Hespanha

# Zero mon increasing 1 x v x 2 x lv

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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) = ￿x￿2 ) exponential stability (mean square when W (x) = ￿x￿2 ) 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 • Lyapunov-based 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 bio-technologies (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, co-variances, 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.

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