# Channel assisted noise propagation in a two-step cascade

Mintu Nandi<EMAIL_ADDRESS>Department of Chemistry, Indian
Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India
Sudip Chattopadhyay<EMAIL_ADDRESS>Department of Chemistry, Indian
Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India
Somshubhro Bandyopadhyay<EMAIL_ADDRESS>Department of Physical Sciences,
Bose Institute, EN 80, Sector V, Bidhan Nagar, Kolkata 700091, India  Suman K
Banik<EMAIL_ADDRESS>Department of Chemical Sciences, Bose Institute,
EN 80, Sector V, Bidhan Nagar, Kolkata 700091, India

###### Abstract

Signal propagation in biochemical networks is characterized by the inherent
randomness in gene expression and fluctuations of the environmental
components, commonly known as intrinsic and extrinsic noise, respectively. We
present a theoretical framework for noise propagation in a generic two-step
cascade (S$\rightarrow$X$\rightarrow$Y) in terms of intrinsic and extrinsic
noise. We identify different channels of noise transmission that regulate the
individual as well as the overall noise properties of each component. Our
analysis shows that the intrinsic noise of S alleviates the overall noise and
information transmission capacity along the cascade. On the other hand, the
intrinsic noise of X and Y acts as a bottleneck of information transmission.

##  I Introduction

Noise plays an important role in shaping the dynamics of various cellular
processes, including gene expression, transcriptional regulation, signal
transduction pathways, and developmental pathways Kærn  _et al._ (2005); Eldar
and Elowitz (2010); Tsimring (2014). These cellular processes adopt diverse
strategies in response to the inevitable influence of noise. Comprehension of
the precise impact of noise on cellular processes is a complex endeavour and
needs individual attention. In this context, exploring the influence of noise
on regulatory cascades could reveal crucial insights into the role of noise in
cellular functioning. Regulatory cascades are a common occurrence in
biological systems, manifesting in various forms, one of which is
transcriptional cascades in Escherichia coli and Saccharomyces cerevisiae
Shen-Orr  _et al._ (2002); Rosenfeld and Alon (2003). Moreover, cascades play
a pivotal role in directing the temporal sequencing of gene expression,
contributing to critical processes like sporulation McAdams and Shapiro (2003)
and flagella formation Kalir  _et al._ (2001). In more complex organisms like
Drosophila and sea urchins, the developmental programs heavily rely on
precisely orchestrated cascaded processes to achieve intricate temporal
coordination of events Arnone and Davidson (1997); Davidson  _et al._ (2002).

To address the propagation of noise in a cascade of gene expression, we
consider the generic two-step cascade (TSC) S$\rightarrow$X$\rightarrow$Y
representing gene regulatory network (GRN). Here, S, a transcription factor
(TF), regulates the production of protein X which in turn regulates the
production of protein Y. For associated kinetics and corresponding master
equation we refer to Fig. 1a,b. TSC is a recurring architecture frequently
found in more complex networks, often associated with additional control
mechanisms Alon (2006); Ferrell (2022). Recent experimental investigation in
synthetic transcriptional cascades having multiple stages of repression
highlights the ability to exhibit ultrasensitive responses and low-pass
filtering, resulting in robust behavior in input fluctuations Hooshangi  _et
al._ (2005). Furthermore, noise properties in steady-state using a framework
of two-stage transcriptional cascades were examined Blake  _et al._ (2003);
Bruggeman  _et al._ (2009); Maity  _et al._ (2015).

The present communication characterises the noise propagation mechanism using
the notion of noise decomposition, viz., intrinsic and extrinsic noise. The
intrinsic noise is an inherent property of the system whereas the extrinsic
noise is due to environmental fluctuations Elowitz  _et al._ (2002). However,
the characteristic distinction between intrinsic and extrinsic noise depends
on the definition of the system Swain  _et al._ (2002); Paulsson (2004, 2005);
Hilfinger and Paulsson (2011). In a TSC, the intrinsic noise of each component
(S, X , and Y) arises due to low copy number. On the other hand, the extrinsic
noise in a component, say X, is accumulated due to noise coming from S. In
this context, S is an extrinsic variable (“environment”) for X. Similarly, for
Y, S and X both act as extrinsic variables (Fig. 1c).

Figure 1: (color online) (a) Schematic of the kinetics of a generic two-step
cascade. The synthesis and degradation of S is denoted by $f_{s}=\alpha_{s}$
and $g_{s}(s)=\beta_{s}s$, respectively, and resembles a simple birth-death
process. $\alpha_{s}$ and $\beta_{s}$ are synthesis and degradation rate
constants, with unit (molecules/V)min-1 and min-1, respectively. Synthesis of
X and Y is characterized by ($s^{n}/(K_{sx}^{n}+s^{n})$ and
$x^{n}/(K_{xy}^{n}+x^{n})$), respectively that captures the binding of the
upstream regulator to the promoter of the downstream gene Alon (2006).
$K_{ij}$ refers to the binding affinity of the upstream protein $i$ to the
promoter of the downstream gene $j$ and $n$ is the Hill coefficient. The
binding affinity $K_{ij}$ (expressed in molecules/V) defines a threshold in
the concentration of $i$ necessary to significantly express the gene product
$j$ Alon (2006). The rate constants $\alpha_{x}$ and $\alpha_{y}$ with unit
(molecules/V)min-1 take care of the synthesis of X and Y, respectively. The
degradation of X and Y are linear functions with rate constants $\beta_{x}$
and $\beta_{y}$, respectively, with unit min-1. In the text, $s$, $x$, and $y$
account for the copy number/volume of S, X, and Y, respectively. (b) Master
equation corresponding to the kinetics given in (a) Gardiner (2009). $f_{i}$
and $g_{i}$ correspond to the synthesis and degradation functions of the
system components, respectively. $\mathbb{E}^{\pm 1}$ are step operators whose
action on a function of $s$, $x$, and $y$ gives a step-up ($\mathbb{E}^{+1}$)
or step-down ($\mathbb{E}^{-1}$) in copy number. (c) Schematic of noise
propagation from S to Y via X. The analytical expression of various intrinsic
and extrinsic noise components are shown in Appendix C. Here, $\langle
s\rangle$, $\langle x\rangle$, and $\langle y\rangle$ stand for the mean copy
number of S, X, and Y, respectively, at steady state.

Investigating the role of intrinsic and extrinsic noise components in cellular
functions, both in terms of their drawbacks Kærn  _et al._ (2005) and
advantagesRao  _et al._ (2002); Eldar and Elowitz (2010); Herranz and Cohen
(2010); Tsimring (2014), is therefore of great significance. Previous studies
have suggested that both noise components (intrinsic and extrinsic) can act as
degrading factors in information transmission in cellular responses Tkačik
_et al._ (2008); Cheong  _et al._ (2011); Uda  _et al._ (2013); Selimkhanov
_et al._ (2014); Voliotis  _et al._ (2014); Garner  _et al._ (2017); Potter
_et al._ (2017); Suderman  _et al._ (2017); Granados  _et al._ (2018).
However, a recent communication shows enhancement of information transmission
due to cell-to-cell variability (extrinsic noise in this context) in skeletal
muscle Wada  _et al._ (2020). We note that mutual information (MI), a measure
of statistical dependency between two variables in information theory
quantifies information transmission capacity Shannon (1948); Shannon and
Weaver (1963); Cover and Thomas (1991).

Regulatory cascades in biological systems have long been recognized for their
critical role in transmitting information and orchestrating various cellular
processes. However, the extent to which intrinsic and extrinsic noise
contribute to the transmission of information within these cascades remains
relatively unexplored. The present study aims to address this gap by
considering the intrinsic and extrinsic noise components as well as
decomposing the extrinsic noise of the final gene product Y (Fig. 1c). Our
analysis identifies distinct noise processing channels that actively govern
the propagation of noise along the cascade. We also delve into the impact of
each decomposed noise term on information transmission along the cascade.
Despite the apparent simplicity in the architecture, the TSC reveals an
intricate relationship between noise and information, offering valuable
insights into the underlying mechanisms that govern cellular functions and
dynamics.

##  II The model and methods

The stochastic kinetics of a generic TSC is given by

$\displaystyle s\stackrel{{\scriptstyle
f_{s}}}{{\longrightarrow}}s+1,\;\;\;x\stackrel{{\scriptstyle
f_{x}(s)}}{{\longrightarrow}}x+1,\;\;\;y\stackrel{{\scriptstyle
f_{y}(x)}}{{\longrightarrow}}y+1,$ (1a) $\displaystyle
s\stackrel{{\scriptstyle
g_{s}(s)}}{{\longrightarrow}}s-1,\;\;\;x\stackrel{{\scriptstyle
g_{x}(x)}}{{\longrightarrow}}x-1,\;\;\;y\stackrel{{\scriptstyle
g_{y}(y)}}{{\longrightarrow}}y-1,$ (1b)

where $s$, $x$, and $y$ stand for the copy number/volume of S, X, and Y,
respectively. Here $f_{i}$ and $g_{i}$ ($i\equiv s,x,y$) correspond to the
synthesis and degradation functions of the system components, respectively.
The explicit expressions of $f_{i}$ and $g_{i}$ are outlined in Fig. 1a. The
master equation following the stochastic kinetics outlined in Eq. (1) is
Gardiner (2009)

$\displaystyle\frac{dP(s,x,y;t)}{dt}$ $\displaystyle=$
$\displaystyle\sum_{i=s,x,y}\left[(\mathbb{E}^{+1}-1)g_{i}P(s,x,y;t)\right.$
(2) $\displaystyle\left.+(\mathbb{E}^{-1}-1)f_{i}P(s,x,y;t)\right].$

In the above equation, $\mathbb{E}^{\pm}$ refers to the step operator which
either step-up ($\mathbb{E}^{+1}$) or step-down ($\mathbb{E}^{-1}$) the copy
numbers of the respective components by unity. We employ van Kampen’s system
size expansion van Kampen (2007) to derive the Lyapunov equation (see Appendix
A) which provides the statistical moments associated with the TSC (see
Appendix B).

###  II.1 Noise decomposition in a two-step cascade

In a GRN, the noise associated with the $i$-th node is measured by the square
of coefficient of variation, $\eta_{i}^{2}=\sigma_{i}^{2}/\langle
i\rangle^{2}$. As per the noise decomposition formalism $\eta_{i}^{2}$ can be
written as $\eta_{i}^{2}=\eta_{int,i}^{2}+\eta_{ext,i}^{2}$ Elowitz  _et al._
(2002); Swain  _et al._ (2002); Paulsson (2005); Hilfinger and Paulsson (2011)
where $\eta_{int,i}^{2}$ and $\eta_{ext,i}^{2}$ are intrinsic and extrinsic
noise, respectively. The intrinsic noise $\eta_{int,i}^{2}$ arises due to the
birth-death processes whereas the extrinsic noise $\eta_{ext,i}^{2}$ is fed
from its upstream regulator. The noise associated with the nodes S, X and Y of
a TSC is (see Appendix C)

$\displaystyle\eta_{s}^{2}$ $\displaystyle=$ $\displaystyle\eta_{int,s}^{2},$
(3) $\displaystyle\eta_{x}^{2}$ $\displaystyle=$
$\displaystyle\eta_{int,x}^{2}+\eta_{ext,x}^{2},$ (4)
$\displaystyle\eta_{y}^{2}$ $\displaystyle=$
$\displaystyle\eta_{int,y}^{2}+\eta_{ext,y1}^{2}+\eta_{ext,y2}^{2}.$ (5)

In Eq. (3), S contains only intrinsic noise ($\eta_{int,s}^{2}$) due to the
Poisson kinetics. The noise associated with X, however, contains both
intrinsic ($\eta_{int,x}^{2}$) and extrinsic parts ($\eta_{ext,x}^{2}$). The
extrinsic noise $\eta_{ext,x}^{2}$ arises due to the propagation of noise from
S to X, i.e., from $\eta_{int,s}^{2}$. Similarly, the total noise of Y,
$\eta_{y}^{2}$, has intrinsic noise $\eta_{int,y}^{2}$ and a summation of two
extrinsic noises $\eta_{ext,y1}^{2}$ and $\eta_{ext,y2}^{2}$. The source of
extrinsic noise $\eta_{ext,y1}^{2}$ is the intrinsic noise $\eta_{int,x}^{2}$.
On the other hand, $\eta_{ext,y2}^{2}$ is generated due to the contribution of
$\eta_{ext,x}^{2}$. We refer to Fig. 1c for the flow of noise along the
cascade. We note that the noise decomposition technique is general for Hill
coefficient $n\geqslant 1$ and can be extended to a linear cascade with nodes
$\geqslant 3$.

For $n=1$ the explicit expressions of the extrinsic noise given in Eqs. (3-5)
are (see Appendix C)

$\displaystyle\eta_{ext,x}^{2}$ $\displaystyle=$
$\displaystyle\phi_{x}^{int,s}\eta_{int,s}^{2},$ (6)
$\displaystyle\eta_{ext,y1}^{2}$ $\displaystyle=$
$\displaystyle\phi_{y}^{int,x}\eta_{int,x}^{2},$ (7)
$\displaystyle\eta_{ext,y2}^{2}$ $\displaystyle=$
$\displaystyle\phi_{y}^{ext,x}\eta_{ext,x}^{2}=\phi_{y}^{ext,x}\phi_{x}^{int,s}\eta_{int,s}^{2},$
(8)

where,

$\displaystyle\phi_{x}^{int,s}$ $\displaystyle=$
$\displaystyle\tau_{x,s}\left(\frac{K_{sx}}{K_{sx}+\langle
s\rangle}\right)^{2},$ (9) $\displaystyle\phi_{y}^{int,x}$ $\displaystyle=$
$\displaystyle\tau_{y,x}\left(\frac{K_{xy}}{K_{xy}+\langle
x\rangle}\right)^{2},$ (10) $\displaystyle\phi_{y}^{ext,x}$ $\displaystyle=$
$\displaystyle\tau_{sy,x}^{-1}\phi_{y}^{int,x}.$ (11)

Eqs. (6-8) suggest that the quantity $\phi$ measures the fraction of upstream
noise propagated to its corresponding downstream node. Depending on the
cascade architecture and the model parameters, $\phi$ quantitates the pool of
noise that flows downstream (see Eqs. (9-11)). To be specific,
$\phi_{x}^{int,s}$ quantifies the fraction of intrinsic noise of S transmitted
to X that builds up the extrinsic noise pool of X, $\eta_{ext,x}^{2}$ (Fig.
1c). $\phi_{y}^{int,x}$ measures the fraction of intrinsic noise of X
propagated to Y to generate the extrinsic noise pool $\eta_{ext,y1}^{2}$ of Y
(Fig. 1c). Similarly, $\phi_{y}^{ext,x}$ measures the fraction of extrinsic
noise of X that flows down to Y and generates the second part of the extrinsic
noise pool $\eta_{ext,y2}^{2}$ of Y (Fig. 1c). The measure $\phi$, thus,
characterizes the noise transmission channels between different nodes of the
cascade and identifies how noise propagation influences various species of the
linear cascade. It is important to mention that $\phi$ always remains $<1$
independent of various model parameters.

In Eqs. (9-11), $\tau$-s are the scaled time scale Paulsson (2004, 2005)
defined as $\tau_{i,j}:=\beta_{i}/(\beta_{i}+\beta_{j})$ and
$\tau_{ij,k}:=(\beta_{i}+\beta_{j})/(\beta_{i}+\beta_{j}+\beta_{k})$ where
$\beta_{i}$ ($i\in\\{s,x,y\\}$) corresponds to the degradation rate constant
of the $i$-th species. The expressions of $\tau$-s indicate that
$\tau_{x,s}<1$, $\tau_{y,x}<1$, and $\tau_{sy,x}<1$. Moreover, the second
factors in Eqs. (9,10) are also less than unity. This results in
$\phi_{x}^{int,s}<1$ and $\phi_{y}^{int,x}<1$. On the other hand, although
$\tau_{sy,x}^{-1}>1$, we assume that this factor will not overpower
$\phi_{y}^{int,x}$, which is much less than one, and hence leads to
$\phi_{y}^{ext,x}<1$. This assumption remains valid as long as the separation
of degradation time scale is maintained, i.e.,
$\beta_{s}\ll\beta_{x}\ll\beta_{y}$. The separation of degradation time scale
maintains a maximum level of noise propagation along the cascade Maity  _et
al._ (2015).

The noise propagation from S to X opens up a single noise propagation channel
characterized by $\phi_{x}^{int,s}$. On the other hand, noise transmission
from X to Y is regulated by two different channels $\phi_{y}^{int,x}$ and
$\phi_{y}^{ext,x}$. Noise flow from S to Y via X is characterized by
$\phi_{x}^{int,s}$ and $\phi_{y}^{ext,x}$. We note that an increase of cascade
length opens up multiple new channels of noise propagation. For example, in a
cascade S$\rightarrow$X$\rightarrow$Y$\rightarrow$Z, additional downstream
channels open up for noise propagation from Y to Z.

Figure 2: (color online) (a,b) Variations in channel capacity and noise
processing capacity $\phi$ as a function of $K_{sx}$ and $K_{xy}$. The
parameters used are $\beta_{s}=0.1$ min-1, $\beta_{x}=1.0$ min-1,
$\beta_{y}=10.0$ min-1, $\langle s\rangle=50.0$ molecules/V, $\langle
x\rangle=100.0$ molecules/V, $\langle y\rangle=100.0$ molecules/V, and $n=1$.
For (a) $K_{xy}=\langle x\rangle$ and for (b) $K_{sx}=\langle s\rangle$. (c-e)
Variations in channel capacity and intrinsic noise of S, X, and Y as a
function of mean copy number. The parameters used are $\beta_{s}=0.1$ min-1,
$\beta_{x}=1.0$ min-1, $\beta_{y}=10.0$ min-1, $K_{sx}=\langle s\rangle$,
$K_{xy}=\langle x\rangle$, and $n=1$. For (c) $\langle x\rangle=100.0$
molecules/V and $\langle y\rangle=100.0$ molecules/V. For (d) $\langle
s\rangle=50.0$ molecules/V and $\langle y\rangle=100.0$ molecules/V, and for
(e) $\langle s\rangle=50.0$ molecules/V and $\langle x\rangle=100.0$
molecules/V. In (a-e), the lines are due to theoretical expression and the
symbols are generated by numerical simulation using stochastic simulation
algorithm Gillespie (1976, 1977). (f) Schematic presentation of channel
capacities in a TSC and the role of intrinsic noise on information
propagation. The pointed and blunt arrowheads stand for activation and
repression, respectively.

##  III Results and discussion

The notion of mutual information Shannon (1948); Shannon and Weaver (1963) is
used to address the interplay between noise and information transmission. The
mutual information between two random variables $i$ and $j$ is written as
$I(j;i):=\sum_{i,j}p(i,j)\log_{2}[p(i,j)/p(i)p(j)]$, where $p(i)$ and $p(j)$
are marginal probability distributions and $p(i,j)$ refer to the joint
probability distribution of the variables Cover and Thomas (1991). For a
bivariate system $i$ and $j$ obeying Gaussian statistics, the channel capacity
is $\mathbb{C}_{ij}:=I(i;j)=(1/2)\log_{2}[1+\mathbb{S}_{ij}]$, where
$\mathbb{S}_{ij}$ ($=\eta_{ij}^{4}/(\eta_{i}^{2}\eta_{j|i}^{2})$) is the
signal-to-noise ratio (SNR) of the respective channels Tostevin and ten Wolde
(2010). Here $\eta_{ji}^{2}$, $\eta_{i}^{2}$, and $\eta_{j|i}^{2}$ stand for
the normalised covariance ($\sigma_{ji}^{2}/(\langle j\rangle\langle
i\rangle)$), variance $(\sigma_{i}^{2}/\langle i\rangle^{2}$) and conditional
variance ($\sigma_{j|i}^{2}/\langle j\rangle^{2}$), respectively (see Appendix
D for the explicit expressions of $\mathbb{S}_{ij}$ and $\mathbb{C}_{ij}$). In
the present scenario, we have three possible information processing channels,
i.e., S$\rightarrow$X, X$\rightarrow$Y, and the overall channel
S$\rightarrow$X$\rightarrow$Y.

For $\beta_{s}\ll\beta_{x}\ll\beta_{y}$ the scaled time scales $\tau$-s become
approximately equal to 1 (see Appendix D). In this limit the analytical
expressions of channel capacities are,

$\displaystyle\mathbb{C}_{sx}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left[1+\frac{\phi_{x}^{int,s}\eta_{int,s}^{2}}{\eta_{int,x}^{2}}\right],$
(12) $\displaystyle\mathbb{C}_{sy}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left[1+\frac{\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}}{\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2}}\right],$
(13) $\displaystyle\mathbb{C}_{xy}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left[1+\frac{A}{B}\right],$ (14)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\phi_{y}^{int,x}\eta_{int,x}^{4}+(\phi_{x}^{int,s}\eta_{int,s}^{2}+2\eta_{int,x}^{2})\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\eta_{int,x}^{2}\eta_{int,y}^{2}+(\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2})\phi_{x}^{int,s}\eta_{int,s}^{2}$
$\displaystyle-\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}\eta_{int,x}^{2}.$

As the input signal S acts as an extrinsic variable for both X and Y, increase
in $\phi_{x}^{int,s}$ amplifies all three channel capacities,
$\mathbb{C}_{sx}$, $\mathbb{C}_{sy}$, and $\mathbb{C}_{xy}$ (see Eqs.
(12-14)). Notably, $\mathbb{C}_{sx}$ exhibits the most rapid increase in
response to variations in $\phi_{x}^{int,s}$ compared to $\mathbb{C}_{sy}$ and
$\mathbb{C}_{xy}$ (Fig. 2a). On the other hand, for X$\rightarrow$Y channel, X
acts as an extrinsic variable for Y. Therefore, an increase in
$\phi_{y}^{int,x}$ and $\phi_{y}^{ext,x}$ leads to enhancement of the channel
capacities $\mathbb{C}_{sy}$ and $\mathbb{C}_{xy}$ (see Eqs. (13,14) and Fig.
2b). These findings illustrate that the magnitude of $\phi$-s, which
characterizes noise propagation along each channel, is closely associated with
the information transmission capacity of each channel. $\phi$ thus serves as
an indicator of the noise processing capacity of individual noise processing
channels along the regulatory edges.

The noise processing capacity $\phi$ generates an extrinsic noise pool at the
downstream node of each channel which enhances information transmission along
the respective channel. Therefore, it can be argued that the contribution of
extrinsic noise at the downstream node plays a positive role in facilitating
information transmission from an upstream node. Eq. (12-14) further reveals
that the intrinsic noise $\eta_{int,s}^{2}$ enhances all the three channel
capacities (Fig. 2c,f). The input noise $\eta_{int,s}^{2}$ thus facilitates
information transmission downstream as it acts as an extrinsic variable for X
and Y. In other words, the more the magnitude of $\eta_{int,s}^{2}$, more the
downstream nodes X and Y sense the input noise as information.

The influence of the intrinsic noise of X ($\eta_{int,x}^{2}$) on
$\mathbb{C}_{sx}$ and $\mathbb{C}_{sy}$ is repressing in nature (Fig. 2d,f).
The increase in intrinsic noise of X acts as a constraint, limiting X’s
ability to accurately detect the signal S and consequently, the information
transmission from S to X ($\mathbb{C}_{sx}$) diminishes. The information
transmission from S to Y ($\mathbb{C}_{sy}$) also diminishes as Y gains
information from S via X. The information transmission from X to Y
($\mathbb{C}_{xy}$) increases due to $\eta_{int,x}^{2}$ (Fig. 2d,f) as
$\eta_{int,x}^{2}$ acts as an extrinsic variable for the channel
X$\rightarrow$Y. The intrinsic noise of Y, $\eta_{int,y}^{2}$, increases the
variability of Y. As a consequence, $\eta_{int,y}^{2}$ acts as a potent
limiting factor for which the information gained by Y from both S and X
($\mathbb{C}_{sy}$ and $\mathbb{C}_{xy}$) is reduced (Fig. 2e,f).

##  IV Conclusion

To summarize, we presented a theoretical analysis of noise propagation in a
TSC on the basis of noise decomposition, and demonstrated the presence of
distinct noise processing channels between the system components S, X, and Y.
The extent of noise propagation along these channels depends on the noise
processing capacity $\phi$. The intrinsic noise of the signal
$\eta_{int,s}^{2}$ together with noise processing capacities ($\phi$) act as
extrinsic variables and alleviates the overall information transmission along
the cascade. On the other hand, the intrinsic noise of X, $\eta_{int,x}^{2}$
put a limit in information transmission along the channels S$\rightarrow$X and
S$\rightarrow$X$\rightarrow$Y and act as bottleneck that reduce information
transfer capacities. Similarly, intrinsic noise of Y, $\eta_{int,y}^{2}$
hinders information flow along the channels S$\rightarrow$X$\rightarrow$Y and
X$\rightarrow$Y. Thus, noise propagation along the TSC depends on noise
processing capacity of each noise processing channels which along with
intrinsic noises modulate the information transmission capacity of the system.

The interplay between intrinsic and extrinsic noise constitutes an optimum
signal transduction machinery. When there is a need to precisely sense and
detect the signal for adaptability, the genes within the cascade exhibit
dynamics in which the molecular intrinsic fluctuations become negligible. This
ensures that the inherent noise of the system does not hinder the accurate
reception of the signal. Conversely, when the signal is detrimental to the
cellular response, the cascade’s gene expression dynamics adapt in a manner
that increases the intrinsic noise of the gene products. Such strategy can be
seen as a protective mechanism, where the system becomes less responsive to
external fluctuations thereby prioritizing robustness that might otherwise
disrupt its normal functioning.

###### Acknowledgements.

MN thanks SERB, India, for National Post-Doctoral Fellowship
[PDF/2022/001807].

##  Appendix A System size expansion and Lyapunov equation

In the following, we use the notation $\mathbf{N}$ to represent a three
dimensional vector of copy numbers of the components, i.e.,
$\mathbf{N}\in\\{n_{s}\equiv s,n_{x}\equiv x,n_{y}\equiv y\\}$. We rewrite Eq.
(2) in terms of macroscopic concentration of each component defined as
$c_{i}:=\lim\limits_{\Omega\to\infty}(n_{i}/\Omega)$ with $i\in\\{s,x,y\\}$
Hayot and Jayaprakash (2004); van Kampen (2007) where $\Omega$ is the reaction
volume,

$\displaystyle\frac{dP(\boldsymbol{N};t)}{dt}$ $\displaystyle=$
$\displaystyle\Omega\sum_{i=s,x,y}\left[(\mathbb{E}_{i}^{+1}-1)g_{i}(n_{i}/\Omega)P(\boldsymbol{N};t)\right.$
(15)
$\displaystyle\left.+(\mathbb{E}_{i}^{-1}-1)f_{i}(\boldsymbol{N}/\Omega)P(\boldsymbol{N};t)\right].$

In the above equation, $f_{i}(\boldsymbol{N}/\Omega)$ represents the
generalised form of any reaction network. For TSC it specifically translates
to $f_{s}$, $f_{x}(n_{s}/\Omega)$, and $f_{y}(n_{x}/\Omega)$, representing the
production of S, X, and Y, respectively. In the following, we provide the
pedagogical formulation of van Kampen’s system size expansion van Kampen
(2007) to solve the Eq. (15). We now expand $n_{i}$ in $\Omega$ space of
$\mathcal{O}(\Omega^{1/2})$

$n_{i}=\Omega c_{i}+\Omega^{1/2}\epsilon_{i},$ (16)

where $\epsilon_{i}$ represents the fluctuations around $n_{i}$. Using Eq.
(16) in the left hand side of Eq. (15) we have

$\frac{dP(\boldsymbol{N};t)}{dt}=\frac{d\widetilde{P}(\boldsymbol{\epsilon};t)}{dt}-\Omega^{-1/2}\sum_{i=s,x,y}\frac{dc_{i}}{dt}\frac{d\widetilde{P}(\boldsymbol{\epsilon};t)}{d\epsilon_{i}},$
(17)

with $\boldsymbol{\epsilon}\in\\{\epsilon_{s},\epsilon_{x},\epsilon_{y}\\}$.
Eq. (17) accounts for the time derivative of $P(\boldsymbol{N};t)$ at fixed
$n_{i}$ so that Eq. (16) becomes $(d\epsilon_{i}/dt)=\Omega^{-1/2}(dc_{i}/dt)$
Elf and Ehrenberg (2003); Hayot and Jayaprakash (2004). Expansion of the step
operator $\mathbb{E}^{\pm}$ in Eq. (15) provides Elf and Ehrenberg (2003);
Hayot and Jayaprakash (2004)

$\displaystyle\mathbb{E}_{i}^{+1}$ $\displaystyle=$ $\displaystyle
1+\Omega^{-1/2}(\partial/\partial\epsilon_{i})+(1/2)\Omega^{-1}(\partial^{2}/\partial\epsilon_{i}^{2}),$
(18) $\displaystyle\mathbb{E}_{i}^{-1}$ $\displaystyle=$ $\displaystyle
1-\Omega^{-1/2}(\partial/\partial\epsilon_{i})+(1/2)\Omega^{-1}(\partial^{2}/\partial\epsilon_{i}^{2}).$
(19)

Moreover, the Taylor expansion of the synthesis and degradation functions in
Eq. (15) around their macroscopic values give Elf and Ehrenberg (2003),

$\displaystyle f_{i}\left(\frac{\boldsymbol{N}}{\Omega}\right)$
$\displaystyle=$ $\displaystyle
f_{i}(\boldsymbol{c})+\Omega^{-1/2}\sum_{j=s,x,y}\epsilon_{j}\frac{\partial
f_{i}(\boldsymbol{c})}{\partial c_{j}},$ (20) $\displaystyle
g_{i}\left(\frac{n_{i}}{\Omega}\right)$ $\displaystyle=$ $\displaystyle
g_{i}(c_{i})+\Omega^{-1/2}\epsilon_{i}\frac{\partial g_{i}(c_{i})}{\partial
c_{i}},$ (21)

with $\boldsymbol{c}\in\\{c_{s},c_{x},c_{y}\\}$. Inserting
$P(\boldsymbol{N};t)=\widetilde{P}(\boldsymbol{\epsilon};t)$ and Eqs. (18-21)
into the right-hand side of Eq. (15) we have

$\displaystyle\frac{d\widetilde{P}(\boldsymbol{\epsilon};t)}{dt}-\Omega^{-1/2}\sum_{i=s,x,y}\frac{dc_{i}}{dt}\frac{d\widetilde{P}(\boldsymbol{\epsilon};t)}{d\epsilon_{i}}$
$\displaystyle=\Omega\sum_{i=s,x,y}\left[\left(-\Omega^{-1/2}\frac{\partial}{\partial\epsilon_{i}}+\frac{1}{2}\Omega^{-1}\frac{\partial^{2}}{\partial\epsilon_{i}^{2}}\right)\left(f_{i}(\boldsymbol{c})+\Omega^{-1/2}\sum_{j=s,x,y}\epsilon_{j}\frac{\partial
f_{i}(\boldsymbol{c})}{\partial
c_{j}}\right)\widetilde{P}(\boldsymbol{\epsilon};t)\right.$
$\displaystyle\left.+\left(\Omega^{-1/2}\frac{\partial}{\partial\epsilon_{i}}+\frac{1}{2}\Omega^{-1}\frac{\partial^{2}}{\partial\epsilon_{i}^{2}}\right)\left(g_{i}(c_{i})+\Omega^{-1/2}\epsilon_{i}\frac{\partial
g_{i}(c_{i})}{\partial
c_{i}}\right)\widetilde{P}(\boldsymbol{\epsilon};t)\right].$ (22)

In the above equation, there are two dominant terms with
$\mathcal{O}(\Omega^{1/2})$ and $\mathcal{O}(\Omega^{0})$. Collecting the
terms of $\mathcal{O}(\Omega^{1/2})$ from both sides results in the usual
deterministic rate equation for TSC motif in terms of concentrations $c_{i}$,

$\frac{dc_{i}}{dt}=f_{i}(\boldsymbol{c})-g_{i}(c_{i}).$ (23)

Using the definition of $c_{i}$, Eq. (23) can be recast in terms of copy
number $n_{i}$ as $dn_{i}/dt=f_{i}(\boldsymbol{N})-g_{i}(n_{i})$. For the
components S, X, and Y, the deterministic equations become,

$\displaystyle\frac{ds}{dt}$ $\displaystyle=$ $\displaystyle f_{s}-g_{s}(s),$
(24) $\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
f_{x}(s)-g_{x}(x),$ (25) $\displaystyle\frac{dy}{dt}$ $\displaystyle=$
$\displaystyle f_{y}(x)-g_{y}(y),$ (26)

where we have used $n_{s}\equiv s$, $n_{x}\equiv x$, and $n_{y}\equiv y$. On
the other hand, collecting terms of $\mathcal{O}(\Omega^{0})$ from both sides
of Eq. (22) results in linear Fokker-Planck equation (FPE) for the
fluctuations $\epsilon_{i}$,

$\displaystyle\frac{\partial\widetilde{P}(\boldsymbol{\epsilon};t)}{\partial
t}$ $\displaystyle=$
$\displaystyle-\sum_{i,j=s,x,y}J_{ij}\frac{\partial}{\partial\epsilon_{i}}\epsilon_{j}\widetilde{P}(\boldsymbol{\epsilon};t)$
(27)
$\displaystyle+\frac{1}{2}\sum_{i,j=s,x,y}D_{ij}\frac{\partial^{2}}{\partial\epsilon_{i}\partial\epsilon_{j}}\widetilde{P}(\boldsymbol{\epsilon};t),$

where,

$\displaystyle J_{ij}$ $\displaystyle=$ $\displaystyle U_{ij}-V_{ij},$
$\displaystyle U_{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial
f_{i}(\boldsymbol{c})}{\partial c_{j}}~{}~{}~{}~{}~{}~{}\forall i,j,$
$\displaystyle V_{ij}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{\partial g_{i}(c_{i})}{\partial
c_{i}}&i=j,\\\ 0&i\neq j,\end{cases}$ $\displaystyle D_{ij}$ $\displaystyle=$
$\displaystyle\begin{cases}f_{i}(\boldsymbol{c})+g_{i}(c_{i})&i=j,\\\ 0&i\neq
j.\end{cases}$ (28)

The average value, $\langle\epsilon_{k}\rangle$ at steady state can be
obtained by multiplying both sides of the FPE (Eq. (27)) with $\epsilon_{k}$
and integrating over all $\boldsymbol{\epsilon}$ van Kampen (2007), one
obtains at steady state,

$\displaystyle\frac{\partial\langle\epsilon_{k}\rangle}{\partial t}$
$\displaystyle=$ $\displaystyle\sum_{i=s,x,y}\langle
J_{ki}\rangle\langle\epsilon_{i}\rangle,$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=s,x,y}\langle J_{ki}\rangle\langle\epsilon_{i}\rangle,$
(29)

which leads to $\langle\epsilon_{k}\rangle=0$. Again multiplying both sides of
the FPE with $\epsilon_{k}\epsilon_{l}$ and integrating over all
$\boldsymbol{\epsilon}$ van Kampen (2007), at steady state we obtain,

$\frac{\partial\langle\epsilon_{k}\epsilon_{l}\rangle}{\partial
t}=\sum_{i=s,x,y}\langle
J_{ki}\rangle\langle\epsilon_{i}\epsilon_{l}\rangle+\sum_{j=s,x,y}\langle
J_{lj}\rangle\langle\epsilon_{k}\epsilon_{j}\rangle+\langle D_{kl}\rangle.$
(30)

The elements of covariance matrix for $\boldsymbol{\epsilon}$ can be written
as
$\xi_{kl}=\langle\langle\epsilon_{k}\epsilon_{l}\rangle\rangle=\langle\epsilon_{k}\epsilon_{l}\rangle-\langle\epsilon_{k}\rangle\langle\epsilon_{l}\rangle=\langle\epsilon_{k}\epsilon_{l}\rangle$
as $\langle\boldsymbol{\epsilon}\rangle=0$. Thus Eq. (30) can be recast in the
matrix form Elf and Ehrenberg (2003),

$\frac{\partial\boldsymbol{\xi}}{\partial
t}=\boldsymbol{J}\boldsymbol{\xi}+\boldsymbol{\xi}\boldsymbol{J}^{T}+\boldsymbol{D},$
(31)

where, $\boldsymbol{J}=\\{\langle J_{ij}\rangle\\}$ and
$\boldsymbol{D}=\\{\langle D_{ij}\rangle\\}$ are Jacobian matrix and diffusion
matrix, respectively. $\boldsymbol{J}^{T}$ refers to the transpose of Jacobian
matrix. At steady state, $\partial\boldsymbol{\xi}/\partial t=0$, which leads
to the Lyapunov equation for $\boldsymbol{\xi}$,

$\boldsymbol{J}\boldsymbol{\xi}+\boldsymbol{\xi}\boldsymbol{J}^{T}+\boldsymbol{D}=0.$
(32)

Using steady state condition in Eq. (16), the covariance in terms of copy
number becomes,

$\displaystyle\sigma_{ij}^{2}$ $\displaystyle=$ $\displaystyle\langle
n_{i}n_{j}\rangle-\langle n_{i}\rangle\langle n_{j}\rangle,$ (33)
$\displaystyle=$ $\displaystyle\Omega\langle\epsilon_{i}\epsilon_{j}\rangle,$

where we have used
$\langle\epsilon_{i}c_{j}\rangle=\langle\epsilon_{i}\rangle\langle
c_{j}\rangle$ and $\langle c_{i}c_{j}\rangle=\langle c_{i}\rangle\langle
c_{j}\rangle$. Using Eq. (33) in Eq. (32) we have,

$\boldsymbol{J}\boldsymbol{\sigma}+\boldsymbol{\sigma}\boldsymbol{J}^{T}+\Omega\boldsymbol{D}=0.$
(34)

Here, $\boldsymbol{\sigma}$ is the covariance matrix with elements
$\sigma_{ij}^{2}$. The Jacobian matrix and the diffusion matrix are expressed
in terms of concentrations $c_{i}$ in Eq. (28). Using the definition
$c_{i}:=\lim\limits_{\Omega\to\infty}(n_{i}/\Omega)$ we transform the Jacobian
matrix in terms of copy number $n_{i}$ and obtain $\boldsymbol{J}=\\{\langle
J_{ij}\rangle\\}=\\{\langle U_{ij}\rangle-\langle V_{ij}\rangle\\}$, where

$\displaystyle U_{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial
f_{i}(\boldsymbol{N})}{\partial
n_{j}}=f_{i,j}^{\prime}(\boldsymbol{N})~{}~{}~{}~{}~{}~{}\forall~{}i,j,$
$\displaystyle\langle U_{ij}\rangle$ $\displaystyle=$ $\displaystyle
f_{i,j}^{\prime}(\langle\boldsymbol{N}\rangle),$ $\displaystyle V_{ij}$
$\displaystyle=$ $\displaystyle\begin{cases}\frac{\partial
g_{i}(n_{i})}{\partial n_{i}}=g_{i,i}^{\prime}(n_{i})&i=j\\\ 0&i\neq
j\end{cases},$ $\displaystyle\langle V_{ij}\rangle$ $\displaystyle=$
$\displaystyle\begin{cases}g_{i,i}^{\prime}(\langle n_{i}\rangle)&i=j\\\
0&i\neq j\end{cases}.$ (35)

The third term in Eq. (34) can be written as
$\boldsymbol{\Xi}=\Omega\boldsymbol{D}$, where $\boldsymbol{\Xi}$ is the
diffusion matrix at steady state in terms of the copy number $n_{i}$. The
explicit expression of the diagonal elements of $\boldsymbol{\Xi}$ (when
$i=j$) is derived using Eq. (28),

$\displaystyle\Xi_{ij}$ $\displaystyle=$ $\displaystyle\Omega\langle
D_{ij}\rangle,$ (36) $\displaystyle=$ $\displaystyle\Omega\langle
f_{i}(\boldsymbol{\phi})\rangle+\Omega\langle g_{i}(\phi_{i})\rangle,$
$\displaystyle=$ $\displaystyle
f_{i}(\langle\boldsymbol{N}\rangle)+g_{i}(\langle n_{i}\rangle).$

For $i\neq j$, $\Xi_{ij}=0$. We now rewrite the Lyapunov equation in terms of
copy number as,

$\boldsymbol{J}\boldsymbol{\sigma}+\boldsymbol{\sigma}\boldsymbol{J}^{T}+\boldsymbol{\Xi}=0.$
(37)

##  Appendix B Solution of Lyapunov equation

Using the explicit functional forms of $f$-s and $g$-s (see Fig. 1a) in Eq.
(35) we have the Jacobian matrix for the TSC at steady state,

$\displaystyle\mathbf{J}=\left(\begin{array}[]{ccc}-g^{\prime}_{s,s}(\langle
s\rangle)&0&0\\\ f^{\prime}_{x,s}(\langle s\rangle)&-g^{\prime}_{x,x}(\langle
x\rangle)&0\\\ 0&f^{\prime}_{y,x}(\langle x\rangle)&-g^{\prime}_{y,y}(\langle
y\rangle)\end{array}\right).$

Here, $g_{s,s}^{\prime}(\langle s\rangle)$ stands for the differentiation of
$g_{s}(s)$ with respect to $s$ and evaluated at $s=\langle s\rangle$, and so
on. In the following, we write $g_{s,s}^{\prime}(\langle s\rangle)$,
$g_{x,x}^{\prime}(\langle x\rangle)$, etc as $g_{s,s}^{\prime}$,
$g_{x,x}^{\prime}$, etc, respectively. Similarly, using Eq. (36), the
diffusion matrix can be written as,

$\displaystyle\mathbf{\Xi}=\left(\begin{array}[]{ccc}2g_{s}(\langle
s\rangle)&0&0\\\ 0&2g_{x}(\langle x\rangle)&0\\\ 0&0&2g_{y}(\langle
y\rangle)\end{array}\right).$

While writing the diffusion matrix, we take into account Eqs. (24-26) at
steady state which yield $f_{s}=g_{s}(\langle s\rangle)$, $f_{x}(\langle
s\rangle)=g_{x}(\langle x\rangle)$, and $f_{y}(\langle x\rangle)=g_{y}(\langle
y\rangle)$. Using the expressions of $\mathbf{J}$ and $\mathbf{\Xi}$ in the
Lyapunov equation (37) we have the following analytical expressions of
variance and covariance for the system components at steady state for $n=1$,

$\displaystyle\sigma_{s}^{2}$ $\displaystyle=$ $\displaystyle\langle
s\rangle,$ (40) $\displaystyle\sigma_{x}^{2}$ $\displaystyle=$
$\displaystyle\langle
x\rangle+\frac{f_{x,s}^{\prime^{2}}}{\beta_{x}(\beta_{s}+\beta_{x})}\langle
s\rangle,$ (41) $\displaystyle\sigma_{y}^{2}$ $\displaystyle=$
$\displaystyle\langle
y\rangle+\frac{f_{y,x}^{\prime^{2}}}{\beta_{y}(\beta_{x}+\beta_{y})}\langle
x\rangle$ (42)
$\displaystyle+\frac{(\beta_{s}+\beta_{x}+\beta_{y})f_{y,x}^{\prime^{2}}}{\beta_{y}(\beta_{s}+\beta_{y})(\beta_{x}+\beta_{y})}\times\frac{f_{x,s}^{\prime^{2}}}{\beta_{x}(\beta_{s}+\beta_{x})}\langle
s\rangle,$ $\displaystyle\sigma_{sx}^{2}$ $\displaystyle=$
$\displaystyle\frac{f_{x,s}^{\prime}}{\beta_{s}+\beta_{x}}\langle s\rangle,$
(43) $\displaystyle\sigma_{sy}^{2}$ $\displaystyle=$
$\displaystyle\frac{f_{x,s}^{\prime}f_{y,x}^{\prime}}{(\beta_{s}+\beta_{x})(\beta_{s}+\beta_{y})}\langle
s\rangle,$ (44) $\displaystyle\sigma_{xy}^{2}$ $\displaystyle=$
$\displaystyle\frac{f_{y,x}^{\prime}}{\beta_{x}+\beta_{y}}\langle x\rangle$
(45)
$\displaystyle+\frac{(\beta_{s}+\beta_{x}+\beta_{y})f_{x,s}^{\prime^{2}}f_{y,x}^{\prime}}{\beta_{x}(\beta_{s}+\beta_{x})(\beta_{s}+\beta_{y})(\beta_{x}+\beta_{y})}\langle
s\rangle.$

##  Appendix C Noise decomposition

The noise associated with each component (S, X, and Y) is measured by the
coefficient of variation (CV). The square of CV for $i$-th component
($i\in\\{s,x,y\\}$) is defined as $\eta_{i}^{2}:=\sigma_{i}^{2}/\langle
i\rangle^{2}$. In the rest of the calculation, we use square of CV, i.e., CV2
as a metric to quantify the noise. The explicit expressions of noise
associated with each component thus becomes,

$\displaystyle\eta_{s}^{2}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{1}{\langle s\rangle}}_{\eta_{int,s}^{2}},$
(46) $\displaystyle\eta_{x}^{2}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{1}{\langle
x\rangle}}_{\eta_{int,x}^{2}}+\overbrace{\underbrace{\frac{f_{x,s}^{\prime^{2}}\langle
s\rangle^{2}}{\beta_{x}(\beta_{s}+\beta_{x})\langle
x\rangle^{2}}}_{\phi_{x}^{int,s}}\times\underbrace{\frac{1}{\langle
s\rangle}}_{\eta_{int,s}^{2}}}^{\eta_{ext,x}^{2}},$ (47)
$\displaystyle\eta_{y}^{2}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{1}{\langle
y\rangle}}_{\eta_{int,y}^{2}}+\overbrace{\overbrace{\underbrace{\frac{f_{y,x}^{\prime^{2}}\langle
x\rangle^{2}}{\beta_{y}(\beta_{x}+\beta_{y})\langle
y\rangle^{2}}}_{\phi_{y}^{int,x}}\times\underbrace{\frac{1}{\langle
x\rangle}}_{\eta_{int,x}^{2}}}^{\eta_{ext,y1}^{2}}+\overbrace{\underbrace{\frac{f_{y,x}^{\prime^{2}}(\beta_{s}+\beta_{x}+\beta_{y})\langle
x\rangle^{2}}{\beta_{y}(\beta_{s}+\beta_{y})(\beta_{x}+\beta_{y})\langle
y\rangle^{2}}}_{\phi_{y}^{ext,x}}\times\underbrace{\underbrace{\frac{f_{x,s}^{\prime^{2}}\langle
s\rangle^{2}}{\beta_{x}(\beta_{s}+\beta_{x})\langle
x\rangle^{2}}}_{\phi_{x}^{int,s}}\times\underbrace{\frac{1}{\langle
s\rangle}}_{\eta_{int,s}^{2}}}_{\eta_{ext,x}^{2}}}^{\eta_{ext,y2}^{2}}}^{\eta_{ext,y}^{2}},$
(48)

where, $\eta_{int,i}^{2}$ and $\eta_{ext,i}^{2}$ stand for intrinsic and
extrinsic noise of the component $i$, respectively. In Eqs. (47-48), the
extrinsic noises are further decomposed into more specific terms. The
extrinsic noise of X is expressed as
$\eta_{ext,x}^{2}=\phi_{x}^{int,s}\eta_{int,s}^{2}$. Similarly, the extrinsic
noise of Y is $\eta_{ext,y}^{2}=\eta_{ext,y1}^{2}+\eta_{ext,y2}^{2}$, where,
$\eta_{ext,y1}^{2}=\phi_{y}^{int,x}\eta_{int,x}^{2}$ and
$\eta_{ext,y2}^{2}=\phi_{y}^{ext,x}\eta_{ext,x}^{2}=\phi_{y}^{ext,x}\phi_{x}^{int,s}\eta_{int,s}^{2}$.
The detailed extrinsic noise components offer significant insights into the
mechanism of noise propagation. Using the expressions of $f$ and $g$ given in
Fig. 1(A), we write the explicit analytical forms of $\phi$-s for $n=1$.

$\displaystyle\phi_{x}^{int,s}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{\beta_{x}}{\beta_{s}+\beta_{x}}}_{\tau_{x,s}}\left(\frac{K_{sx}}{K_{sx}+\langle
s\rangle}\right)^{2},$ (49) $\displaystyle\phi_{y}^{int,x}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{\beta_{y}}{\beta_{x}+\beta_{y}}}_{\tau_{y,x}}\left(\frac{K_{xy}}{K_{xy}+\langle
x\rangle}\right)^{2},$ (50) $\displaystyle\phi_{y}^{ext,x}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{\beta_{s}+\beta_{x}+\beta_{y}}{\beta_{s}+\beta_{y}}}_{\tau_{sy,x}^{-1}}\frac{\beta_{y}}{\beta_{x}+\beta_{y}}\left(\frac{K_{xy}}{K_{xy}+\langle
x\rangle}\right)^{2},$ (51)

where we have used the notation $\tau$ to represent the scaled time scale.

##  Appendix D Analytical expressions of channel capacity

Under Gaussian channel approximation, the mutual information $I(i;j)$ provides
the measure of channel capacity $\mathbb{C}_{ij}$ Cover and Thomas (1991),
which can be written in terms of signal-to-noise ratio (SNR) as Tostevin and
ten Wolde (2010), $\mathbb{C}_{ij}:=I(i;j)=(1/2)\log_{2}(1+\mathbb{S}_{ij})$,
where $\mathbb{S}_{ij}=\eta_{ij}^{4}/(\eta_{i}^{2}\eta_{j|i}^{2})$ stands for
the SNR. In the expression of SNR, we use
$\eta_{ij}^{4}=\sigma_{ij}^{4}/(\langle i\rangle^{2}\langle j\rangle^{2})$,
$\eta_{i}^{2}=\sigma_{i}^{2}/\langle i\rangle^{2}$, and
$\eta_{j|i}^{2}=\sigma_{j|i}^{2}/\langle j\rangle^{2}$, where
$\sigma_{j|i}^{2}=\sigma_{j}^{2}-(\sigma_{ij}^{4}/\sigma_{i}^{2})$. The
conditional relation can be written as
$\eta_{j|i}^{2}=\eta_{j}^{2}-(\eta_{ij}^{4}/\eta_{i}^{2})$. Using these
definitions together with Eqs. (43-51), we have

$\displaystyle\eta_{sx}^{4}$ $\displaystyle=$
$\displaystyle\tau_{x,s}\phi_{x}^{int,s}\eta_{int,s}^{4},$ (52)
$\displaystyle\eta_{sy}^{4}$ $\displaystyle=$
$\displaystyle\tau_{x,s}\tau_{y,s}\tau_{xy,s}\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{4},$
(53) $\displaystyle\eta_{xy}^{4}$ $\displaystyle=$
$\displaystyle\tau_{y,x}\phi_{x}^{int,s}\phi_{y}^{int,x}\eta_{int,x}^{4}+\tau_{y,s}\tau_{xy,s}^{-1}(\phi_{x}^{int,s})^{2}\phi_{y}^{ext,x}\eta_{int,s}^{4}$
(54)
$\displaystyle+2\tau_{y,x}\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}\eta_{int,x}^{2},$

where, $\tau_{i,j}=\beta_{i}/(\beta_{i}+\beta_{j})$ and
$\tau_{ij,k}=(\beta_{i}+\beta_{j})/(\beta_{i}+\beta_{j}+\beta_{k})$. The
explicit expressions of SNR for the TSC motif thus become

$\displaystyle\mathbb{S}_{sx}$ $\displaystyle=$
$\displaystyle\frac{\tau_{x,s}\phi_{x}^{int,s}\eta_{int,s}^{2}}{\eta_{int,x}^{2}+(1-\tau_{x,s})\phi_{x}^{int,s}\eta_{int,s}^{2}},$
(55) $\displaystyle\mathbb{S}_{sy}$ $\displaystyle=$
$\displaystyle\frac{\tau_{x,s}\tau_{y,s}\tau_{xy,s}\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}}{\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2}+(1-\tau_{x,s}\tau_{y,s}\tau_{xy,s})\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}},$
(56) $\displaystyle\mathbb{S}_{xy}$ $\displaystyle=$
$\displaystyle\frac{\tau_{y,x}\phi_{y}^{int,x}\eta_{int,x}^{4}+\tau_{y,s}\tau_{xy,s}^{-1}(\phi_{x}^{int,s})^{2}\phi_{y}^{ext,x}\eta_{int,s}^{4}+2\tau_{y,x}\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}\eta_{int,x}^{2}}{\left[\displaystyle\begin{multlined}\eta_{int,x}^{2}\eta_{int,y}^{2}+\phi_{x}^{int,s}\eta_{int,s}^{2}(\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2})+(1-\tau_{y,x})\phi_{y}^{int,x}\eta_{int,x}^{4}\\\
+(1-\tau_{y,s}\tau_{xy,s}^{-1})(\phi_{x}^{int,s})^{2}\phi_{y}^{ext,x}\eta_{int,s}^{4}+(1-2\tau_{y,x})\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}\eta_{int,x}^{2}\end{multlined}\eta_{int,x}^{2}\eta_{int,y}^{2}+\phi_{x}^{int,s}\eta_{int,s}^{2}(\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2})+(1-\tau_{y,x})\phi_{y}^{int,x}\eta_{int,x}^{4}\\\
+(1-\tau_{y,s}\tau_{xy,s}^{-1})(\phi_{x}^{int,s})^{2}\phi_{y}^{ext,x}\eta_{int,s}^{4}+(1-2\tau_{y,x})\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}\eta_{int,x}^{2}\right]}.$
(59)

In the present study, we use separation of degradation time scales
$\beta_{s}\ll\beta_{x}\ll\beta_{y}$. The inequality in $\beta$-s results in
$\tau_{x,s}\approx 1$, $\tau_{y,s}\approx 1$, $\tau_{xy,s}\approx 1$, and
$\tau_{y,x}\approx 1$, for which we approximate $1-\tau_{x,s}\approx 0$,
$1-\tau_{x,s}\tau_{y,s}\tau_{xy,s}\approx 0$,
$1-\tau_{y,s}\tau_{xy,s}^{-1}\approx 0$, and $1-2\tau_{y,x}\approx-1$. Using
these approximations, the expressions of SNR yield,

$\displaystyle\mathbb{S}_{sx}$ $\displaystyle=$
$\displaystyle\frac{\phi_{x}^{int,s}\eta_{int,s}^{2}}{\eta_{int,x}^{2}},$ (60)
$\displaystyle\mathbb{S}_{sy}$ $\displaystyle=$
$\displaystyle\frac{\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}}{\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2}},$
(61) $\displaystyle\mathbb{S}_{xy}$ $\displaystyle=$
$\displaystyle\frac{A}{B},$ (62)

where,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\phi_{y}^{int,x}\eta_{int,x}^{4}+(\phi_{x}^{int,s}\eta_{int,s}^{2}+2\eta_{int,x}^{2})\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\eta_{int,x}^{2}\eta_{int,y}^{2}+(\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2})\phi_{x}^{int,s}\eta_{int,s}^{2}$
$\displaystyle-\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}\eta_{int,x}^{2}.$

Using Eqs. (60-62), the channel capacities is written as,

$\displaystyle\mathbb{C}_{sx}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left[1+\frac{\phi_{x}^{int,s}\eta_{int,s}^{2}}{\eta_{int,x}^{2}}\right],$
(63) $\displaystyle\mathbb{C}_{sy}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left[1+\frac{\phi_{x}^{int,s}\phi_{y}^{ext,x}\eta_{int,s}^{2}}{\eta_{int,y}^{2}+\phi_{y}^{int,x}\eta_{int,x}^{2}}\right],$
(64) $\displaystyle\mathbb{C}_{xy}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left[1+\frac{A}{B}\right].$ (65)

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