Synthetic Biology

Solutions for a better future: Mutate one gene and create a perfect cell factory!?

• Results: the mutant dies in a few hours.
• Reason: genes have multiple roles, changing one alters many functions.
• The cell is a circuit of interconnected elements (proteins/genes/metabolites/RNA/DNA/…)

Understanding circuitry

Changing paradigm

Changes in Biology

Satellites in Biology

Meta-networks

By 2050 ?

• Given a seed with

  • the DNA sequence,
    
  • the composition of the medium/organelles,
    
  • environmental parameters,

• Computers will be able to

  • grow the organism numerically… in advanced speed.
    
  • predict the impact of environmental and molecular changes.

• Numerical experiments (faster and less costly) will be a precursor to lab experiments.

Transcriptional Network

⇒ Each cell has an encoded network (circuit) in DNA.

• \(gene \rightarrow RNA \rightarrow protein\)
  
• Some protein (TF) can modify RNA production of target genes (TG)

Expression data

Data matrix (illustrative)

Spearman’s correlation

i.e. a Pearson’s correlation but on ranks, to better detect monotonic dependencies

Co‑expression networks

Compute squared Spearman correlation for all pairs

• why squaring correlations?
• why is the matrix symmetric? diagonal irrelevant?

Co‑expression networks: False‑positive trends

Let \(I(X,Y)\) be an information measure.

• If \(X_i \rightarrow X_k \rightarrow X_j\) (or \(X_i \leftarrow X_k \rightarrow X_j\)),

• then \(I(X_i,X_k)\) and \(I(X_k,X_j)\) are high

• and \(I(X_i,X_j)\) is likely high as well, suggesting a false link between \(X_i\) and \(X_j\).

• \[ \begin{array}{rcl} & fire\\ \swarrow & ? & \searrow\\ firemen \quad \quad& \longleftrightarrow &\quad victims\end{array} \]

Improving Co‑expression Networks

• Compute co-expression matrix
• Apply an arc-elimination algorithm on the matrix, e.g.
  • RelNet (thresholding) 
  • ARACNE
    
  • CLR
    
  • MRNET
• But direct thresholding (RelNet) usually fail, why?

ARACNE

Algorithm for the Recon. of Accurate Cellular NEtwork

Three possible indirect‑interaction motifs (three variables):

• \(X_j \rightarrow X_k \rightarrow X_i\)
  

• \(X_j \leftarrow X_k \rightarrow X_i\)
  

• \(X_j \rightarrow X_k \leftarrow X_i\)

• By the data‑processing inequality

  \[ I(X_j;X_i) <= I(X_j;X_k) \quad\text{and}\quad I(X_j;X_i) <= I(X_k;X_i) \]

• For every triangle, eliminate the weakest link.

ARACNE: False‑negative trends

• How about quaternary interactions?
• How about feed-back and feed-forward loops?
• ARACNE will remove genuine edges, leading to false negatives.

MRNET

Minimum Redundancy Network (Meyer et al., 2007)

For each target variable \(X_j\), select predictors using MRMR (minimum redundancy – maximum relevance):

\[ X_i^{\text{MRMR}} = \arg\max_{X_i \in X\setminus S}\bigl(u_i - z_i\bigr) \]

Iteratively build the network.

Example MRMR (1)

Example MRMR (2)

Example MRMR (3)

Example MRMR (4)

CLR – Context Likelihood of Relatedness

\[ w_{ij}= \sqrt{z_{i}^{2}+z_{j}^{2}} \]

where \(z_i\) and \(z_j\) are z-scores

\[ z_{i}= \max\!\left(0,\, \frac{I(X_{i};X_{j})-\mu_{i}}{\sigma_{i}}\right) \]

with \(\mu_{i}\) and \(\sigma_{i}\) the mean and standard deviation of the distribution of informations \(I(X_{i},X_{k})\), \(k=1,\dots,n\).

CLR (2)

principle: Each gene is choosing his bests friends

{Hayete et al. 2007}

Problem Formalization

Network Inference Objective
> Given a dataset \(D_{m}\), infer a network \(\hat T\) that has the highest similarity \(S\) with the known (or partially known) true network \(T\). \[ \hat{T}^{\max}= \arg\max_{\hat{T}} S(\hat{T},T) \]

• Confusion matrix

  edge	actual positive	actual negative
  inferred positive	TP	FP
  inferred negative	FN	TN

Weighted Network

Inferred networks are weighted:

→ Infinity of networks (depending on the threshold).
Which confusion matrix do we use?

ROC Curves

Precision–Recall

One network - One score

• Correlation with partial truth

• AUROC

• AUPR10 and AUPR20
  

• F‑Score (weighted harmonic mean of precision and recall)

  \[ 0 \le F = \frac{2\;pre*rec}{pre+rec} \le 1 \]

MiNET Bioconductor Package

Citation: Meyer et al. 2008

library(minet)
res <- validate(my.net, true.net)
show.pr(res, col = "green", type = "b")
auc.pr(res)
show.roc(res, col = "green", type = "b")
auc.roc(res)

Experimental Framework

Results

Wins/losses statistically significant on

Graphs for Guilt-by-association

A graph G is a mathematical object that allows us to represent networks (interconnected elements).

• Formally, (G = (V, E))

  • V is a set of vertices, E is a set of edges.
  • An edge is a pair of vertices (e.g. {1,2}).

• Guilt-by-association:

  • Social clubs
  • ECG (brain areas)
  • Functional gene sets (predictions on unknown genes)!

The power of using graphs

Examples:

• Bio: How many genes are targeted by a TF?
  CS: What is the degree of a vertex?
• Bio: Minimum number of steps needed to convert pyruvate into lactate?
  CS: What is the shortest path between two vertices?
• Bio: The maximal set of interacting proteins?
  CS: What is the maximally connected subgraph of G?

R package igraph

There are R packages for dealing with graphs, like igraph.

> library(igraph)
> g <- graph.formula(A-B, A-C, B-C, B-D, B-E, D-E)
V(g)
Vertex sequence:
[1] "A" "B" "C" "D" "E"
E(g)
Edge sequence:
[1] B -- A
[2] C -- A
[3] C -- B
[4] D -- B
[5] E -- B
[6] D -- E
plot(g)

Types of Graphs

> g <- graph.formula(A-+B, A+-C, B-+C, B+-D, B+-E, D-+E)
plot(g)
is.weighted(g)
# [1] FALSE
E(g)$weight <- runif(ecount(g))
is.weighted(g)
# [1] TRUE

Coding Adjacencies

Adjacency matrix:
  A B C D E
A . 1 . . .
B . . 1 . .
C 1 . . . .
D . 1 . . 1
E . 1 . . .

Adjacency list:
A -> B C
B -> C
C -> D
D -> B E
E -> B

• less memory intensive using lists
• but faster computation using matrices
• both can be used to create graphs:
  • graph_from_edgelist(el,...)
  • graph_from_adjacency_matrix(mat,...)

Graph dimensions

> g <- graph.formula(A-B,A-C,
        B-C,B-D,B-E,D-E)
> gorder(g)
[1] 5
> gsize(g)
[1] 6
> edge_density(g, loops = FALSE)
[1] 0.6

> shortest.paths(g,v="A",to="D")
    D
  A 2
> get.shortest.paths(g,from="A",to="D")
  $vpath
  $vpath[[1]]
  [1] 1 2 4
> diameter(g)
  [1] 2

Degree

> degree(g)
  A B C D E 
  2 4 2 2 2
> degree(g,mode="in")
  A B C D E 
  1 3 1 0 1 
> degree(g,mode="out")
  A B C D E 
  1 1 1 2 1 

• The degree of a vertex is the number of incident edges
• Out-degree: out of a vertex
• In-degree: toward a vertex

Clusters and Communities in a graph

> clusters(g)
$membership
[1] 1 1 1 1 1

$csize
[1] 5

$no
[1] 1

• Graph partitioning (Min dist in - max dist out):

  > fastgreedy.community(g)
  Graph community structure calculated with the fast greedy algorithm
  Number of communities (best split): 2
  Modularity (best split): 0.1111111
  Membership vector:
  A B C D E 
  1 1 1 2 2

Drawing a graph

> g <- graph.formula(A-+B,A-+C,A-+D,
       B-+F,B-+G,C-+H,C-+I)
> par(mfrow=c(1,3))
> igraph.options(vertex.size=25, edge.arrow.size=0.5)
> plot(g,layout=layout.circle)
> plot(g,layout=layout.reingold.tilford)
> plot(g,layout=layout.reingold.tilford(g,circular=T))