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Unformatted text preview: ≥ 0
LV (x) ≤ −W (x) V ( x) ≥ W ( x) ≥ 0
LV (x) ≤ −µV + c !
!
α 2 ( x 0 )
P ∃t : x(t) ≥ M ≤ α1 (M )
P x(t) → 0 = 1 almost sure (a.s.) ! asymptotic stability ∞
0 E W x(t) dt < ∞
E W x ( t) ≤e −µt c
V ( x0 ) +
µ stochastic stability
(mean square when
W (x) = x2 )
exponential stability
(mean square when
W (x) = x2 ) Example #1: Degradation Regulation
λon x dt
g 0 x g k x 1
k dx λoﬀ dt
Assume λon x 0: For every m V g, x
pg x m
LV g, x
µV x c ! 1, p0 , p1 , µ, c E xt m e 0 such that µt x0 m c
µ all moments are bounded Example #1: Degradation Regulation
λon x dt
g 0 x g k x E xt 1
k m e µt x0 c
µ m all moments are bounded dx λoﬀ dt
λon x dt
g 0 x g k x k bounded
decay rate 1
d xh
α xh λoﬀ dt
Assume λon radially unbounded: For every m
V g, x
x m 1 pg x m
LV g, x
µxm c ! 1
T 1, p0 , p1 , µ, c T Ext
0 m dt c 0 such that V g 0 ,x 0
T , T Talk Outline
Examples Modeling/Analysis tools • Biology / degradation regulation • Lyapunovbased analysis • Biology / transcription regulation • Moments dynamics (ex) students: D. Antunes (IST), A. Mesquita (UCSB), Y. Xu
(Advertising.com), A. Singh (UCSD) collaborators: M. Khammash (UCSB), C. Silvestre (IST)
acknowledgements: NSF, Institute for Collaborative biotechnologies (ARO),
AFOSR (STTR program)
disclaimer: ! This is an overview, technical details in papers referenced in
bottom right corner… http://www.ece.ucsb.edu/~hespanha Example III:(Unregulated) Gene Expression
Gene expression " process by which a gene (encoded in the DNA) produces proteins:
http://en.wikipedia.org transcription
(constant rate) mRNA #$ X + mRNA translation
mRNA #$ *
X #$ * mRNA decay
protein decay one transcription event * #$ mRNA Example III:(Unregulated) Gene Expression
http://en.wikipedia.org transcription
(constant rate) mRNA #$ X + mRNA translation
mRNA #$ *
X #$ * mRNA decay one transcription event * #$ mRNA protein decay x(t) " number of proteins at time t
prob. of one transcription
event in (t, t+dt] Kdt # of proteins produced per
transcription event
x → x + N d x dt x → x − 1 prob. of one decay
event in (t, t+dt] equivalent to Gillespie’s
stochastic simulation
algorithm (SSA) Example III:(Unregulated) Gene Expression
http://en.wikipedia.org transcription
(constant rate) mRNA #$ X + mRNA translation
mRNA decay mRNA #$ *
X #$ * one transcription event * #$ mRNA protein decay How to go beyond stability/bounds and
study the dynamics of means, variances, covariances, etc.?
x(t) " number of proteins at time t
prob. of one transcription
event in (t, t+dt] Kdt # of proteins produced per
transcription event
x → x + N d x dt x → x − 1 prob. of one decay
event in (t, t+dt] equivalent to Gillespie’s
stochastic simulation
algorithm (SSA) Moment Dynamics
http://en.wikipedia.org transcription
event Kdt decay
event x → x + N
d
E V (x) = E (LV...
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This note was uploaded on 09/24/2013 for the course ECON 202 taught by Professor Smith during the Fall '13 term at IUPUI.
 Fall '13
 Smith

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