Transcriptional Regulatory Network

1
Transcriptional Regulatory Network

• A Stochastic Differential Equation (SDE) Model
  for Quantifying Transcriptional Regulatory
  Network in Saccharomyces cerevisiae

2
Goal

• In systems biology or genetic network, four
  models are widely used Linear Differential
  Equation, S System, Boolean Network and Bayesian
  Network.
• Goal Apply stochastic differential equation
  (SDE) to model the dynamic stochastic patterns in
  genetic network?
• With the use of SDE, it enables not only
  transform linear differential equation model into
  statistical model, but also combine with Bayesian
  network to be applied to small sample problem in
  our future work?
• Our SDE model is basic but the test results agree
  very well with the observed dynamic expression
  patterns.

3
e.g. S System

• Dynamics of biological systems at the level of
  pools of molecular species can be described
  mathematically by the S-system as follows (Voit,
  2000) (power function)
• where X is n-dimensional components or pools, the
  parameter vector p consists of rate constants, aj
  and ßj , and kinetic orders, gij and hij . f is
  the set of net rate equations, which consists of
  the influxes and the effluxes. The m-dimensional
  independent variables in the S-system equations
  are expressed as Xnj , j 1, . . . ,m.

4
Dynamic Stochastic Model (1/4)

• Let Nt denote the exact amount of target gene
  mRNA at time t.
• From time t to t?t, the dynamic transcription
  and degradation process is
• (Nt?t Nt )/Nt ?Nt / Nt
• (gt ?) ?t et,?t
• where gt is the transcription rate, ? is the
  degradation rate, et,?t is the noise or random
  error captured by normal distribution N(0,s2?t).

5
Dynamic Stochastic Model (2/4)

• Let ?t ? 0, we have the SDE as follows,
• where Wt is the standard Brownian motion, which
  represents the source of uncertainty of random
  error, and s is the fluctuation.

6
Dynamic Stochastic Model (3/4)

• Since Nt could not be directly measured, we
  measure the signal intensity St instead, which is
  proportional to Nt.
• Let Xt log(St B) be the expression level of
  the gene, where B is the background intensity.
• Without loss of generality, we might assume that
  Xt logNt.

7
Dynamic Stochastic Model (4/4)

• From Itô rule, we have
• 
  (1)
• where XtlogNt denotes the expression level of
  the gene,
• P.S. In Chen, et al. (2004) Bioinfomatics,
  their model is as follows

8
Regulatory Functions (1/2)

• At time t, the transcription rate gt of the
  target gene depends on the n regulators through
  regulatory functions.
• Let Xit be the expression level of the i-th
  regulator at time t. The regulatory function of a
  specific regulator i is described as a sigmoid
  function
• 
  (2)
• where the mean and s.d. are estimated by
  and

9
Regulatory Functions (2/2)

• Let
  (3)
• Combing equations (1) and (3), we have
• where c0 c0 -?/2 s2/2, and ci is the
  contribution of regulator i to the
  transcriptional regulatory network.

10
Statistical Method (1/2)

• From time tj to tj1, j1,2,,m, we have
• where i1,2,,n denote the n regulators, fi
  is the regulatory functions and Ztj are i.i.d.
  normally distributed.
• For the m samples at time tj, j1,2,,m

11
Statistical Method (2/2)

• where
• is observed, and s are unknown
  but can be
• estimated by simple linear regression.
• But in Chen et al. (2004), Y is unknown and the
  statistical inference is incomplete.

12
Computational Approach

• Initially for n1.
• For a given gene, choose the regulator with
  the highest log likelihood value from the
  candidate regulator pool.
• For nn1.
• Choose the new one when it combines with the
  former regulators can achieve the highest log
  likelihood.
• For each n, calculate the AIC value.
• If AIC starts to increase, stop and the
  former regulators are the ones we want.

13
Result

• Result.1,2
• http//www.csie.ntu.edu.tw/b89x035/yeast/

14
Result

• Result.3,4

15
Result

• Result.5,6

16
Result

• Result.best.1,2

17
Result

• Result.best.3

18
Result

• Result.worst

19
Result

• Result.worst

20
Twenty specific target genes
Table 1. Twenty specific target genes with
corresponding putative regulators and associated
regulatory abilities
21
Ten best-fitting target genes
Table 2. (a) Ten best-fitting target genes with
corresponding putative regulators and associated
regulatory abilities
22
Ten worst-fitting target genes
(b) Ten worst-fitting target genes with
corresponding putative regulators and associated
regulatory abilities
23
The statistical summary of regulators of the SDE
mode
24
Future Work

• Associate with Bayesian Networks to apply in the
  small sample analysis.
• Add the nonlinear terms to model the cross
  interaction.
• Add the exponential terms to model the powers
  law, e.g. associate with S system.
• When it comes to the phase change, we can add the
  Boolean functions, State space models (SSMs).