Rate-Distortion Scenario for the Emergence and Evolution of Noisy Molecular Codes

We discuss, in terms of rate-distortion theory, the fitness of molecular codes as the problem of designing an optimal information channel. The fitness is governed by an interplay between the cost and quality of the channel, which induces smoothness in the code. By incorporating this code fitness into population dynamics models, we suggest that the emergence and evolution of molecular codes may be explained by simple channel design considerations.

Figure 1

Molecular codes as noisy information channels. The code relates the space of meanings (left) with that of symbols (right). The channel is a three-stage Markov process (blue arrows): (i) A meaning $\alpha$ is encoded as a symbol $i$ by the encoder $e_{\alpha i}$ (ii) $i$ is read as $j$ by the reader $r_{ij}$ (iii) $j$ is decoded as $\omega$ by the decoder $d_{j\omega }$. The distance between the original and the reconstructed meanings is $c_{\alpha \omega }$ (red arrow).

Figure 2

Emergence and evolution of molecular codes. (a) A code relates $m=2$ meanings $\{1,2\}$ and $s=2$ symbols $\{1,2\}$. The encoder $e_{\alpha i}$ has 4 entries and is constrained to a 2D square by the 2 conservation relations $e_{11}+e_{12}=e_{21}+e_{22}=1$. The reader is $r_{ij}=(1-2\epsilon ){\delta }_{ij}+\epsilon$ with the misreading probability $\epsilon =0.1$. At low mutation rates the population peaks at the optimal encoder (illustrated as sharp peaks). Below the critical gain ${\kappa }_c=1.56$ the state is noncoding with $e_{\alpha i}=\frac{1}{2}$. Above ${\kappa }_c$ a coding state evolves. (b) The channel cost $I$ increases from 0 at the coding transition, $\kappa ={\kappa }_c$, while the distortion $D$ decreases. The average order parameter $⟨|\delta e|⟩$ increases continuously from 0 at the second-order coding transition. The fitness $H$ (plotted is $-H$) increases to an asymptotic value. (c) A code that relates 8 meanings and 6 symbols. The distance is $c_{\alpha \omega }=\min (|\alpha -\omega |,8-|\alpha -\omega |)$ and the reader is defined by a probability of 0.98 that $i$ is read as $i$ and 0.01 that it is read as one of its two neighbors on the symbol graph. (d) The optimal encoder $e_{\alpha i}$ is plotted as color-coded $6\times 8$ arrays at increasing gains $k$. Below ${\kappa }_c=0.52$ (top left) the encoder is $e_{\alpha i}=1/6$ with uncorrelated symbols and meanings. A coding state emerges at ${\kappa }_c$. The symbol-meaning correlation increases with $\kappa$ until every meaning $\alpha$ is encoded by exactly one symbol $i$ (bottom right). The optimal code is smooth; i.e., close meanings are encoded by close symbols, as manifested by the continuous diagonal shape of the encoder. (e) Quasispecies dynamics of the code from (a) with mutation rate $\mu =5\times {10}^{-5}$. Below ${\kappa }_c=1.56$, the population distribution $\Psi$ is smeared around the noncoding optimum. Above ${\kappa }_c$, a coding state appears, $\Psi$ sharpens and migrates towards the one-to-one code $e_{11}=e_{22}=1$. (f) A coding transition in the TRN, when a universal transcription factor splits into distinct species when the gain $\kappa$ increases.