Multidimensional biochemical information processing of dynamical patterns

Cells receive signaling molecules by receptors and relay information via sensory networks so that they can respond properly depending on the type of signal. Recent studies have shown that cells can extract multidimensional information from dynamical concentration patterns of signaling molecules. We herein study how biochemical systems can process multidimensional information embedded in dynamical patterns. We model the decoding networks by linear response functions, and optimize the functions with the calculus of variations to maximize the mutual information between patterns and output. We find that, when the noise intensity is lower, decoders with different linear response functions, i.e., distinct decoders, can extract much information. However, when the noise intensity is higher, distinct decoders do not provide the maximum amount of information. This indicates that, when transmitting information by dynamical patterns, embedding information in multiple patterns is not optimal when the noise intensity is very large. Furthermore, we explore the biochemical implementations of these decoders using control theory and demonstrate that these decoders can be implemented biochemically through the modification of cascade-type networks, which are prevalent in actual signaling pathways.

Figure 1

(a) Relation between input $u\left(t\right)$ and output $x\left(t\right)$ in a linear system through linear response function $h\left(t\right)$. (b) Decoding of dynamical patterns by two decoding systems. Input signal $w\left(t\right)$ is received by $N$ types of receptors, and the signal is processed by subsequent internal molecular decoders ($N=2$ in this figure). The linear response function of the $i\mathrm{th}$ decoder is given by $h_i\left(t\right)$. Each decoder outputs results by $x_i\left(T\right)$. (c) Dynamical pattern $w\left(t\right)$ as a sum of basis functions ${\eta }_j\left(t\right)$ at intensities $v_j$.

Figure 2

Basis sets for the $M=2$ case [solid and dashed lines denote ${\eta }_1\left(t\right)$ and ${\eta }_2\left(t\right)$, respectively]. (a) Slow and fast patterns (basis set A). (b) Constant and oscillation patterns (basis set B).

Figure 3

Information transmission model assumed in different mutual information. (a) Information is embedded in two distinct basis functions, transmitted through one common channel, and decoded by two decoders. (b) Information is embedded in two distinct basis functions, transmitted through two designated channels, and decoded by two decoders. (c) Information is embedded in two identical basis functions, transmitted through one channel, and decoded by two identical decoders. In (a)–(c), $\oplus$ and $\otimes$ denote addition and multiplication operations, respectively.

Figure 4

(a), (b) Mutual information as a function of noise intensity $D$ by basis set; (a) basis set A [Fig. 2] and (b) basis set B [Fig. 2]. Solid, dashed, dotted, and dotted-dashed lines denote $I^{\mathrm{full}}$, $I^{\mathrm{decor}}$, $I^{\mathrm{sll}}$, and $I^{\mathrm{dual}}$, respectively. In (a) and (b), the insets highlight $I^{\mathrm{dual}}-I^{\mathrm{full}}$ as a function of $D$. (c) Mutual information quantities $I^{\mathrm{dual}}$ (solid line) and $I^{\mathrm{dup}}$ (dashed line) as functions of $D$. Note that $I^{\mathrm{dual}}$ and $I^{\mathrm{dup}}$ do not depend on the basis set. In all panels, parameters are $T=1$ and ${\sigma }_{v_1}^2={\sigma }_{v_2}^2=1$.

Figure 5

(a), (b) Optimal linear response function $h_i\left(t\right)$ of the full decoder with basis set A for different noise intensities; (a) $D=0.1$ and (b) $D=1.0$, where all other parameters are the same. (c) Optimal linear response function $h_i\left(t\right)$ of the decorrelating decoder with $D=0.1$. In all panels, the solid and dashed lines denote $h_1\left(t\right)$ and $h_2\left(t\right)$, respectively. For all $h_i\left(t\right)$ shown, the functions that are horizontally symmetric with respect to $h_i=0$ are also optimal solutions. We set ${\sigma }_{x_1}^2={\sigma }_{x_2}^2=1$, and these parameters affect only the magnitude of the functions. The other parameters are identical to those in Fig. 4.

Figure 6

(a) Molecular realization of $h_1\left(t\right)$ in Fig. 5, and (b) its linear response function. In (a), arrows and bar-headed arrows denote activation and inhibition, respectively. In (b), the dashed line is the linear response function of the realization network and the solid line is the target optimal linear response function shown in Fig. 5. (c) Molecular realization of $h_2\left(t\right)$ in Fig. 5 and (d) its linear response function. (e) Reduced molecular realization for the full network shown in (a) and (f) its linear response function. In (c)–(f), the meanings of the arrows and lines are the same as in (a) and (b).

Figure 7

(a), (b) Network representations of (a) $P\left(x_i^T\right|\mathit{v})$ and (b) $P\left(x_i^T\right|v_i)$ for $N=M$. (c) Basis function ${\eta }_2\left(t\right)$ of basis set A [Fig. 2] (solid line) and its Fourier series approximation (dashed line).