Multiplexing Biochemical Signals

In this Letter we show that living cells can multiplex biochemical signals, i.e., transmit multiple signals through the same signaling pathway simultaneously, and yet respond to them very specifically. We demonstrate how two binary input signals can be encoded in the concentration of a common signaling protein, which is then decoded such that each of the two output signals provides reliable information about one corresponding input. Under biologically relevant conditions the network can reach the maximum amount of information that can be transmitted, which is 2 bits.

Figure 1

(a) Biochemical multiplexing: $N$ different signals are encoded in a common pathway $\mathcal{V}$, and then decoded such that each output species ${\mathcal{X}}_i$ responds to the corresponding input ${\mathcal{S}}_i$. (b) Multiplexing is a mapping problem. The states of two inputs ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ are mapped onto the concentration of $\mathcal{V}$, which is then mapped onto states of the output species ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$; we require that the two lowest (highest) levels of $X_i$ correspond to the lowest (highest) level of $S_i$. The dashed arrow denotes a mapping that violates this requirement; levels of $V$ and $X_i$ are colored according to input pattern $\mathbf{s}=(s_1,s_2)$. (c) The 3 distinct mappings of $\mathbf{s}$ to $v$; in panel (b) mapping $C$ is shown.

Figure 2

Decoding $V$ using an incoherent feedforward loop. (a) Network architecture. (b) The values of $V$ corresponding to the four input patterns $\mathbf{s}$ for mapping $C$ [see Fig. 1c], with thresholds $K_{\alpha }$, $K_{\beta }$, $K_{\gamma }$ [see Eqs. (1, 2)]. (c) $X_1(V)$. (d) $X_2(V)$. The nonmonotonicity of $X_2(V)$ swaps the states corresponding to $(1,1)$ and $(1,0)$ in the mapping from $v$ to $x_2$.

Figure 3

(a) An encoding scheme where two ligands ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ competitively bind to a protein $\mathcal{V}$, which can be in an active state $A$ or an inactive state $I$. (b) Energy levels of $\mathcal{V}$ in the active or inactive state with no ligand, ligand ${\mathcal{S}}_1$, or ligand ${\mathcal{S}}_2$ bound; $e^{-E_i^I}\equiv S_i/K_i^I$; $e^{-E_i^A}\equiv e^{-\Delta E_0}{S}_i/K_i^A$. (c) Input-output relation $V(S_1,S_2)$ corresponding to the energy levels in (b), yielding mapping $C$ [Fig. 1c]; symbols ■, ▴, ●, ▾ correspond to states in Fig. 2.

Figure 4

The transmitted information $I\equiv I(S_1,X_1)+I(S_2,X_2)$ as a function of $V^{\max }$ for four different values of $X^{\max }$. $I$ can reach the maximum value of 2 bits provided $V^{\max }$ and $X^{\max }$ are large enough. Typical values for the optimized parameters are $K_{\alpha }\sim V^{\max }$, $K_{\beta }\sim 0.3V^{\max }$, $K_{\gamma }\sim 0.5X^{\max }$, $\Delta E_0\approx -2k_{B}T$, $S_1^{(0)}=S_2^{(0)}=0$, $K_1^I\gg S_1^{(1)}$, $K_2^I\sim S_2^{(1)}$, $K_1^A\ll S_1^{(1)}$, $K_2^A<S_2^{(1)}$.