# Robustness

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Gene expression is subject to stochastic fluctuations, thus parameters can vary from cell to cell. It has been proposed, that gene expression is robust against fluctuations in parameters. Here we want to explain the concept of robustness at the example of negative autoregulation (NAR). To this end we consider the dynamics of NAR with high Hill coefficients. The differential equation describing the dynamics is given by \begin{align} \frac{dY}{dt}=\frac{\beta_{max}}{1+(Y/K)^n}-\alpha Y \end{align} If we plot the Hill function for several Hill coefficients and the degradation rate in the same diagram we have

As the degradation rate $\alpha$ varies the steady state concentration of $Y_{st}$ moves. The smaller the Hill coefficient, the higher is the sensitivity of the steady state $Y_{st}$ to paramter changes in $\alpha$.

The robustness of negative autoregualtion was shown in the beautiful experimental work of Attila Becskei and Luis Serrano. They calculated that the stability of NAR is doubled with respect to an unregulated system. Single cell measurements in a NAR circuit revealed a very small number of cells that deviate from the mean steady state concentration compared to unregulated genes (see further reading below).

##  Parameter sensitivity

To analyse the relationship between two paramters (e.g. $Y_{st}$ and production rate $\beta$) we calculate the parameter sensitivity coefficient. The general parameter sensitivity coefficient $S(A,B)$ of property $A$ with respect to parameter $B$ is given by the relative change in $A$ for a small relative change in $B$.

\begin{align} S(A,B)=\frac{\Delta A /A}{\Delta B/B}=\frac{B}{A}\frac{dA}{dB} \end{align}

As an example we calculate the sensitivity coefficient for $S(Y_{st},\beta)$. We calculated the steady state for the case of NAR by neglecting the term $1$ in the dereasing Hill function, which was plausible for $Y/K >>1$, and were then able to caluclate the solution of the bernoulli differential equation. We use this steady state \begin{align} Y_{st}= \left[ \frac{\beta}{\alpha}K^n \right]^{1/(n+1)} \end{align} and compute the sensitivity coefficient \begin{align} S(Y_{st},\beta)&= \frac{\beta}{Y_{st}} \frac{d}{d \beta}\left[ \frac{\beta}{\alpha}K^n \right]^{1/(n+1)}\\ &= \frac{\beta}{Y_{st}} \left( \frac{K^n}{\alpha} \right)^{1/(n+1)} \frac{d}{d \beta} \beta^{1/(n+1)}\\ &= \frac{\beta}{Y_{st}(n+1)} \left( \frac{K^n}{\alpha} \right)^{1/(n+1)} \beta^{1/(n+1)-1}\\ &= \frac{1}{Y_{st}(n+1)} \left( \frac{K^n}{\alpha} \right)^{1/(n+1)} \beta^{1/(n+1)}\\ \end{align} so the sensitivity coefficient is \begin{align} S(Y_{st},\beta)&= \frac{1}{n+1} \end{align} This is the same answer that we derived intuitively from the diagram. The sensitivity decreases with increasing Hill coefficient $n$.

Video Lecture:

• Uri Alon - Introduction to Systems Biology Lecture 2