I Introduction
Molecular communication (MC) uses molecules as carriers of information. It is a prevalent mechanism for communication among living organisms in nature. Molecular communication is envisioned to be applicable in a wide range of engineering and medical applications, especially as a means to realize communications among engineered biological nanomachines (see [pierobon2010physical, nakano2012molecular, guo2015molecular, survey:medicine]). In diffusion based MC, molecules are released into the medium by molecular transmitters. Information is coded by the transmitter in the type, number or the release time of the molecules. The released molecules randomly diffuse in the medium, with some reaching the molecular receivers. The receiver decodes the message sent by the transmitter based on the number of sensed molecules. See for instance [kuran2011modulation, arjmandi2013diffusion, srinivas2012molecular] for some modulation schemes proposed in the literature.
Broadly speaking, two main types of diffusionbased MC exists: microscale MC and macroscale MC. In microscale MC, a small number of molecules are released into the medium by transmitters. The movement of these molecules are random and follow Brownian motion. On the other hand, in macroscale MC, a large number of molecules are released into the medium (the number of released molecules is in the order of moles). Macroscale MC is of interest as an alternative to electromagnetic wave based systems in certain media [guo2015molecular, farsad2016comprehensive]
. In macroscale MC, instead of focusing on the location of individual molecules, by the law of large numbers, the
average concentration of molecules is described by the Fick’s law of diffusion, which is a deterministic differential equation. Everything is described by deterministic equations until this point. However, various sources of noise or imperfection could be considered for molecular transmitters and receivers [farsad2016comprehensive, gohari2016information]. In particular, the literature on MC usually associates a measurement noise to the receivers. The variance of this measurement noise is considered to depend on the concentration level of molecules measured by the receiver
[einolghozati2012collective]. In particular, a commonly used measurement noise is the socalled “Poisson particle counting noise” where the measurement is assumed to follow a Poisson distribution whose mean is proportional to the concentration of molecules at the time of measurement.
One of the unique features of MC with no parallel in classical wireless communication is chemical reactions: different types of molecules can react with each other and form new types of molecules in the medium. This feature of MC poses a main challenge in macroscale MC since equations describing chemical reaction with diffusing particles are nonlinearpartial differential equations with no closedform solutions[noclosedform:1, noclosedform:2, noclosedform:3]. In other words, there is no analytical closedform formula that describes the variations in the concentration of molecules over time and space for given boundary conditions, inputs to the medium and system parameters such as the diffusion and reaction rate constants. As these equations have no closed form solutions, they are commonly solved by numerical methods. While numerical techniques (such as the finite difference method and the finite element method) can provide accurate solutions for a given input and system parameters, they require extensive computational resources. More importantly, these solutions are not generalizable: assume that we have solved the reactiondiffusion equations for one particular release pattern of molecules by the transmitters into the medium. Next, if we make some changes to the release pattern of molecules, we should start all over again and resolve the reactiondiffusion equations again for the new release pattern. In other words, numerical methods provide little insight into how the solution would change if the input or parameters of the medium change. This has a major drawback for the design of coding and modulation schemes, where we wish to optimize and find the best possible release pattern. In other words, a tractable (even if approximate) model for the solution of the reactiondiffusion systems is required to design codes or modulation schemes. Only with such a model can one formally prove optimality of certain waveforms for a given communication problem.
In this paper, we apply the perturbation theory to MC in order to study and design modulation schemes for reactiondiffusion based mediums. Perturbation theory is a set of mathematical methods for finding approximate solutions. It was originally proposed to calculate motions of planets in the solar system, which is an intractable problem (as the socalled threebody problem shows). The theory has been broadly applied in different scientific areas such as physics and mechanics for solving algebraic equations with no analytical solutions or intractable differential equations. While the perturbation theory has been used in the study of differential equations in general, to the best of our knowledge it has not been hitherto utilized in molecular communication or in the context of the particular chemical reactions that arise in it. Just as nonlinear functions can be locally approximated by the first terms of their Taylor series, perturbation theory shows that nonlinear reactiondiffusion equations can be also approximated by their lower order terms. Our main contribution is a proposal to design coding and modulation schemes based on the lower order terms, which admit closed form solutions. In other words, we provide an analytically tractable (approximate) model for studying chemical reaction in molecular communication. The accuracy of the model can be increasingly improved by including more terms from the Taylor series.
To illustrate our technique, we consider a number of examples and apply our technique to these problems. These examples are aimed to be illustrative and with distinctive focus in order to cover different scenarios where chemical reaction is either facilitating or impeding effective communication. In particular, we consider three examples. In each case, we find the optimal waveforms (that minimize the error probability) in the low rate reaction regimes. In all of the examples we consider, we illustrate that in the low reaction rate regime, transmitters should release molecules in a bursty fashion at a finite number of time instances. In other words, it is not necessary to use continuous waveforms to transmit molecules over the time. The three examples are as follows:

In Example I we consider a communication system with two transmitters and one receiver. Each transmitter sends a type of molecule (reactant) and the receiver measures the concentration of products in the reaction. The goal is to compute the best transmission waveform for each transmitter so that the receiver has minimum error probability to decode the messages of both transmitters.

In Example II, we consider a communication system with one transmitter and one receiver. Transmitter can send molecules of types and , while the receiver can only measure molecules of type . The medium has molecules of a different type that can react with both and . The transmitter releases molecules of type in order to “clean up” the medium of molecules of type so that molecules of type can reach the receiver without being dissolved through reaction with as they travel from the transmitter to the receiver.

In Example III, we consider a twoway communication system with two transceiver nodes. In other words, each node aims to both send and receiver information from the other node. Transmitters use different molecule types and these molecules can react with each other. The chemical reaction in this scenario is destructive because it reduces the concentration of molecules at the two transceivers.
Related works:
The math literature on nonlinear partial differential equations (PDEs) studies different aspects of these equations, such as existence and uniqueness of solution, semianalytical solutions, numerical solutions, etc. Since most PDEs do not have closedform solutions, showing existence and in some cases uniqueness of solution are important topics in the theory of PDEs. Unlike ordinary differential equations (ODE) which have a straightforward theorem for existence and uniqueness of solution, there is no general method for PDEs to show the existence and uniqueness of solution. For instance, the existence and smoothness of solutions for the NavierStokes equations that describe the motion of a fluid in space is one of the fundamental open problems in physics. Calculus of variation, monotonicity methods, fixedpoint theorems
[evans2010partial], etc are some known tools for proving existence and uniqueness for some classes of PDE. Semianalytical techniques are series expansion methods in which the solution is expressed as an infinite series of explicit functions. Semianalytical techniques include Adomian decomposition method, Lyapunov artificial small parameter method, homotopy perturbation method and perturbation methods [liao2003beyond, liao2012homotopy].There are many numerical methods for solving PDEs which are classified in terms of complexity and stability of the solution. Finite elements method, finite difference method, spectral finite element method, meshfree finite element method, discontinuous Galerkin finite element method are some examples of numerical techniques for solving PDEs
[numericalmethode:1, noclosedform:2]. For example, in the finite difference method, we partition the domain using a mesh and approximate derivatives with finite differences computed over the mesh. One of the challenges of using this method is determining the appropriate mesh size to guarantee the stability of the solution.^{1}^{1}1 Our own experiment with this method indicates that choosing the appropriate mesh size is particularly challenging when we have a reactiondiffusion system involving molecules that have very different diffusion constants (especially when one diffusion constant is more than ten times another diffusion constant).There are some prior work in the literature on chemical physics that utilize the perturbation theory. For instance, average survival time of diffusion influenced chemical reactions is studied in [pagitsas1992perturbation:e:1:1]. In [kryvohuz2014nonlinear:e:1:2] some classes of chemical kinetics (describe by ODEs) have been solved by the perturbation theory. However, the setup and approach in these problems differ from the one encountered in MC.
Due to their evident importance to MC, chemical reactions have been subject of various studies. A molecular communication system involves transmitters, receivers and the molecular media, and chemical reactions relates to each of these individual building blocks. Transmitters could use chemical reactions to produce signaling molecules [bi2019chemical:j:9]. This is a chemical reaction inside the transmitter unit. On the receiver side, one might have ligand or other types of receptors on the surface of the receiver which react with incoming molecules. These receptors have been the subject of various studies, e.g., see [Ligand1, Ligand0, Ligand1, Ligand2, Ligand3, kuscu2018modeling:e:2:3:2, ahmadzadeh2016comprehensive:e:2:3:1, chou2015impact:e:2:2:3, chou2014molecular:e:2:2:4]. Finally, chemical reactions have also been considered in the communication medium. It is this aspect of chemical reactions that is of interest in this work. It is pointed out that chemical reactions could be used to suppress signaling molecules and thereby reduce reduce intersymbol interference (ISI) and the signaldependent measurement noise [noel2014improving:j:11, cho2017effective:j:12], or to amplify the signal [nakano2011repeater:F:22]. Complicated propagation patterns can be obtained by using chemical reactions[nakano2015molecular:F:19], and negative molecular signals could be implemented using chemical reactions [farsad2016molecular:F:16, wang2014transmit:F:20, mosayebi2017type:F:21]. Notably, chemical reactions are shown to be beneficial for coding and modulation design [farsad2016molecular:F:16, farahnak2018medium:j:16, jamali2018diffusive:j:17, nakano2015molecular:F:19, wang2014transmit:F:20, mosayebi2017type:F:21].^{2}^{2}2The three examples given in this paper differ from the previously reported applications of chemical reaction in modulation design. Since solving the reactiondiffusion equations is intractable, these works either use numerical simulations to back up the presented ideas or else simplify the reactiondiffusion process via idealized assumptions about chemical reactions. Authors in [cao2019chemical:j:ro]
provide an iterative numerical algorithm for solving reaction diffusion equations. In a recent work, a neural network is used as the decoder in a medium with chemical reactions
[farsad2017novel:e:2:2:2].This paper is organized as follows: in Section II, the perturbation method is introduced. In Section III, we consider three scenarios of MC (which utilize chemical reactions) and solve the reactiondiffusion equations for them. In Section LABEL:sec:Val, we validate our model through numerical simulation. In Section LABEL:Sec:Mod, we design modulation schemes for the settings of Section III. Finally concluding remarks and future work are given in Section LABEL:Conclusion_and_Future_Work.
Notations and Remarks: The Laplacian operator is shown by . For functions and , we define index convolution as follows:
(1) 
For two functions and , convolution in both time and space is denoted by and defined as follows:
(2) 
Ii Solving ReactionDiffusion Equations via the Perturbation Method
The perturbation method provides an explicit analytical solution in the form of an infinite series of functions. One can approximate the solution by taking a finite number of terms from this series. In this paper we use the perturbation method to obtain an analytical model for the reactiondiffusion equations which can be used for molecular communication.
For simplicity, assume that and are the density of molecules of type and at location and time . The evolution of and are governed by second order reactiondiffusion differential equations. These equations express temporal variation of and , i.e., and in terms of spatial variations of and (usually the densities and their second order partial derivatives and ). These equations also involve diffusion constants, reaction rates, and external inputs and to the medium. The system of partial equations is nonlinear due to the chemical reaction between molecules. Letting denote the reaction rate (with corresponding to the noreaction case), we can express the solutions and as
(3) 
for some functions and . This can be understood as the “Taylor expansion” of and in terms of . Perturbation theory works by showing that and are analytical functions in reaction rate and the expansion in (3) is valid for for some radius .
The functions and correspond to the noreaction case and can be found directly (the equations for and disentangle from each other in case of no chemical reaction). As we will demonstrate, the functions and can be found recursively from , , …, and , , …, by replacing the form of the solution given in (3) into the differential equations and matching the powers of on both sides of the equation.
Iii Modelling of ReactionDiffusion Media
We illustrate our technique via three specific examples.
Iiia Example 1: Reaction for Production of Signalling Molecules
Consider two molecular transmitters and which release molecules and respectively, and a molecular receiver which receives molecules of type . The transmitters and receiver are placed on a onedimensional line, with , and being located at , and respectively. We assume that the transmitters and receivers are small in size and view them as points on the real line; the transmitter and receiver are considered to be transparent, not impeding the diffusion of the released molecules. We also do not assume any boundary condition for the medium and allow molecules to diffuse over the entire real line.
Molecules of type and react with each other and produce molecules of type as follows:
(4) 
The receiver measures concentration of molecules at its location in order to detect the messages of the two transmitters. This is depicted in Figure 1.
The following equation describes the system dynamic:
(5)  
(6)  
(7) 
where are the input signals of the transmitters, and and are the diffusion constants and the rate of reaction respectively. The initial conditions are set to zero.
For the special case of , i.e., when there is no reaction, the above system of equations is linear and tractable.
Consider a solution of the form given in (3): assume that and for some Taylor series coefficients and . By substituting these expressions in (5), we obtain
(8) 
and
(9) 
Equations (8) and (9) could be viewed as power series in for fixed values of and . Matching the coefficients of on both sides of the equation, we obtain the following:
(10) 
For we obtain
(11) 
where we used the index convolution defined in (1).
The functions and could be find recursively by first computing and from (10), and then using (11) to compute and from and for .
In order to solve (10), let be the impulse response of the heat equation with diffusion coefficient , i.e.,
(12) 
We have
(13) 
Similarly, we define and for diffusion coefficients and respectively. Then, the solutions of (10), (11) are as follows:
(14) 
and for ,
(15) 
where stands for the two dimensional convolution (see (2)). From equation (7), the density of molecule is equal to
(16) 
To sum this up, we take a solution of the form and . By substituting these expressions in the reactiondiffusion differential equation (i.e., equations (8) and (9)) and matching the powers of , we obtained equations (10) and (11). The functions and could be found recursively. As a result the functions and exist. However, one still needs to show that the power series and are convergent to functions that satisfy the original reactiondiffusion differential equation. Perturbation theory does not provide a general recipe for showing this convergence, and it should be done in a case by case basis. We show the convergence for Example I in details. Similar ideas could be used for other examples.
To show that the power series and are convergent to functions that satisfy the original reactiondiffusion differential equation, we consider time for some fixed and prove that
(17) 
uniformly converge. Similarly we prove that the corresponding power series for molecules of type also converge. Uniform convergence of the above series over all are proven in Appendix LABEL:AppA0.
Using the above model, we design a modulation scheme in Section LABEL:Sec:Mod.
IiiB Example 2: Reaction for Channel Amplification
While the previous example assumes diffusion in a onedimensional medium, we consider a twodimensional medium in this section. Moreover, the previous section used reaction as a means to produce molecules that are detected by the receiver. However, reaction may be used for other purposes as well. For instance, it may be used to enhance the channel between the transmitter and the receiver. This concept is considered in the example below.
Consider the following example with one transmitter and one receiver. The receiver can only measure the density of molecules of type at its location. The transmitter is also able to release molecules of types and into the medium. Assume that there is an enzyme in the medium (outside of our control) which reacts with molecules of type . If the level of enzyme is high, molecules of type are mostly dissolved before reaching the receiver. To overcome this, the transmitter may release molecules of a different type , which would also react with the enzyme and thereby reduce the concentration of in the medium. This “cleaning” of the medium from molecules of type would enhance the channel from the transmitter to the receiver. More specifically, assume that the medium is governed by the following chemical reactions:
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