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Topology Proteins Software Summary

Topological Data Analysis

Peter Bubenik

University of FloridaDepartment of Mathematics,[email protected]

http://people.clas.ufl.edu/peterbubenik/

British Applied Mathematics ColloquiumUniversity of Oxford

April 6, 2016

Peter Bubenik Topological Data Analysis

[email protected]

http://people.clas.ufl.edu/peterbubenik/

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Topology Proteins Software Summary Basics Persistent homology

Topological Data Analysis

Idea

Use topology to summarize and learn from the “shape” of data.


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Simplicial complexes from point data

The Cech construction


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Homology of simplicial complexes

Definition

Homology in degree k is given by k-cycles modulo thek-boundaries.


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Persistence

Main idea

Vary a parameter and keep track of when features appear anddisappear.

radius = 0Peter Bubenik Topological Data Analysis

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Persistence

Main idea



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Mathematical encoding

We have an increasing sequence of simplicial complexes

X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xm

called a filtered simplicial complex.

Apply homology.

We get a sequence of vector spaces and linear maps

V0 → V1 → V2 → · · · → Vm

called a persistence module.


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Persistence module to Barcode

V0 → V1 → V2 → V3 → V4 → V5 → V6 → V7 → · · · → Vm

Fundamental Theorem of Persistent Homology

There exists a choice of bases for the vector spaces Vi such thateach map is determined by a bipartite matching of basis vectors.

Get a barcode:

2 3 4 5 6 7 8 9 10 11 12


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Barcode to Persistence Landscape

Barcode:

0 2 4 6 8 10 12 14

Persistence Landscape:

2 4 6 8 10 12 14

2

4

6

0

λ1

λ2

λ3

λk = 0,

for k ≥ 4


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Topology Proteins Software Summary Topological Data Analysis Machine Learning

Maltose Binding Protein, two conformations

V. Kovacev-Nikolic, P. Bubenik, D. Nikolic, and G. Heo. Using persistenthomology and dynamical distances to analyze protein binding. StatisticalApplications in Genetics and Molecular Biology, 15 (2016) no. 1, 19–38.


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Maltose Binding Protein Data

The Data

Fourteen MBP structures from the Protein Data Bank.

7 closed conformations

7 open conformations

X-ray crystallography: coordinates of atoms.

Represent each amino acid residue by its Cα atom.

Have 14 sets of 370 points in R3.

The Goal

Can we use topological data analysis to distinguish the open andclosed conformations?


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Filtered simplicial complex


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Average persistence landscapes

H1 closed

0.70.60.50.40.30.20.180

60

40

0.08

0.06

0.1

0.04

0.02

0

0.12

0.14

20

H1 open

0.550.50.450.40.350.30.250.20.150.10.05

60

40

20

0.06

0.04

0.02

0

0.12

0.14

0.16

0.1

0.08


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Clustering of protein conformations

H1

0

8

14

11 10 13912

1

6354

2 7

1

12 1113

10 148

9

5

6 3472

12 14

765

23

14 811

13910

11

3

13 1214

89

10

45

6 71

2

L1-no

Figure 1: Distance analysis based on the 2-Landscape distance shows a separation between theclosed (blue) and the open (red) MBP conformation for degree 0 (left) and degree 1 (right) persistenthomology. Similar results hold for degree 2. The projection of the 14 × 14 distance matrix ontothe 3D space is attained via Isomap.

4.3 Statistical Inference

To measure the statistical significance of visually observed differences between the closed

and the open conformation we use a permutation test. For each degree, we calculate

fourteen sample values of the random variable X from Equation (3). The permutation

test carried out at the significance level of 0.05 yields a p-value of 5.83 × 10−4 for both

homology in degree 0 and in degree 1. We obtain the same p-value since in both cases the

observed statistic was the most extreme among all 1716 possible permutations. Hence, at

14

Projection of the L2 distance matrix to R3 using Isomap.


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Classification of protein conformations

H1

x

y

z

x

y

z

H0 H2

z

xy

- closed - open - support vector

SVM on Isomap embedded 3D coordinates in the metric space induced by the 2-Landscape distance

H1

x

y

z

x

y

z

H0 H2

z

xy

- closed - open - support vector

SVM on Isomap embedded 3D coordinates in the metric space induced by the 2-Landscape distance


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Software

Persistent Homology software:

JavaPlex

PHAT, DIPHA

Perseus

Dionysus

CHOMP

GUDHI

Persistence Landscape software:

The Persistence Landscape Toolbox

the R package TDA


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Topological Data Analysis Summary

Raw Data Clean data

Filteredsimplicialcomplex

Persistencemodule

Topologicalsummary

Statisticsand

MachineLearning

Preprocess

Transform

Homology


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