# Optimality and Evolution

A liquid culture of E. coli contains ~$10^9$cells/ml. If we inoculate a liquid culture of 10ml with a single bacteria, the culture contains ~$10^{10}$ individuals. Assuming that the probability for a mutation is $1/10^9$ per letter and that the single bacteria devided $10^{10}$ times, there are 10 bacteria with a mutation in the same letter of the genome. If we consider that the change of a single letter in the DNA sequence of the promoter can abolish the affinity for the transcription factor, we can imagine that the franscription network is very plastic. Arrows in the transcription network can be gained and lost very rapidly.

The production of a protein bears costs for the cell. Resources that are consumed to form proteins are restricted, so is the benefit that these proteins provide in a certain environment. Cells in exponential growth phase are under continuous selection pressure and individuals, that distribute resources more efficiently, than their competitors have a selection advantage. Gene expression levels of bacterial cells like Escherichia Coli, are therefore fine tuned by evolution. To quantitatively determine the optimal expression level of lac proteins $(Z)$, Dekel and Alon measured explicitly the cost function $\eta(Z)$ and the benefit function $B(Z)$ of the lac system, to determine the fitness $g$ (growth rate). \begin{align} g=- \eta (Z) + B(Z) %\eta (Z)=\frac{\eta_0 Z}{1-Z/M}\\ %B(Z)=\delta \frac{ZL}{K_y+L} \end{align} Serial dilution experiments under different lactose concentrations showed, that bacteria can adapt the expression level to the environment after $300-500$ generations. The expression level of the lac operon is then a solution to a cost-benefit optimisation problem, that maximizes growth [Erez Dekel $\&$ Uri Alon - Nature 2005].

However, for unadapted individuals responses to a certain environment might be nonoptimal. Under DNA stress caused by antibiotics like fluroquinolones, that inhibit DNA replication and transcription, growth was shown to be nonoptimal [Bollenbach et. al. Cell 2009]. Since the number of ribosomes in the cell is not reduced by the same factor, as the amount of DNA, DNA stress leads to an imbalance between these components. Addition of antibiotics, that inhibit protein synthesis can correct this imbalance and therefore increases the growth rate. These observations can explain drug interactions of this kind, where addition of a second antibiotic can help the bacteria to grow are called suppressive [Bollenbach et. al. Cell 2009].

Optimality was also studied in metabolic networks. Reconstructed metabolic networks from annotated genome sequences allow to represent complex biological networks in silico in form of a stoichiometric matrix $\mathbf{S}$.

The representation of a metabolic network consisting of $m$ metabolites and $r$ reactions is given by

\begin{equation} \frac{dX_i}{dt}=\sum\limits_{j=1}^{r}S_{ij}v_j \hspace{1cm} for \hspace{1cm} i=1, \dots , m. \end{equation}

The matrix $\mathbf{S}$ is determined by the stoichiometric coefficients known from chemical reactions. The vector $\mathbf{v}$ contains the reaction velocities. The following example illustrates the derivation. Columns of $\mathbf{S}$ represent reactions, rows represent metabolites.

Network Stoichiometric Matrix
$\overset{v_1}{\rightarrow} 1S_1 \overset{v_2}{\rightarrow} 2S_2 \overset{v_3}{\rightarrow} 1S_3 \overset{v_4}{\rightarrow}$ $\mathbf{S}=\begin{pmatrix} 1 & -1 & 0 & 0\\ 0 & 2 & -2 & 0\\ 0 & 0 & 1 & -1\\ \end{pmatrix}$
In the first reaction $S_1$ is produced. In the second reaction one molecule $S_1$ is consumed to form two molecules $S_2$ and so on.

Constraint based optimization is used to predict optimal growth rates in steady state. \begin{equation} \mathbf{S} \cdot \mathbf{v}= \mathbf{0} \tag{1} \end{equation} Since the system of equations in equ. (1) is under-determined, there exists a whole subspace of solutions. The growth rate can be predicted by picking the solution that maximizes biomass production (growth).

In mathematical terms this can be achieved by maximizing the objective function $f(\mathbf{v})$

\begin{equation} max \overset{!}{=}f(\mathbf{v})= \mathbf{c}^T \mathbf{v} \end{equation} subject to the constraints (1) and \begin{equation} v_i^{min} \leq v_i \leq v_i^{max} \end{equation} The vector $\mathbf{c}$ describes the biomass composition of the organism. The reaction velocities are constrained by the maximal reaction rate of the enzymes, the expression level of the enzyme and thermodynamic considerations (e.g. $v_i$ is restricted to be positive because the reaction is irreversible). Mathematical problems of this kind can be solved with the simplex algorithm.

The in silico predicted growth rates were consistent with experimental data in many cases. For the other cases like growth on maltate, acetate and succinate, it was shown, that after sufficient time, E. coli adapted from nonoptimal growth to the in silico predicted optimal growth rate [Ibarra, R. U., Edwards, J. S. $\&$ Palsson, B. O. Nature 2002].

Overall, these studies show that gene expression levels can be quickly fine tuned by evolution to maximize growth rate.