Reverse engineering gene networks using singular value decomposition and robust regression M.K.Stephen Yeung Jesper Tegner James 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: x i = concentration of mRNA (reflects expression level of genes) λ i = self-degradation rates b i = external stimuli ξ i = noise W ij = type and strength of effect of jth gene on ith gene

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 enough

In M experiments with N genes, each time apply stimuli (b 1,…,b N ) to the genes measure concentrations of N mRNAs (x 1,…,x N ) using a microarray You get: subscript i = mRNA number superscript j = experiment number

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(-λ i ) 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 (X T ) -1, but mostly M << N (this is our goal: M = log(N))

Therefore, use SVD (to find least squares sol.): Here, U and V are orthogonal (U T = U -1 ) and W is diag(w 1,…,w N ) with w i the singular values of X Suppose all w i = 0 are in the beginning, so w i = 0 for i = 1…L and w i ≠ 0 (i=L+1...L+N)

Then the least squares (L2) solution to the problem is: With 1/w j replaced by 0 if w j = 0 So this formula tries to match every datapoint as closely as possible to the solution.

But all possible solutions are: with C = (c ij ) NxN where c ij = 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.

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 sparse thus: search c ij 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)

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(N 4 ) (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 Computeby interpolation Do this M times

Test 1 Then apply algorithm to become approximation of A Computed error (with the computed A):

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

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