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We consider the motion of a spot under the influence of chemotaxis. We propose a two-component reaction diffusion system with a global coupling term and a Keller-Segel type chemotaxis term. For the system, we derive the equation of motion of the spot and the time evolution equation of the tensors. We show the existence of an upper limit for the velocity and a critical intensity for the chemotaxis, over which there is no circular motion. The chemotaxis suppresses the range of velocity for the circular motion. This braking effect on velocity originates from the refractory period behind the rear interface of the spot and the negative chemotactic velocity. The physical interpretation of the results and its plausibility are discussed.

The behaviors of artificial and biological microswimmers such as oil droplets, bimetallic nanorods, catalytic Janus colloids, liposomes, flagellated bacteria, and

In a two-dimensional reaction diffusion (RD) system, the droplet is often referred to as a spot solution. In order to systematically describe the motion of spots in an RD system, the time evolution equation of the spot was derived and the mechanism of elastic collision of moving spots was clarified in a previous study [

In addition to the Marangoni effect, which plays an important role in the motion of oil droplets, chemotaxis is an important property of cell migration; it is important in mass transfer and immunological response in biology. In inflammatory response, the neutrophils among blood cells have a remarkable migration potency (chemotaxis) and can change their form by generating pseudopods toward the antigen. In biophylaxis, several chemokines (chemoattractants) are released from the macrophages and mast cells. Then other immunocompetent cells (neutrophils) respond to the gradient of the chemoattractant. Consequently, the immunocompetent cells move unidirectionally to the source point of the antigen [

The mathematical model for chemotaxis was first proposed by Keller and Segel [

As described above, with the recent increase in importance of chemotaxis in biology, medicine, and cytoengineering [

We first consider the following three-component RD system with an activator

When the second term on the right hand side of (

In the presence of the second term on the right hand side of (

The time evolution equation for

In (

In the absence of chemotaxis, (

Boundary layer and interface. The solid and dotted curves correspond to

Although the bifurcation from a motionless spot to a straight moving spot and the collision of two moving spots were studied in [

In this study, we assume that the localized domain (spot) of an activator exists under global feedback in the system described by (

As

We first consider the velocity of the interface in one dimension in (

In two dimensions, the velocity of a flat interface directed along

When the motion of the interface of the spot is slow compared with the relaxation rate of the chemotactic substance, the left hand side of (

In this section, we derive the equation of motion of the spot. In order to describe the deformations of the spot, tensors are introduced. The tensors depend on time, and the time evolution equation of the tensors are derived following [

We firstly describe the position and velocity of a deformed spot. The motion of the spot consists of two components: the motion of the center of gravity and the motion of the interface relative to the center of gravity. The center of gravity is denoted by

For a general function

We derive the equation of motion of a spot from (

In the above expansions, we omitted the term proportional to

When the spot is in a stationary motionless state, we assume that the spot is a circle with radius

Dependence of

When the spot moves with an infinitesimal velocity,

In order to obtain

In the above process, we iteratively derived

Using (

By carrying the integral over

After practical calculations of

Equation (

Dependence of

In the derivation of (

In order to express the deformation of the spot, we introduce tensors. We rewrite the vector form (

We first introduce a second-rank tensor

Next, in order to describe the head-tail asymmetric deformation, we first define

The terms

In the previous subsection, we defined the tensors and described the equation of motion of the spot by using tensors, including the time-dependent tensor

In order to derive the equation of motion of tensors, we calculate the first-order time derivative of

We first derive the equation of motion of

Next, we consider the time evolution equation of

Using (

We summarize the coefficients in (

The dependence of

Dependence of

Henceforth, we drop the primes on the parameters in the full tensor model (

In the following subsections, we consider the stationary solution and phase diagrams. For this, we rewrite

Using (

For the reduced and full tensor models, we consider the stationary solution of the moving spot, which moves straight in the

When

For the stationary states

Phase diagram of spot motion.

In the previous subsection, we showed that a spot appears in the circular motion when

We first consider the case when

Dependence of the critical velocity

Next, we consider the case when

In order to verify the above theoretical results, the simulation results for the reduced tensor model are shown in each figure. The results obtained via simulation agree well with the theoretical results. In Figures

The dependence of the stationary velocity and radius of the circular motion on

Dependence of velocity and radius on

From the above analyses on (

In this subsection, we examine the effect of

The phase diagrams of the spot motion are shown in Figure

Phase diagram of spot motion. Data are obtained by the simulation of the full tensor model.

The phase diagram of spot motion in the

Phase diagram of spot motion in the

The effects of

Dependence of velocity and radius on

The dependence of the critical velocity

Dependence of the critical velocity

In this section, we discuss the physical origins of the braking effect observed in the previous section. For our RD system, we derived the time evolution equation of

In the absence of chemotaxis, that is,

In the presence of chemotaxis, that is,

In this study, we considered the motion of a spot solution in two dimensions under the influence of chemotaxis. Starting from a three-component RD system, we proposed a two-component (an activator and a chemotactic substance) RD system with a global coupling term. We remark that, in our model system, the spot secretes the chemotactic substance from the inside and the motion of the spot is influenced by the chemotaxis. Thus, the model is an autonomous system. The chemotaxis term is of the Keller-Segel type, and the chemotactic velocity is opposite to the traveling direction. The reason for the opposite direction is that the system involves autosecretion, and the gradient of the chemotactic substance at the leading interface is negative. Although there have been several studies on the motion of spots under the influence of chemotaxis, the chemoattractant was supplied from the outside [

For the RD system, by employing the method proposed in [

In practical experiments, the candidates of autonomous systems including chemotaxis are

Here, we derive the velocity of the interface in one dimension, given by (

The stationary solution of (

We show that in the limit

We first derive

Next, we derive

Here, we give the detailed derivation of (

Substituting

Next, we derive

The expansions of

We give the coefficients in (

The author declares that there are no conflicts of interest.