Physics 380, 2011: Homework 5

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Back to Physics 380, 2011: Information Processing in Biology.

1. For the system discussed in class, where a protein is produced at a constant rate $\alpha$ and is degraded with the rate $rn$, write down the master equation and solve (analytically) for the steady state of it. What is $P_{n}$?
2. Consider a biochemical system for production of a protein that is somewhat different from what we explored in class: number of proteins $n$ increases by one with a rate $\alpha$ and it decreases with the rate $rn^{2}$. This happens when the protein is involved in a self-degradation, as is widely believed to be true for development (Eldar et al., 2003) Derive the deterministic equation describing the system. Then derive the equations describing its stochastic dynamics: the master, the Langevin, and the Fokker-Planck equations. Let's explore how well all four of these descriptions agree with each other. Write programs to solve all four of these equations (see below for suggestions). Start your simulations with $n(t=0)=50$ and $\alpha =50$ and $r=1/50$. Explore different values of $\alpha ,r$ and answer the following questions: (a) What are the conditions, under which all four different simulation techniques largely agree with each other? (b) Which simulations are closer to each other: Langevin vs. Fokker-Planck or Langevin vs. master equation? (c) Under which conditions is each of the two pairs close to each other? Illustrate your answers with graphs. The following should be helpful when you write your simulations code:
   • Solve the deterministic chemical kinetics equation numerically using the Euler stepping method. That is: $n(t+\Delta t)=n(t)+{\frac {dn}{dt}}\Delta t$
   • For the master equation, formally the array of probabilities $P_{n}$ is infinite dimensional. But the probabilities $P_{n}$ for $n\gg {\bar {n}}+5{\sqrt {\bar {n}}}$, where ${\bar {n}}$ is the average value of $n$ obtained from the equivalent deterministic equation, are extremely small. So we may keep track of a finite set of probabilities $P_{n}$ that are not too much larger than the the expected deterministic mean. Then one can do Euler stepping again, now for each of $P_{n}$, to calculate the dynamics of each of these probabilities.
   • For the Fokker-Planck equation, one can specify $P(n,t)$ at discretized values of $n$ and approximate spatial derivatives with finite differences, as you did in calculus. And then use the Eurler method again.
   • Finally, for the Langevin equation, we get a single random trajectory $n(t)$ at a time. One must then generate many such random trajectories and histogram them to get an approximation to $P(n,t)$.
   • Compare distributions from Langevin, Fokker-Planck, and master simulations by plotting them on top of each other.
3. Consider the following system: a DNA has a transcription factor binding site, which switches randomly between a bound (transcription is on) and an unbound (transcription is off) states. The off-to-on rate is $r_{\rm {on}}$ and the on-to-off rate is $r_{\rm {off}}$. When the gene is on, it produces proteins randomly with the rate $\alpha$. When it is off, there's no production. The proteins degrade with the rate $rn$. In the limit when the degradation is slow ($r\ll r_{\rm {on}},r\ll r_{\rm {off}}$), calculate (analytically) the variance of protein number in the steady state (using the law of large numbers mantra). Do the same in the limit of fast degradation ($r\gg r_{\rm {on}},r\gg r_{\rm {off}}$). Discuss how your results compare (qualitatively) with the predictions regarding the effect of the propagating noise (Paulsson, 2004) on the protein coefficient of variation. For Graduate Students: Derive (analytically, or find an analytical solution online) the coefficient of variation for the protein for this system for arbitrary parameters. Compare to Paulsson's result.
4. Graduate Students: In class we discussed a coupled system for mRNA and protein production, and we wrote an appropriate Langevin description for this system. Now let's consider a slight complication of this system, similar to the Problem 2 above. Namely, let's assume that the protein is involved in self-induced degradation, while the mRNA still degrades linearly $\left\{{\begin{array}{l}{\frac {dx}{dt}}=Ky-r_{x}x^{2}\\{\frac {dy}{dt}}=\alpha -r_{y}y\end{array}}\right.$. Write the system of two coupled Langevin equations for this system, and simulate it for different values of creation/destruction rates to verify numerically if the general formula for noise propagation we discussed in class holds: ${\rm {c.v.}}_{x}^{2}={\frac {1}{<x>H_{xx}}}+{\rm {c.v.}}_{y}^{2}{\frac {H_{yx}^{2}}{H_{xx}^{2}}}{\frac {H_{xx}/\tau _{x}}{H_{yy}/\tau _{y}+H_{xx}/\tau _{x}}}$. Think about how you would estimate the average molecular life-times (and note that these may depend on instantaneous molecular copy numbers).