Physics 380, 2011: Lecture 19

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Back to Physics 380, 2011: Information Processing in Biology. This is the last lecture following the article by Detwiler et al., 2000.

Student Presentation

Zhenya Botezat will present the paper Vergassola et al., 2007.

unfinished

Main lecture

1. Why are there cascades of enzymatic amplifiers? The total time delay is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max \tau_i} , and the gain is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod g_i} , while for each specific amplifier Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau_ig_i\le {\rm const}} (aka, the gain-bandwidth tradeoff). So one can have faster, stronger amplifiers with cascades.
2. But one should be careful since each new step introduces extra noise.
   • We should have the amplified fluctuations of the input be larger than the intrinsic fluctuations of the amplifier.
   • Since, at minimum, we have input fluctuations (but no input signal), we should, at minimum, have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \frac{d\omega}{2\pi} \frac{k_{XE}S_{\delta E_a}(\omega)}{k_{XX}^2+\omega^2} > \int \frac{d\omega}{2\pi} \frac{\Omega}{k_{XX}^2+\omega^2}}
   • This sets the minimum value of $k_{XE}$.
   • Plugging in the numbers, this sets the limit on the minimum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{\rm tot}} and hence on the minimum gain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{\rm tot}>\frac{1}{\tau k_a}\frac{1}{\tau k_d \bar{E}_d}\frac{\bar{E}_a}{<\delta E_a^2>}} .
3. Why are the enzymatic amplifiers so great? They are tunable. Let's look at one simple version of the tunability: linear negative feedback.