This preview shows page 1. Sign up to view the full content.
Unformatted text preview: )(x)
dt x → x − 1 (LV )(x) = K V (x + N) − V (x)
+ d x V (x − 1) − V (x) dE [ x]
= K E[N] − d E[x]
dt
dE [ x2 ]
= K E[N2 ] + (2K E[N] + d)E[x] − 2d E[x2 ]
dt
One can show that d x dt (Unregulated) Gene Expression
http://en.wikipedia.org transcription
event Kdt x → x + N d x dt
decay
event x → x − 1 Thus, at steadystate, • measure of stochastic fluctuations in protein level x
(normalized by mean population)
• intrinsic noise (solely due to random protein
expression/degradation) AutoRegulated Gene Expression
http://en.wikipedia.org * #$ mRNA transcription
(nonconstant rate) mRNA #$ X + mRNA translation
mRNA #$ *
X #$ * mRNA decay
protein decay Protein production rate is a function of the current protein molecule count through
transcription regulation:
transcriptional response
(stochastic rate at which
transcription events occur) • Altering the RNA polymerase specificity for a
given promoter or set of promoters
• Binding to noncoding sequences on the
DNA to impede RNA polymerase's progress g(x) dt x → x + N d x dt x → x − 1 AutoRegulatory Negative Feedback
http://en.wikipedia.org transcription g(x) dt
event x → x + N d x dt
decay
event x → x − 1 negative feedback % protein production rate is a decreasing function
of the protein molecule count transcriptional
response • Common form of auto regulation
(e.g., half of the repressors in E. Coli)
• Experimentally shown to exhibit noise
reduction ability g(x)
x Moment Dynamics
http://en.wikipedia.org transcription g(x) dt
event d x dt x → x + N
d
E V (x) = E (LV )(x)
dt decay
event x → x − 1
(LV )(x) = g (x) V (x + N) − V (x)
+ d x V (x − 1) − V (x) dE [ x]
= E[N]E[g (x)] − d E[x]
dt
dE [ x2 ]
= E[N2 ]E[g (x)] + 2 E[N]E[g (x)x] + dE[x] − 2d E[x2 ]
dt ! When g(x) is an afﬁne function we still get a ﬁnite system of linear equations
! When g(x) is a polynomial, we get a closed but inﬁnite system of linear equation
(general property of polynomial SHSs)
! For other g(x), one generally does not get a closed system of equations AutoRegulated Gene Expression
http://en.wikipedia.org transcription g(x) dt
event x → x + N d x dt
decay
event x → x − 1 Approximate Analysis Methods
Distributionbased: assume a specific type of distribution (Normal, LogNormal,
Poisson, etc.) and force dynamics to be compatible with this type of distribution
Large numbers/large volume: take the limit as volume $ & and assume
concentrations do not $ 0
Derivative matching: force solutions of approximate dynamics to match exact
equation locally in time
Linearization: Linearize transcriptional response around steadystate value of the
mean AutoRegulatory Negative Feedback
http://en.wikipedia.org transcription g(x) dt
event x → x + N d x dt
decay
event x → x − 1 For a transcriptional response approximately linear around steadystate mean
steadystate
population mean protein...
View
Full
Document
 Fall '13
 Smith

Click to edit the document details