Carissa Bleker, Ashley Cliff, Sergio Garciaweb.eecs.utk.edu/~cphill25/cs594_spring2017/... ·...

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Carissa Bleker, Ashley Cliff, Sergio Garcia

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Test Questions

1. Which thresholding method did we try to use?

2. Name a type of calculated edge.

3. Name one type of graph that can be used to represent metabolic

networks.

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Ashley Cliff● Bredesen Center Student (DSE)

○ Advisor: Dan Jacobson (ORNL)● Central College, Pella, IA

○ BA: Physics & Computer Science● From: Decorah, IA

○ Population: ~8,000

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Carissa Bleker

Bredesen Center Student (DSE)Advisor: Dr Langston

Stellenbosch University:● BScHons Mathematics

From Cape Town, South Africa● Population of 3.7 million● Not the tip of Africa…

Mojo & Amper

Cape Town Cape L’Agulhas

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Sergio Garcia

● From: Murcia (Population 439K), Spain.

● Pursuing PhD in Chemical and Biomolecular Engineering.

● I play the piano.

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Presentation Outline

1. Background

2. Building Graphs from Real Data

3. Time Varying Graphs

4. Network Analysis Tools

5. Common Networks in Molecular Biology

6. Higher Level Biological Examples

7. Yeast Life Cycle Time Varying Graph

8. Issues

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1. Background

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The Discipline of Systems Biology

● Systems biology is the computational and mathematical modeling of complex biological systems.

● Focus on complex interactions within biological systems, using a holistic approach (holism instead of the more traditional reductionism) to biological research.

● Model and discover emergent properties of cells, tissues and organisms functioning as a system.

● Applications: Disease, biocatalysis, waste management, ….

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Cell

Communities

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High-throughput Collection of Biological Data

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2. Building Graphs from Real Data

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Building Graphs from Real (Dirty) Data

● Formatting ○ File format - tab vs space vs comma vs no format○ Remove corrupted lines, odd characters (#, %)○ Create ‘simple’ Dimacs or adjacency matrices

● Data cleaning○ Duplicate data○ Are zeros actually zeros○ Missing values (remove, ignore, impute)○ Standardizing/Normalizing

These steps could take longer than the graph analysis - never assume your data is clean/formatted

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Determining Vertices and Edges

● What are the vertices?○ genes, locations, molecules, …

● And edges?○ interactions, covariation, proximity, ...○ roads, wired connection○ Calculated edges - correlation○ Directed, weighted

● Multigraphs○ Ex: Vertices - cities, Edges - roads (blue), direct flight (red), etc..

● What different insights are gained by changing or swapping how we identify vertices and edges?

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Calculated Edges

● Pearson correlation coefficient

○ Pearson p-value

● Spearman (rank based Pearson)

○ Spearman p-value

● Cosine

● Euclidian

● Mutual information

● ...

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Calculate Edges

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Thresholding

A correlation analysis or matrix will generate a complete graph. Thresholding aims to separate signal from noise.

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Thresholding

After: we only have significant edges/associations between vertices.

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Thresholding: Spectral methods

• Weaker (smaller weighted) edges connect dissimilar clusters of the graph

• As t (the threshold) is increased:→ weaker edges are removed→ dissimilar clusters are less connected→ the number of “nearly-disconnected” clusters increases

Perkins, A. D., & Langston, M. A. (2009). Threshold selection in gene co-expression networks using spectral graph theory techniques. Bmc Bioinformatics, 10(11), S4.

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Thresholding: Spectral methods

Finding “nearly-disconnected” clusters: ● Extract the largest connected component● Laplacian of G:

● Sort the values of eigenvector of the second smallest eigenvalue

● Results in an ascending step like function, and each step corresponds to a transition from one cluster to another

Perkins, A. D., & Langston, M. A. (2009). Threshold selection in gene co-expression networks using spectral graph theory techniques. Bmc Bioinformatics, 10(11), S4.

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• Select t that maximises the number of "nearly-disconnected" components, and therefore minimises the number of edges connecting dissimilar parts of the network.

Thresholding: Spectral methods

Perkins, A. D., & Langston, M. A. (2009). Threshold selection in gene co-expression networks using spectral graph theory techniques. Bmc Bioinformatics, 10(11), S4.

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Thresholding: Random Matrix Theory

Based on the nearest neighbor spacing distribution (NNSD) of eigenvalues from the adjacency/correlation matrix.

NNSD: differences between subsequent (ordered) eigenvalues

Jalan, S., & Bandyopadhyay, J. N. (2007). Random matrix analysis of complex networks. Physical Review E, 76(4), 046107.

random network scale-free network small-world network

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Thresholding: Random Matrix Theory

Based on the nearest neighbor spacing distribution (NNSD) of eigenvalues from the adjacency/correlation matrix.

NNSD: differences between subsequent (ordered) eigenvalues

NNSD of eigenvalues of a random matrix can be approximated by Wigner Surmise

NNSD of a non-random matrix appears Poisson

Iterate over t until we find the point of transition of the NNSD

Gibson, S. M., Ficklin, S. P., Isaacson, S., Luo, F., Feltus, F. A., & Smith, M. C. (2013). Massive-scale gene co-expression network construction and robustness testing using random matrix theory. PLoS One, 8(2), e55871.

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Thresholding: Random Matrix Theory

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3. Time-varying Graphs

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Time-varying Graphs

● Dynamic undirected graphs with fixed underlying vertex set○ Edges change over time, vertices do not

● AKA time evolving graphs (TEG)

● Useful for time based data ○ gene expression over time, ○ congestion at intersections, etc

● Use simple metrics (or more complicated) to ‘label’ the differences between time steps

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4. Network Analysis Tools

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Network Properties

● Density: Some biological networks are sparse

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Network Properties

● Clustering coefficient

N= |V|; Ei = |edges between neighbors of i|; ki = degree of i;

Average clustering coefficient for the metabolic networks of 43 organisms (colors represent different taxonomic domains). N is the number of nodes.The diamonds correspond to a scale free network with the same number of nodes and edges.

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Network Properties

● Other:○ Diameter: Shortest distance between the two most distant nodes

in the network. The diameter of metabolic networks is conserved even across distant organisms.

○ Average path length: Average shortest paths.○ Degree Distribution: Number of nodes with a certain degree.

Biological networks tend to follow a power law.○ ...

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Complex Network Models

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Node Centralities and Ranking

● Degree centrality: Nodes with high degree centrality are hubs. While biological networks are robust against perturbation, the removal of hubs often leads to system failure.

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Node Centralities and Ranking

● Other:○ Closeness Centrality: Indicates important nodes that can

communicate quickly with other nodes. Used to identify key central metabolites and extract the core metabolic network.

○ Betweenness Centrality, nodes that appear in many shortest paths rank higher. Metabolites controlling flux between two modules. In telecommunication networks such node would have higher control.

○ Eigenvector Centrality, ranks higher the nodes that are connected to important neighbors. Used to identify pairs of genes that cause sickness/death.

○ ....

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Clustering

● Clusters are parts of a graph that are highly associated

● In biology clusters are of interest for a number of reasons:○ Finding co-regulated vertices○ Finding vertices that are part of the same process○ Hypothesising functionality on unannotated vertices

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Underlying Idea: A random walk on a transition graph that starts within a cluster, is more likely to stay within that cluster than to leave it.

Clustering Algorithms: Markov Clustering

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Two steps on a transition matrix:

1. Matrix square- Simulates random walks through the graph

2. Elementwise matrix squaring- Strengthens strong transition probabilities, and weakens low

probabilities

Clustering Algorithms: Markov Clustering

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Transition matrix

Pi, j = P(i | j)

= probability of walking from j to i

Each column consists of the probabilities of each way you can leave that node, and sums to 1

Clustering Algorithms: Markov Clustering

1 2 3 4 5

1 P1, 1 P1, 2 . . P1, 5

2 P2,1 . .

3 . . .

4 . . .

5 P5, 1 P5, 5

M =

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Transition matrix multiplication

(M2) i, j = ∑k Pi, k Pk, j = ∑kP(walk to i from j through k)

(M2) 2, 3 = Probability of walking to 2 from 3, over all 2-step paths

Clustering Algorithms: Markov Clustering

P1, 1 P1, 2 . . P1, 5

P2,1 . .

. . .

. . .

P5, 1 P5, 5

P1, 1 P1, 2 . . P1, 5

P2,1 . .

. . .

. . .

P5, 1 P5, 5

XM2 =

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Clustering Algorithms: Markov Clustering

G is a graph

add self-loops to G

set parameter I

set M the stochastic matrix of G

while (change > ε){

M’ = M x M

M’ = ГI (M’)

make M’ stochastic

change = M - M’

M’ = M

}

Clustering is the components of M’

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Clustering Algorithms: Markov Clustering

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Clustering Algorithms: Markov Clustering

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Clustering Algorithms: Markov Clustering

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Clustering Algorithms: Markov Clustering

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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Clustering Algorithms: Paraclique

g = 3

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5. Common Networks in Molecular Biology

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Metabolic Networks● Network of chemical reactions enabling the conversion of substrates

into energy and biomass. Can be represented by:○ Simple graphs○ (Directed) bipartite graphs.○ (Directed) hypergraphs

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Hierarchical Modularity in Metabolic Networks

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Protein-protein Interaction Networks● Represent how different proteins operate in coordination with

others to enable biological processes within the cell.

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Gene Co-expression Networks

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6. Higher Level Biological Examples

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Epilepsy Seizure Prediction

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fMRI in Schizophrenia

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● WHO child cause of death for 194 countries in 2013

● Method:

○ Pearson correlation between countries, across COD categories

● Graph:

○ Vertices - Countries

○ Edges - Similarities in COD

● Threshold at 0.95

● Markov clustering

Health Disparities

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Africa

Americas

East Mediterranean

Europe

South East Asia

Western Pacific

Health Disparities Graph

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Health Disparities Clustering

Africa

Americas

East Mediterranean

Europe

South East Asia

Western Pacific

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7. Yeast Life Cycle Time Varying Graph

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Want to know how gene-gene associations change over the life cycle of a yeast cell.

Yeast Life Cycle

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Yeast Life Cycle Data

● Yeast gene expression data collected from synchronised cultures

● 24 time points over 10 minute increments● 6,178 genes

Time points

Genes

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Expression Over Time

Time (s)

Nor

mal

ized

exp

ress

ion

valu

e

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Yeast Example Process

1. Removed genes with:

○ More than 4 missing values○ Low variance over all time points (<0.6)

2. Calculated all-to-all pairwise Spearman correlations for time steps

3. Spectral thresholding - did not work

○ Hard cut off of 0.8 for all graphs

4. Metric calculations

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Variance

Variance

Num

ber o

f gen

es

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Time-varying Co-expression Network

t-1 t-2 ... t-M

G-1

G-2

::

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Time-varying Co-expression Network

t-1 t-2 ... t-M

G-1

G-2

::

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Time-varying Co-expression Network

t-1 t-2 ... t-M

G-1

G-2

::

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Time-varying Co-expression Network

t-1 t-2 ... t-M

G-1

G-2

::

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Time-varying Co-expression Network

t-1 t-2 ... t-M

G-1

G-2

::

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Time-varying Co-expression Network

t-1 t-2 ... t-M

G-1

G-2

::

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Time Graphs - Time 1

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Time Graphs- Time 2

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Time Graphs- Time 3

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Time Graphs- Time 4

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Time Graphs- Time 5

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8. Issues

● Figure out how to threshold

● Better metrics to pinpoint differences in time based graphs

● Network validation, particularly for less studied systems

● Noise in high-throughput data

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Test Questions

1. Which thresholding method did we try to use?

2. Name a type of calculated edge.

3. Name one type of graph that can be used to represent metabolic

networks.