Intrinsic Noise in Nonlinear Gene Regulation Inference
1
Du
Chao
& Wing H.
2
Wong
1 Department of Statistics, University of Virginia. 2 Department of Statistics, Stanford University.
Intrinsic Noise: Modeling and Inference
Transcriptional Regulation: An Overview
• The abundance of each species (RNA or protein) is treated as discrete
random variable that represents the copy number of molecules.
Regulation
Transcription
Gene 1
RNA 1
Translation
Translation
Transcription
Gene n
RNA n
Protein 2
∅
Translation
Protein n
Degradation
Degradation
∅
RN A
xi
i
∅
Degradation
Degradation
∅
ri x i
Production
⌧i Fi (X)
X = {X1 , X2 , · · · , XM }
P rotein
yi
i
∅
• The properties of gene regulation system are determined by
the rates of transcription, translation and degradation, which
are determined by the abundance of RNAs and proteins.
• The key question is how to infer the regulation relationships
between genes, based on the expression data.
Gene i
X
X0
0
W (X ! X )P (X) +
Pmi
bi0 + j=1 bij Sij
Pmi
Fi (X) =
1 + j=1 cij Sij
Sij =
Y
n
bij =?
cij =?
Xk kij
k
∅
X0
W (X 0 ! X)P (X 0 )
W (X ! X 0 )
X0
i Xi
{X1 , · · · , Xi + 1, · · · , XM }
{X1 , · · · , Xi , · · · , XM }
{X1 , · · · , Xi
At the steady state, we can
utilize the moments generating
function of X to construct a
series of linear functions of
unknown parameters.
=⌧i e
i
h
0
⇥
bi0 E e
k Xk
P
k
⇤
+
mi
X
k Xk
⇥
cij E gnij (Xi )Sij Xi
mi
X
+
⇤
H(X))W (X ! X ) = 0
j=1
⇤
i Xi
0
E (H(X )
h ⇥
P
i E Xi e k
⌧i F (X)
1, · · · , XM }
⇥
dP (X)
=0
dt
Solution:
Problem of Interests:
X
X
∅
How to infer the unknown
parameters and terms in the
rational function based on the
observed distribution of X?
• Modern single-cell level experiment allows us to study the
gene regulation system as a stochastic system.
dP (X)
=
dt
Xi
Degradation
∅
Protein i
Degradation
RNA 2
i fi (y)
yi
Translation
RNA i
Degradation
Gene 2
Transcription
Degradation
Degradation
∅
Transcription
Protein 1
xi
• The system can be modeled by a multivariate birth-death process whose
evolution is described by Kolmogorov forward equation (master equation)
j=1
⇥
bij E Sij e
P
k
nij
k Xk
e
P
⇤i
k
k Xk
⇤i
rk 2 R
[1]
Procedure of Inference:
1. Propose candidate
model (rational function)
How to Account for the Physical Nature of Cellular System?
2. Collect single-cell expression
data from different steady states
(via perturbation experiments)
3. Estimate the moments of gene
expression at steady states, construct
linear functions of unknown parameters.
4. Estimate unknown parameters
using convex optimization.
1. Quasi-Steady-State Assumption
Production
• As RNA metabolism is often much slower (or faster) than protein
metabolism, we may assume that the concentrations of protein (or RNA)
species always reach “quasi-steady state” instantly.
Gene i
⌧i fi (x)
Degradation
• This assumption allows us to simplify the model to only include RNA (or
protein) species.
Example: Genetic Toggle Switch
i xi
• Genetic toggle switch refers to a bistable gene-regulation module.
∅
• It exists in natural gene circuit and has been constructed artificially in the lab [3].
• Can be formed by two repressor genes.
Pmi
2. Production Rates as a Nonlinear Rational Function
bi0 + j=1 bij Sij
P mi
⌧i fi (x) = ⌧i
1 + j=1 cij Sij
• The production rates are determined by the concentrations of different
transcription factors, which are protein molecules or complexes formed
by several protein molecules.
0  bi0  1
n
xk kij
F1 (X1 , X2 )
0.5
=
1 + X22
0  bij  cij
k
Gene 1
0.01X1
Production
Gene 2
F2 (X1 , X2 )
0.5
=
1 + X12
Degradation
• By the law of thermodynamic, the production rates can be modeled as a
nonlinear rational function [2]:
Production
∅
Example:
0.5 + x1 + 0.01x2 x3
f1 (x) =
1 + x1 + 0.9x22 + 0.2x2 x3
x1
x1
0.5
1
0
0.9x22
0.01x2 x3
0.2x2 x3
3. Construct linear functions
X2
X1
Degradation
Sij =
Y
Example:
[4]
Unperturbed Steady-State
( 1,
Perturbed Steady-State
(
2)
1,
2 [0, 1] ⇥ [0, 1]
2)
2 [0, 2] ⇥ 0
0.01X2
b10
b11
b12
∅
⇥
1. Candidate model
b13
b14
b15
c11
c12
c13
Gene 3
NA
P1
NA
Activator
P2
P2
P2
P3
Repressor
Repressor
F1 (X1 , X2 ) =
Gene 1
Gene 2
c14
b10 + b11 X1 + b12 X2 + b13 X12 + b14 X1 X2 + b15 X22
1 + c11 X1 + c12 X2 + c13 X12 + c14 X1 X2 + c15 X22
c15
4. Estimate parameters
2. Collect sample from two steady states
3. Intrinsic Noise
Unperturbed Steady-State
• In a typical ensemble experiment, measured gene expression represents
the ensemble average of huge number of cells. Modern technique
allows us to observe gene expression at single cell level, which reveals
intrinsic noise — stochastic cell-to-cell variation.
• The intrinsic noise, or the observed distribution of gene expression at
single cell level, contains valuable information regarding the underlying
regulation relationship.
X1
X1
X2
X2
X̄
X3
X4
…
X1
…
Ensemble Average Measurements
X3
X4
..
.
Perturbed Steady-State
X2
X3
X4
…
…
(X1 , X2 )
(X1 , X2 = C)
Single-Cell Level Measurements
[1] Source of background image: http://cronodon.com/images/DNA4.jpg.
[2] Bintu, L., Buchler, N. E., Garcia, H. G., Gerland, U., Hwa, T., Kondev, J., and Phillips, R. (2005). Transcriptional regulation by the numbers: models. Current Opinion in Genetics & Development 15: 116–124.
(2000 samples per steady-state)
[3] Gardner, T., Cantor, C., and Collins, J. (2000). Construction of a genetic toggle switch in Escherichia coli. Nature 403: 339–342.
[4] Source of Image http://openwetware.org/images/4/4a/Geneswitch.png.