Hespanha

Of one decay event in t tdt equivalent to gillespies

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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 steady-state, • measure of stochastic fluctuations in protein level x (normalized by mean population) • intrinsic noise (solely due to random protein expression/degradation) Auto-Regulated Gene Expression http://en.wikipedia.org * #$ mRNA transcription (non-constant 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 non-coding sequences on the DNA to impede RNA polymerase's progress g(x) dt x ￿→ x + N d x dt x ￿→ x − 1 Auto-Regulatory 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 Auto-Regulated 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 Distribution-based: 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 steady-state value of the mean Auto-Regulatory 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 steady-state mean steady-state population mean protein...
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