Reverse engineering gene networks using singular value decomposition and robust regression

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Reverse engineering gene networks using singular
value decomposition and robust regression

• M.K.Stephen Yeung
• Jesper Tegner
• James J. Collins

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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

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• 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)
• ?i self-degradation rates
• bi external stimuli
• ?i noise
• Wij type and strength of effect
• of jth gene on ith gene

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(No Transcript)
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• 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

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• 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 number
• superscript j experiment number

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• 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

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• System becomes
• With A W diag(-?i)
• Compute by using several measurements of the
  data for X. (e.g. using interpolation)
• Goal deduce W (or A) from the rest
• If MN, compute (XT)-1, but mostly M ltlt N
• (this is our goal M log(N))

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• 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 X
• Suppose all wi 0 are in the beginning, so wi
  0 for i 1L and wi ? 0 (iL1...LN)

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• 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.

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• But all possible solutions are
• with C (cij)NxN where cij 0 if j gt 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.

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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

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• 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)

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• 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)

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• Thus, to reverse-engineer a network of N genes,
  we only need Mc O(logN) experiments.
• Then Mc ltlt 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)

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Test 1

• Create random connectivity matrixfor each row,
  select k entries to be non-zero
• - k lt kmax ltlt 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

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Test 1

• Then apply algorithm to become approximation of A
• Computed error (with the computed A)

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• Results Mc O(log(N))
• Better than only SVD,
• without regression

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Test 2

• One-dimensional cascade of genes
• Result for N 400
• Mc 70

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Test 3

• Large sparse gene
• network, with ran-
• dom connections,
• external stimuli,
• Results the same
• as in previous tests

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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

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Discussion

• Disadvantages
• Less efficient for small networks (MN)
• No quantification yet of the necessary
  sparseness, though avg. 10 connections is good
  for a network containing gt 200 genes
• Uncertain
• Especially useful with exact data, which we dont
  have

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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