hairpin loop: two bases forming a base pair must be separated by at least 3 other bases in the sequences. The steric constraints are inherently geometric; it is difficult to capture them all with only a few combinatorial restrictions.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Validate predicted structures by reconstruction

Similar work has been done on recovering protein tertiary structures from contact maps. 3

3“Reconstruction of 3D Structures From Protein Contact Maps” by

Vassura et al. this afternoon.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Validate predicted structures by reconstruction

Similar work has been done on recovering protein tertiary structures from contact maps. 3 On a -i2 option, Delta accepts as input a secondary structure (represented by a list of base pairs) then tries to reconstruct a tertiary structure that conforms to the secondary structure.

3“Reconstruction of 3D Structures From Protein Contact Maps” by

Vassura et al. this afternoon.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Validate predicted structures by reconstruction

Similar work has been done on recovering protein tertiary structures from contact maps. 3 On a -i2 option, Delta accepts as input a secondary structure (represented by a list of base pairs) then tries to reconstruct a tertiary structure that conforms to the secondary structure. We validated all 229 local RNA sequences from

• PseudoBase. Each sequence takes only a couple of

seconds.

3“Reconstruction of 3D Structures From Protein Contact Maps” by

Vassura et al. this afternoon.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

A few screenshots of reconstructed structures

CCCCUUUACUUGAGGGAAAUCAAGC :(((::::[[[[[))):::]]]]]: zzzuXuzUUZUZuZZXXXuzuzuz

BSBV1

CCCCCAUCCGGAGGGUUAUCCGGC :(((:::[[[[[))):::]]]]]: ZZUZUZUzzzzxuzXUXZZZZZZ

SBWMV2

UAGGGGCUUACCGAAAUAAGCC :(((:[[[[[))):::]]]]]: ZZZUZuuuzUzzuxZZZUUUU

CcTMV UPD-PK2

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Visualize secondary structures by reconstruction

On a -2d option, Delta folds the sequence on a 2D triangular lattice.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Visualize secondary structures by reconstruction

On a -2d option, Delta folds the sequence on a 2D triangular lattice. Some secondary structures with complex pseudoknots may not have a planar drawing. Planar visualization tools (RnaViz, PseudoViewer, jViz.RNA) use long or crossing edges that distort the geometry. Our lattice visualization preserves geometric properties such as proportion and distance better.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Outline

1

Motivation

2

The Delta Toolset Structure Representation Visualization and Manipulation Folding Simulation

3

Applications Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Measures for thermodynamic stability

For an RNA sequence s and a set of shuffled sequence Xs, compare the minimum free energies (mfe) of s and Xs: the Z-score of s Z(s) = mfe(s) − average(Xs) stdev(Xs) is the number of standard deviations by which mfe(s) drops below the average MFE of the set Xs. The p-value of s is the fraction of sequences in Xs having MFEs lower than mfe(s).

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Estimate mfe with or without pseudoknots

RNAfold and mfold cannot handle pseudoknots. Delta can handle arbitrary pseudoknots.

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary Prediction of RNA Secondary Structures Validation and Visualization of RNA Secondary Structures Thermodynamic Stability Analysis of RNAs

Estimate mfe with or without pseudoknots

RNAfold and mfold cannot handle pseudoknots. Delta can handle arbitrary pseudoknots. For 69 microRNA precursors of M.musculus, we computed Z-score: −1.3 ± 2.3 p-value: 0.33 ± 0.35 confirming Bonnet et al.’s results. For 150 short sequences from PseudoBase, we computed Z-score: −1.9 ± 1.3 p-value: 0.12 ± 0.22

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

Summary

3D triangular lattice, pull moves, simulated annealing All kinds of polymers: RNAs, DNAs, proteins, . . . Structural analysis: prediction, validation, visualization, . . .

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice

Motivation The Delta Toolset Applications Summary

Summary

3D triangular lattice, pull moves, simulated annealing All kinds of polymers: RNAs, DNAs, proteins, . . . Structural analysis: prediction, validation, visualization, . . . Outlook Circular RNAs (found in viroids) and RNA-interaction (miRNAs bind to mRNAs as regulators) Combo moves (each anneal step attempts a group of random moves on nearby elements) Other optimization techniques: genetic algorithm Other applications: protein/RNA design, folding kinetics Web interface

Minghui Jiang, Martin Mayne, Joel Gillespie Delta: a Toolset for Structural Analysis on a 3D Triangular Lattice