Reverse engineering gene networks using singular value decomposition and robust regression

Reverse engineering gene networks using singular value

decomposition and robust regressionM.K.Stephen Yeung

Jesper TegnerJames J. Collins

General idea

Reverse-engineer:• Genome-wide scale• Small amount of data • No prior knowledge• Using SVD for a family of possible

solutions• Using robust regression to choose from

them

If the system is near a steady state, dynamics can be approximated by linear system of N ODEs:

xi = concentration of mRNA

(reflects expression level of genes)λi = self-degradation rates

bi = external stimuli

ξi = noise

Wij = type and strength of effect

of jth gene on ith gene

)()()()()(1

ttbtxWtxtx ii

N

jjijiii

Suppositions made:• No time-dependency in connections

(so W is not time-dependent), and they are not changed by the tests

• System near steady state

• Noise will be discarded, so exact measurements are assumed

• can be calculated exactly enoughX

In M experiments with N genes, • each time apply stimuli (b1,…,bN) to the genes• measure concentrations of N mRNAs (x1,…,xN)

using a microarray

You get:

subscript i = mRNA numbersuperscript j = experiment number

MNNN

M

M

MN

xxx

xxx

xxx

X

21

222

12

121

11

x

Goal is to use as few measurements as possible. By this method (with exact measurements):

M = O(log(N))e.g. in 1st test, the results will be:

System becomes:

With A = W + diag(-λi)Compute by using several measurements of the

data for X. (e.g. using interpolation)Goal = deduce W (or A) from the rest

If M=N, compute (XT)-1, but mostly M << N (this is our goal: M = log(N))

MNMNNNMN BXAX xxxx

NMT

NMT

NNT

NMT BXAX xxxx

X

Therefore, use SVD (to find least squares sol.):

Here, U and V are orthogonal (UT = U-1)and W is diag(w1,…,wN) with wi the singular

values of XSuppose all wi = 0 are in the beginning, so wi = 0

for i = 1…L and wi ≠ 0 (i=L+1...L+N)

NNT

NNNMNMT VWUX xxxx

NM

TNM

T

NNT

NNT

NNiNM

BX

AVwdiagU

xx

xxxx )(

Then the least squares (L2) solution to the problem is:

With 1/wj replaced by 0 if wj = 0

So this formula tries to match every datapoint as closely as possible to the solution.

NNT

jNMMNMN V

wdiagUBXA xxxx0

1

But all possible solutions are:

with C = (cij)NxN where cij = 0 if j > L and otherwise just a scalar coefficient

How to choose from the family of solutions ?The least squares method tries to match

every datapoint as closely as possible → a not-so-sparse matrix with a lot of

small entries.

TCVAA 0

1. Basing on prior biological knowledge,impose this on the solutions.e.g.: when we know 2 genes are related,the solution must reflect this in the matrix

2. Work from the assumption that normal gene networks are sparse, and look for the matrix that is most sparsethus: search cij to maximize the number of zero-entries in A

So:• get as much zero-entries as you can• therefore get a sparse matrix• the non-zero entries form the connections

• fit as much measurements as you can, exactly: “robust regression”(So you suppose exact measurements)

Do this using L1 regression. Thus, when considering

we want to “minimize” A. The L1 regression idea is then to look for the

solution C where is minimal.

This causes as many zeros as possible.

Implementation was done using the simplex method (linear adjustment method)

10 |||| TCVA

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Thus, to reverse-engineer a network of N genes, we “only” need Mc = O(logN) experiments.

Then Mc << N, and the computational cost will be O(N4)

(Brute-force methods would have a cost of O(N!/(k!(N-k)!)) with k non-zero entries)

Test 1• Create random connectivity matrix:

for each row, select k entries to be non-zero- k < kmax << N (to impose sparseness)- non-zero entry random from uniform distrib.

• Do random perturbations• Do measurements while system relaxes back

to its previous steady state → X• Compute by interpolation • Do this M times

X

Test 1

• Then apply algorithm to become approximation of A

• Computed error (with the computed A):A~

otherwise 0

|~| if 1 where

1 1

ijijij

N

i

N

jij

AAe

eE

• Results: Mc = O(log(N))

• Better than only SVD, without regression:

Test 2

• One-dimensional cascade of genes

• Result for N = 400:

Mc = 70

Test 3

• Large sparse gene network, with ran-dom connections, external stimuli,…

• Results the same as in previous tests

Discussion

Advantages:• Very few data needed, in comparison with

neural networks, Bayesian models• No prior knowledge needed• Easy to parallelize, as it recovers the

connectivity matrix row by row (gene by gene)

• Also applicable to protein networks

Discussion

Disadvantages:• Less efficient for small networks (M≈N)• No quantification yet of the necessary

“sparseness”, though avg. 10 connections is good for a network containing > 200 genes

• Uncertain • Especially useful with exact data, which

we don’t have

X

Improvements

• Other algorithms to impose sparseness: alternatives are possible both for L1 (basic criterion) as for simplex (implementation)

• By using a deterministic linear system of ODEs, a lot has been neglected (noise, time delays, nonlinearities)

• Connections could change by experiments;then the use of time-dependent W is

necessary