Physics 434, 2012: Lecture 10

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Back to Physics 434, 2012: Information Processing in Biology.

We are wrapping up all the loose ends for the probability/random walks section of the class. We will return to some of the related questions in the later sections, of course. During this lecture, we also started the new block on information theory, Physics 434, 2012: Lectures 10-11.

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• Langevin equation. If a chemical species is produced in a reaction $F$ and degraded in a reaction 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 G} , and all of reactions are independent, then the mean number of produced particles per time 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 t} 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 (F-G)\Delta t} and the variance 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 (F-G)\Delta t} . If the number of production and degradation events is large, 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 F,G\gg 1} , then these terms can be approximated as Gaussians. We can, therefore, write 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(t+\Delta t)=x(t)+\Delta x=x(t) + (F-G)\Delta t+\sqrt{(F+G)\Delta t}\eta} , where 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 \eta\sim N(0,1)} .
  • Notice that the noise term scales as 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 \sqrt{\Delta t}} and is larger than the deterministic term for small 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 \Delta t}
  • We can transform this into a (stochastic) differential equation by taking a limit 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 \Delta t\to0} , 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 \frac{dx}{dt}=F-G+\sqrt{F+G}\eta} , where 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 \langle \eta\rangle =0} , and 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 \langle \eta(t)\eta(t')\rangle = \delta(t-t')} . Such 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 \eta} is called a Wiener process, after Norbert Wiener, or white noise -- we will understand why later.
• As the last item in this block of the class, we have talked about gradient sensing in another organism, D. discoideum. It's a large bug, and so many of the constraints that E. coli had do not apply. We have introduced the Local Excitation -- Global Inhibition (LEGI) (Levchenko and Iglesias, 2002), which allows an organism to do a spatial, rather than a temporal comparison of chemical concentrations. The question then is: does what we have discussed in this "probability" block of the class apply? We discussed that a comparison is made locally, by a small volume (about a size of a molecular complex), and hence the arrival of molecules is very stochastic. Even for a large organism, the noise may be quite important!