Physics 434, 2012: Lecture 14

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Back to Physics 434, 2012: Information Processing in Biology. Good lecture notes on the subject are in Bialek's draft textbook, version 2011.

Warmup question

  1. E. coli and many other bacteria exhibit a phenomenon of persistence, so elegantly studied in Balaban et al., 2004. Briefly bacteria may choose to switch from a growing to a non-growing phenotype, when they are less sensitive to effects of antibiotics and persist through antibiotic applications. This is advantageous when antibiotics are applied, but they don't grow in the persistent state, making it a loosing choice when there are no antibiotics. Leaving aside the issue of how they actually switch (we will study a related problem a bit later in class), can you predict what should the bacterial strategy be in choosing whether to divide or not to divide? Which considerations should enter the bacterial decision?

Main Lecture

Population biology

  • Once switched, a bacterium remains committed to the state for a while. We will discretize the time and consider the bacterium in making a choice 0 (growing) and 1 (persistent) once per time step. The time step is much longer than the switching time and the division/death time of a single bacterium.
  • Each time step does (1) or doesn't (0) have an application of an antibiotic, and the bacterium can be in the persistent (1) or the growing state (0).
  • For the state of bacteria and the world 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 [bw]=\{[00],[01],[10],[11]\}} , at the end of the time period, the number of the bacteria will be multiplied by the growth rate of the bacteria, which we call 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 \alpha_{bw}} . We choose 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 \alpha_{00}=\alpha^{(0)}} -- growth in unhindered conditions; 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 \alpha_{01}=0} -- growing bacteria die in antibiotic world; 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 \alpha_{10}=\alpha_{11}=1} -- persistent bacteria persist in the world that either has or doesn't have the antibiotics.
  • We suppose that 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 P(w=1)=p} (antibiotic probability), 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 P(w=0)=1-p=q} .
  • If the bacterium can sense presence/absence of antibiotics and switch appropriately, then for 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 \alpha^{(0)}>1} the bacterial growth rate in a single time step 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 \bar{\alpha}=\alpha^{(0)}(1-n_i)+n_i} , 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 n_i=(0,1)} is a random variable describing if antibiotics is present or absent in this time step. Over many trials, the population if 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 K_0} individuals at the beginning will grow 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 K(N)=K_0\prod_{i=1}^N\left[\alpha^{(0)}(1-n_i)+n_i\right]=K_0\prod_{i=1}^N 2^\left[(1-n_i)\log_2\alpha^{(0)}+n_i\right]=K_02^{\Lambda_{\rm best} N}} .
  • The growth rate 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 \Lambda_{\rm best}=\frac{1}{N}\sum \left[(1-n_i)\log_2\alpha^{(0)}+n_i\right]} . By the central limit theorem, 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{1}{N}\sum n_i\to p} . Thus 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 \Lambda\to (1-p)\log_2\alpha^{(0)}+p} . This is the fastest possible growth rate. However, as argued by Kussel and Leibler (2005), sensing the environment is resource-consuming and might not be useful in all situations, and the bacteria might instead prefer to switch randomly, hedging their bets, rather than sensing.
  • Suppose that the bacterium is going to switch randomly at the beginning of every time step. The fraction 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} of the population will be in the persistent state, 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 1-f} in non-persistent. The actual state of the world will be 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 n_i} . Then the growth rate of the colony at time step 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 i} 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 \alpha_i= \alpha^{(0)}(1-f)(1-n_i)+f=\left[\alpha^{(0)}(1-f)+f\right](1-n_i)+fn_i=2^{(1-n_i)\log_2\left[\alpha^{(0)}(1-f)+f\right]+n_i\log_2 f}} . After 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 N} time steps, the whole population 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 K(N)=K_0 \prod_{i=1}^N 2^{(1-n_i)\log_2\left[\alpha^{(0)}(1-f)+f\right]+n_i\log_2 f} \equiv K_0 2^{\Lambda N}} .
  • 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 \Lambda =\frac{1}{N}\sum_i \left\{(1-n_i)\log_2\left[\alpha^{(0)}(1-f)+f\right]+n_i\log_2 f\right\}} . By central limit theorem 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{1}{N}\sum n_i\to p} . Thus 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 \Lambda\to (1-p)\log_2\left[\alpha^{(0)}(1-f)+f\right]+p\log_2 f} .
  • To find the optimum fraction of bacteria that should choose to be persistent at any time, we maximize the growth rate 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 \Lambda} . We do 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 \partial \Lambda/\partial f} . Solving the ensuing equation, we get for the optimal 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} 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_0=\min\left\{\frac{p\alpha^{(0)}}{\alpha^{(0)}-1}, 1\right\}} .
  • 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_0} is proportional to 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 p} -- special case of matching, see Gallistel et al. 2001.
  • When 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 \alpha^{(0)}<1/(1-p)} , that is, the maximum growth is not too fast, 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_0=1} -- it makes sense not to hedge bets and always stay in the persistent state.
  • Plugging the result into the growth rate formula (for 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 \alpha^{(0)}} ), we get: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 \Lambda_{\rm max}= \Lambda_{\rm best}- S[p] -p(1-\log_2(\alpha^{(0)}/(1-\alpha^{(0)})} . Note that the growth rate decreases by 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 S[p]} , the amount of what the bacterium doesn't know about the world. The amount of information about the outside world is actually important in the evolutionary sense: if you know the world, you multiply faster.

Building effective models

  • Information theory provides a measure for characterization of quality of input output relations. But in addition, due to the data processing inequality, it also provides ways of unambiguously reducing dimensionality of the modeled biological system.
  • (Indeed, say we have a large-dimensional signal 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 \vec{s}} and response 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 \vec{r}} . There's a certain mutual information between these 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 I[\vec{s};\vec{r}]=I_0} . If we propose a reduction of the signal and response to 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=f(\vec{s}),g=g(\vec{r})} , then 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 I[f;g]\le I[\vec{s};\vec{r}]} by the data processing inequality.
  • We can, for example, solve the problem like: Which inputs are informative of the outputs (and hence need to be accounted for in a model)? We omit different subsets of the inputs 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 s_i} , calculate 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 I[\vec{s}-s_i;\vec{r}]=I_{\not i}} , and calculate the error due to omitting this signal 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_i=\frac{I_0-I_{\not i}}{I_0}} . Those components that have a 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_{\not i}} can be safely neglected. This type of analysis can be used, for example, to understand which features of the neural code are important. In the subsequent presentation by Farhan, we will hear about using this trick to understand if high precision of neural spikes is important or not.
  • The ARACNE algorithm.
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