Journal of Robotics, Networking and Artificial Life
Vol. 7(1); June (2020), pp. 52–57
DOI: https://doi.org/10.2991/jrnal.k.200512.011; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

An Error Correction Mechanism for Reliable
Chemical Communication Systems
Masashi K. Kajita*
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

ARTICLE INFOABSTRACT

Article History

Chemical communication systems, such as bio-inspired chemical sensory systems or biological cells, sense the environment bydetecting target ligand molecules, which convey environmental information. However, non-target ligands, similar to the target
ones, are ubiquitous in the environment and can hamper accurate information transmission. In this work, we investigate an error
correction mechanism for reliable chemical communications and find an intuitive understanding of how the mechanism canamplify the small difference between the target and non-target ligands. We also demonstrate that the mechanism can balance
accuracy and output intensity. Our approach may provide a method to design reliable chemical communication systems.

Received 25 November 2019
Accepted 10 March 2020
Keywords
Bio-inspired communication
chemical reaction network
chemical sensory systemerror correction
kinetic proofreading© 2020 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Chemical communication systems, such as bio-inspired chemical sensory systems or biological cells, sense the environment bydetecting target ligand molecules, which convey environmental
information [1]. The chemical communication system detects thetarget ligands using receptor molecules, which act as sensors forenvironmental information. However, non-target ligands, similar
to the target ones, are ubiquitous in the environment. Due to thestructural similarity, the non-target ligands have affinities to thereceptors and can attach to the receptor molecules. This nonspecific and undesired interaction may send an erroneous signal
into the system and hamper accurate information transmission.
Even under the effect of non-target ligands, biological cells suchas immune T cells or chemical reaction systems for the translationand transcription of genetic information have high fidelity to thetarget ligands [2–6].
The series of experimental observations suggest that biological systems have some error correction mechanisms to detect the target
ligands for reliable chemical communication. To reveal the underlying mechanism of the biological error correction systems forchemical communications, the zero-order proofreading model [7]was recently proposed as an extended model of the kinetic proofreading model, which is the pioneering works by Hopfield andNinio for a biological error correction mechanism [8,9].
In this paper, we first model chemical sensing. Then we introduce
the zero-order proofreading model [7] and investigate the intuitive understanding of how the model can precisely discriminate
the target ligand from the similar non-target one based on theiraffinity parameters. We further investigate the target detection
Email: masashi.k.kajita@gmail.com*

performance of the zero-order proofreading model under thecondition that the target and non-target ligands exist in theenvironment simultaneously.

2. ZERO-ORDER PROOFREADINGMECHANISM FOR ULTRA-SPECIFICITY
2.1. Modeling of Chemical Sensing
We start with considering the simplest situation that only a single
type of ligand molecule exists in the environment (Figure 1). In thissituation, a chemical communication system or a cell distinguishes
whether the ligand molecule is the target ligand or non-target one
by using receptor molecules, which work as chemical sensors forthe environmental signal. The receptor R can detect the ligand L,
and the ligand–receptor binding and unbinding reactions aredescribed as follows:k1

R + L

 C0 , 
k-1(1)

where k1(s−1M−1) and k−1(s−1) are the reaction constants of thebinding and unbinding reactions, respectively. The complex of thereceptor R and the ligand L is denoted as C0.
We assume that the target and non-target ligands have similar sizes
and structures. Then, we can assume that the difference between thetarget and non-target ligands is only the unbinding rate k−1 (Figure 1).
If we denote the unbinding rates of the target ligand k−1,T(s−1)(Figure 1a) and that of the non-target one k−1,N(s−1) (Figure 1b),
the unbinding rates satisfy k−1,T(s−1) < k−1,N(s−1) because the nontarget ligand is more likely to dissociate from the receptor.

M.K. Kajita / Journal of Robotics, Networking and Artificial Life 7(1) 52–57

ab

Figure 1 | Schematics of a chemical communication system when (a) thetarget ligand molecule or (b) non-target one exists in the environment. The
reaction constants of the binding and unbinding reactions are respectivelyk1(s−1M−1) and k−1*(s−1) where * is T and N for the target ligand and nontarget one, respectively. We assume that k−1,T(s−1) < k−1,N(s−1) because thenon-target ligand is more likely to dissociate from the receptor.

53

two unbinding reactions, C0 → R + L and C1 → R + L, have the samerate k−1 [8–10]. This assumption is valid if the chemical modification of the receptor, which is carried out by the reaction C0 → C1,
only affects the intracellular part of the receptor and does not affectthe unbinding rate k−1. We also note that the unbinding reaction
C1 → R + L is an irreversible reaction for an error correction function, which is called kinetic proofreading [8–10]. The product RP
is covalently modified and activated receptor R, and RP can bedeactivated by an enzyme P. D denotes the complex of RP and P.
The reaction constants of the binding and unbinding reactions arek2(s−1M−1) and k−2(s−1), respectively. The receptor R is generated
from D with rate k3(s−1).

2.3. Model Reduction Based on
Michaelis–Menten Approximation
By assuming that each reaction follows the law of mass action, thedynamics of the zero-order proofreading model [Equation (2)] canbe described by Ordinary Differential Equations (ODEs). To obtainan intuitive understanding of how the zero-order proofreading
model amplifies the small difference among the target and nontarget ligands, we perform a model reduction based on Michaelis–Menten approximation [11].

Figure 2 | Schematic diagram of the zero-order proofreading model[Equation (2)] in a chemical communication system. The receptors detect
the ligands in the environment, then transmit the signal into the insideof the system using the chemical reactions. The system distinguishes
whether the ligand is the target ligand or not by converting the unbinding
rate k−1 into the amount of the product molecules [RP] through the zeroorder proofreading model.

2.2. Zero-order Proofreading Model
If the target and non-target ligands are similar molecules, the difference in the unbinding rates among k−1,T(s−1) and k−1,N(s−1) is small.
For reliable chemical communications, the system must amplify thesmall difference of the unbinding rate k−1. To this end, the following phosphorylation and dephosphorylation cycle was proposed in
Kajita et al. [7] as an intracellular chemical reaction network that
can amplify the small difference in k−1 (see also Figure 2):(2)

After the formation of the ligand–receptor complex C0, the receptor can be converted into the intermediate state of the complex
denoted by C1 with rate w(s−1) via an irreversible reaction. The
product RP can be generated from C1 with rate w. However, someof C1 can be converted back to R + L due to the ligand dissociation from the receptor with rate k−1. Note that we assume that theIf the total concentration of receptor [R]total:= [R] + [RP] + [C0] + [C1]+ [D] is much larger than those of ligand and enzyme, [L]total:= [L] +[C0] + [C1] and [P]total:= [P] + [D], [RP]total ≈ [R]total ≈ [R] + [RP] holds,
where [X] denotes the concentration of molecule X, whose unit is[M]. At the quasi steady state, that is, d[C1]/dt = d[C0]/dt = d[P]/dt= 0, the dynamics of [RP] can be described by the following ODE:[R]total - [R P ]d[R P ][R P ]
= V1
,
- V2
dt
K m,1 + [R]total - [R P ]K 2 + [R P ](3)

where V1:= [L]totalwa/(1 + a), V2:= k3[P]total, Km,1:= K1/(1 + a),
K1:= (k−1 + w)/k1, K2:= (k−2 + k3)/k2, and a:= w/(k−1 + w).
The production speed of [RP] can be decomposed into the positive
and negative fluxes, which are denoted as J1 and J2, respectively. The
definitions are
J1 := V1

[R]total - [R P ],
K m,1 + [R]total - [R P ](4)J 2 := V2[R P ].K 2 + [R P ](5)Thus, the net flux is given by J:= d[RP]/dt = J1 − J2. These Equations(4) and (5) are called Michaelis–Menten equations [11].
The parameters Km,1 and K2 are the effective Michaelis–Menten
constants of the positive and negative fluxes, respectively. Note that
Km,1 and K2 respectively control the ligand and enzyme saturationlevels. At the unsaturated condition, when Km,1 >> [RP]total and K2 >>[RP]total, Equations (4) and (5) become
J1 »V1([R]total - [R P ]), 
K m,1

(6)

M.K. Kajita / Journal of Robotics, Networking and Artificial Life 7(1) 52–57

54

J2 »

V2
[R P ]. 
K2(7)These fluxes are approximately the first-order reactions withrespect to [RP], which are demonstrated in Figure 3a. On theother hand, at the saturated condition, when Km,1 << [RP]total andK2 << [RP]total, Equations (4) and (5) becomeJ1 » V1 ,

(8)

J 2 » V2 .



(9)These approximated equations do not depend on [RP], thus these
are called zero-order reactions with respect to [RP], which are also
demonstrated in Figure 3b.

2.4. An Intuitive Understanding of the
Zero-order Proofreading Mechanism
Next, we clarify the mechanism of how the saturation level
affects the response of the zero-order proofreading model to theunbinding rate k−1. At the steady state, J1 = J2 holds. By solving
J1 = J2 for [RP], we can obtain the steady state concentration of[RP], denoted as [RP]*. Because J1 depends on the unbinding ratek−1, [RP]* also depends on k−1. This means that [RP]* changes
depending on whether the ligand is the target or not. To investigate how the saturation level changes the difference in the steadystate concentrations [RP]* for the target and non-target ligands,
we analyze the dependence of the fluxes, Equations (4) and (5),on the unbinding rate k−1. When the system is in unsaturated
condition (Figure 3a), the small change in k−1 slightly affects J1.
Because of the small effect, the steady-state concentrations [RP]*for the target and non-target ligands do not differ largely. On theother hand, when the system is saturated (Figure 3b), even though
the small difference in k−1 slightly changes the positive flux,aJ1 ≈ V1 = wa [L]total/(1 + a) = w2/(k−1 + 2w), due to the zero-order
reactions, the steady state concentration [RP]* largely moves. Thisis an intuitive understanding of how the zero-order proofreading
model [Equation (2)] [7] can amplify the small difference of theunbinding rate k−1 between the target and non-target ligands bychanging the saturation level.
Note that the highly nonlinear response due to the cycle reaction
composed of the two zero-order reactions is called zero-orderultra-sensitivity [11,12]. Although the original model proposed byGoldbeter and Koshland [12] has the ultra-sensitivity to concentration variables ([L]total and [P]total in the zero-order proofreading
model), the model cannot amplify the unbinding rate k−1. Our representation in Figure 3 clarifies how the zero-order proofreading
model obtains the ultra-specificity to the target ligand by the nonlinearity to the unbinding rate k−1 based on the mechanism of thezero-order reaction.

3. THE CASE OF TWO LIGANDS
IN THE ENVIRONMENT
Let us consider a more realistic condition for chemical communication systems, that is, the system is surrounded by more than
one type of ligand. As the simplest case, we consider that there are
two types of ligands, the target and non-target ligands, exist in theenvironment simultaneously (Figure 4). In this situation, we investigate how the zero-order proofreading mechanism contributes tothe reliable detection of the target ligand.
When there are two ligands, the zero-order proofreading model
can be extended as follows (see also Figure 5):

(10)

b

Figure 3 | Fluxes of the activation and deactivation reaction cycle of thezero-order proofreading model [Equation (2)] when the system is in an
unsaturated condition (a) and a saturated condition (b). The positive flux J1[Equation (4)] with the target ligand (red lines), with the non-target ligand(red dashed lines), and the negative flux J2 [Equation (5)] (blue lines)
are plotted as functions of [RP]. The steady states with the target ligand areplotted as red circles, and the steady states with the non-target one
are plotted as blue rhombuses in the figures. Here, K denotes the effective
Michaelis–Menten constants, which controls the unsaturation level of J1
and J2, respectively, where K = Km,1 = K2. The displayed values of K are set
for the target ligand, and k−1 and k2 are obtained from the values of K. The
other parameters are k−1 = 1 for the target ligand, k−1 = 2 for the non-target
one, w = 1, k−2 = 10, k3 = 1, [R]total = 100, [L]total = 3.5, and [P]total = 1.

where [LT] and [LN] are the target and non-target ligands, respectively. Each ligand has intermediate states, [C0,*] and [C1,*], when it
forms a complex with the receptor R, where * ∈{T, N}. Here, k−1,Tand k−1,N denote the unbinding rate of the target ligand and that of
the non-target one, respectively.

Figure 4 | Schematics of a chemical communication system when thetarget and non-target ligands exist in the environment simultaneously.

M.K. Kajita / Journal of Robotics, Networking and Artificial Life 7(1) 52–57

h :=Figure 5 | Schematics of the zero-order proofreading model when thetarget and non-target ligands exist simultaneously in the environment.

55

FP
.
TP + FPFor a reliable chemical communication, the balance between theaccuracy and the intensity of output is necessary. For example, even
if the error is low, but the output intensity from the sensory system
is quite small, there is a risk that the output signal may disappear
before the subsequent reactions are induced inside of the communication system. To evaluate the intensity of output, we introduce
the output intensity O, which is defined byO :=3.1. Model Reduction based on
Michaelis–Menten Approximation
As we did the model reduction of the zero-order proofreading forthe single type of ligand, we apply the Michaelis–Menten approximation to the zero-order proofreading model for two types ofligands [Equation (10)].
Assume that the dynamics of Equation (10) follows the law of massaction, then the dynamics can be described by ODEs. If we also
assume that the total concentration of receptor [R]total:= [R] + [RP] +∑i,*[Ci,*] + [D] is much larger than those of the target ligand, [LT]total:=[LT] + ∑i,*[Ci,T], non-target ligand, [LT]total:= [LT] + ∑i,*[Ci,T], [LN]total:=[LN] + ∑i,*[Ci,N], and enzyme, [P]total:= [P] + [D], then [R]total ≈ [R] +[RP] holds. At the quasi-steady-state, that is, d[C1,*]/dt = d[C0,*]/dt= d[P]/dt = 0 for * ∈{T, N}, the dynamics of RP can be described by
the following ODE with respect to [RP]:[R]total - [R P ]d[R P ]
= V1,T
K m,T + [R]total - [R P ]dt+ V1, N[R]total - [R P ][R P ],- V2
K m, N + [R]total - [R P ]K 2 + [R P ]



(11)

where V1,T:= [LT]totalaT/(1 + aT), V1,N:= [LN]totalwaN/(1 + aN), Km,T:=
K1,T/(1 + aT), Km,N:= K1,N/(1 + aN), K1,T:= K1,N/(1 + aT), K1,N:= K1,N/(1 + aN), K2:= (k−2 + k3)/k2, aT:= w/(k−1,T + w), and aN:= w/(k−1,N + w).

3.2. The Reliability of the Zero-order
Proofreading Model(14)[R P ]*,[R]total(15)

where [RP]* is the steady-state concentration of [RP] obtained from
Equation (11) at the steady-state condition d[RP]/dt = 0. Hereafter,by evaluating the error η and the output intensity O, we investigate
the performance of the zero-order proofreading model.
To analyze the effect of the absence of non-target ligands, we introduce the fraction of the target ligand [13],r :=[L T ]total,[L T ]total + [L N ]total(16)

and evaluate the dependency of the error η and the output intensity
O on r (Figure 6). Although the error η in the saturated condition(Figure 6a) is slightly higher than those of the unsaturated conditions (Figure 6b), there is a common tendency that the error η
decreases with the increase of the target ligand fraction r. On theother hand, the output intensity O shows a qualitatively different
dependency on r. The output intensity O in the saturated conditionshows all-or-none response to r (Figure 6a). However, the response
becomes gradual with a decrease in the saturation level (Figure 6b).
Consequently, when the target ligand fraction is high (r > 1/2),
a

b

c

d

To evaluate the performance of the target ligand detection, we
introduce the following quantities. First, we introduce the True
Positive (TP) and False Positive (FP) by using the partial flux toproduce the product molecule RP by the target ligand and that by
the non-target one, which are defined respectively by
TP := V1,T
FP := V1, N[R]total - [R P ],
K m,T + [R]total - [R P ](12)[R]total - [R P ].K m, N + [R]total - [R P ](13)

By using TP and FP, we introduce the error η to evaluate thereliability of the target detection, which is defined as

Figure 6 | The error η (red dots) and the output intensity O (blue squares)
are plotted as functions of the target ligand fraction r (a and b) and theunbinding rate ratio ∆ (c and d) for different values of K, where K = Km,T = K2.
The parameters are k−1,T = 1, ∆ = 1/10 (a and b), r = 1/2 (c and d), w = 1,
k−2 = 10, k3 = 1, [R]total = 100, and [L]total = 5. Note that k1 and k2 are obtained
from the value of K, [L*]total for * ∈{T, N} is obtained from r (a and b), and k−1,N
is obtained from ∆ (c and d).

M.K. Kajita / Journal of Robotics, Networking and Artificial Life 7(1) 52–57

56

the saturated system (K = 1) can balance the accuracy and output
intensity (Figure 6a).
To investigate the effect of similarity between the target andnon-target ligands, we introduce the unbinding constant ratio
D :=

k1,T

k1, N

.

(17)

As same as in Figure 6a and b, the error of saturated condition(Figure 6c) is always slightly higher than those of unsaturated condition (Figure 6d). However, these two cases have the same tendency
that the error decreases when the difference between the target andnon-target ligands Δ becomes large. In contrast, the response of theoutput intensity O to Δ again shows a qualitatively different dependency on the unsaturation level K. When the system is saturated(Figure 6c), the response of O to Δ shows an all-or-none response. In
addition, in this saturated condition, the system can generate large
output O even when Δ is small, which means that the system cangenerate a large output without the help of the non-target ligand.
With an increase in the unsaturation level (Figure 6d), the response
becomes more gradual, and the system cannot generate large output
O in the range where the error η is small (for e.g. Δ ∈ [0.0, 0.3]). As
the results, at the saturated condition (Figure 6c), the system canbalance the high accuracy and large output O around Δ ∈ [0.1, 0.2].
We also investigate the dependencies of the error η and output
intensity O on both the target ligand fraction r and the unbinding constant ratio Δ for various saturation levels K (Figure 7), andconfirm that the qualitatively same results in Figure 6 hold for thevarious values of K. Together with the results in Figures 6 and 7,
the zero-order proofreading mechanism with saturating condition

can balance the high accuracy and large output O at the slight costof the error η.

4. SUMMARY AND DISCUSSION
In this work, we analyzed the zero-order proofreading model [7]and obtain an intuitive understanding of how the mechanism
amplifies the unbinding rate from the viewpoint of the balance ofthe positive and negative fluxes in a saturated condition. We further
demonstrated that the zero-order proofreading model is valid forthe case that there are the target and non-target ligands in the environment simultaneously, by revealing that the model with saturated
conditions can balance both the accuracy and output intensity. The
zero-order proofreading mechanism may not only give an insight
into how biological cells perform reliable chemical communications in the complex mixtures of ligands, but also provide a method
to design bio-inspired reliable chemical communication systems.

CONFLICTS OF INTEREST
The author declares no conflicts of interest.

ACKNOWLEDGMENTS
I would like to thank Tetsuya J. Kobayashi and Kazuyuki Aihara
for fruitful discussion. This work was supported by JSPS KAKENHI
Grant Numbers 15H05707 and 18K18147.

REFERENCES
a

b

c

d

Figure 7 | The error η (a and c) and the output intensity O (b and d) areplotted as functions of the target ligand fraction r and the unsaturationlevel K (a and b), and the unbinding rate ratio ∆ and the unsaturationlevel K (c and d), where K = Km,T = K2. The parameters are k−1,T = 1, w = 1,
k−2 = 10, k3 = 1, [R]total = 100, and [L]total = 5. The others are ∆ = 1/10 for (aand b), and r = 1/2 for (c and d). Note that k1, k2, [L*]total for * ∈{T, N}(a and b), and k−1,N (c and d) are obtained in the same manner as in Figure 6.

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AUTHOR INTRODUCTION
Dr. Masashi K. Kajita
He is an Assistant Professor in the
Department of Applied Chemistry andBiotechnology at University of Fukui. He
received his PhD degree from the Universityof Tokyo in 2017. From October 2017
to February 2020, he had worked in the
University of Tokyo as a Research Associate.
Since March 2020, he has been in University of Fukui. His current research interests are biochemical information processing,and data-driven modeling of biological systems.

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