Hespanha

Of x at constant rate k dx x exp k d degradation of

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Unformatted text preview: exp k d degradation of “each protein X” at a constant rate Degradation regulation " feedback mechanism used to regulate the concentration of a protein by destroying protein molecules “in excess” Example #1: Degradation Regulation x(t) = concentration of protein X at time t x k dx x exp k d degradation of “each protein X” at a constant rate production of X at constant rate Degradation regulation " feedback mechanism used to regulate the concentration of a protein by destroying protein molecules “in excess” Suppose:! Gene G produces an enzyme that tags proteins for destruction (e.g., ubiquitination for subsequent degradation by the proteasome) angiebiotech.com x x k k dx G off G on protein is only degraded when the Gene is on Example #1: Degradation Regulation Negative feedback " when the protein X is a transcription factor that activates the gene X binds to G and activates it (X-dependent activation rate) λon x dt g 0 x k g x 1 k dx λoff dt X unbinds to the gene (X-independent deactivation rate) Is this enough to keep the variance bounded? Small? For which gene activation rates λon pxq ? What about higher order moments? Example #1: Degradation Regulation Negative feedback " when the protein X is a transcription factor that activates the gene X binds to G and activates it (X-dependent activation rate) λon x dt g 0 x g k x 1 k dx λoff dt X unbinds to the gene (X-independent deactivation rate) What if the degradation is constrained by the enzyme concentration? λon x dt g x k g 0 x k λoff dt 1 d xh α xh bounded decay rate ODE " Lie Derivative Given scalar-valued function V : Rn → R ￿ ￿ ￿ ￿ ￿ dV x(t) ∂ V x ( t) ￿ = f x ( t) dt ∂x derivative along solution to ODE Lf V Lie derivative of V Basis of “Lyapunov” formal arguments to establish boundedness and stability… E.g., picking V (x) := ￿x￿2 dV x t dt V fx x 0 V xt xt 2 x0 2 ￿x￿2 remains bounded along trajectories ! Generator of a Stochastic Hybrid System λ￿ q, x dt x (q, x) ￿→ φ￿ (q, x) fq x Given scalar-valued function V : Q × Rn → R ￿ ￿￿ ￿ d ￿￿ E V q (t), x(t) = E (LV ) q (t), x(t) dt x & q are discontinuous, ￿ but the expected value is ￿ differentiable Dynkin’s formula (in differential form) where LV (extended) generator of the SHS q, x V q, x fq x x Lie derivative instantaneous variation m λ￿ q, x V φ￿ q, x ￿1 intensity V q, x Reset term Example #1: Degradation Regulation λon x dt g 0 x g k x 1 k dx λoff dt LV g, x V g, x k x gd λon x 1 g λ o ff g V 1, x V 0, x L yapunov Analysis " SHSs λ(x)dt x expected-value notions sample-path notions gxw ￿ ￿ ￿￿ ￿ ￿￿ d ￿￿ E V x(t) = E (LV ) x(t) dt fx x ￿→ φ(x) probability of ||x(t)|| exceeding any given bound M, can be made arbitrarily small by making ||x0|| small class-K functions: (zero at zero & mon. increasing) α1 (￿x￿) ≤ V (x) ≤ α2 (￿x￿) LV (x) ≤ −α3 (￿x￿) ￿ ￿ V ( x)...
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