Chemophoresis as a driving force for intracellular organization: Theory and application to plasmid partitioning.

Biological units such as macromolecules, organelles, and cells are directed to a proper location by gradients of chemicals. We consider a macroscopic element with surface binding sites where chemical adsorption reactions can occur and show that a thermodynamic force generated by chemical gradients acts on the element. By assuming local equilibrium and adopting the grand potential used in thermodynamics, we derive a formula for the "chemophoresis" force, which depends on chemical potential gradients and the Langmuir isotherm. The conditions under which the formula is applicable are shown to occur in intracellular reactions. Further, the role of the chemophoresis in the partitioning of bacterial chromosomal loci/plasmids during cell division is discussed. By performing numerical simulations, we demonstrate that the chemophoresis force can contribute to the regular positioning of plasmids observed in experiments.

1. Introduction

Biological entities such as cells, organelles, and macromolecules often move to appropriate locations under an external chemical gradient. In cell chemotaxis, the process by which an organism senses the presence of an external chemical and responds to it has been elucidated in depth. However, the coordinated motion of entities is not limited to those at the organism level. In recent studies, it has been shown that such directional motion under chemical gradients/localizations plays an important role in organization at an intracellular level1, e.g., microtubule guidance under a RanGTP gradient2–4 or Stathmin gradient5, actin nucleation under an IcsA gradient on the outer membrane of a pathogen Shigella flexneri (actin comet)6,7, bacterial chromosomal locus/plasmid DNA partitioning by a Par system8–24. However, the general mechanism underlying the coordinated motion of organelles or macromolecules under an intracellular gradient/localization is yet to be understood.

The formation of organelle/macromolecule patterns by chemical concentrations under non-equilibrium conditions, first observed during macroscopic morphogenesis25, has recently been observed at the intracellular level as well, and its relevance to intracellular organization processes has been demonstrated in the case of bacterial plasmid DNA partitioning by a Par system11–24, determination of the plane of cell division in bacteria using the Min system26–36, etc. These studies have discussed how positional information given by the chemical concentration gradient/localization is generated and maintained. However, there are few studies on the role of chemical gradients/localization in the coordinated motion, transport, and positioning of organelles or macromolecules under non-equilibrium conditions.

In the present paper, we present a physical mechanism that can explain the coordinated motion and positioning. According to the mechanism, in the presence of a chemical gradient resulting from a reaction, a macroscopic element consisting of a number of reaction sites is generally subjected to a force. By considering this element to be a scaffold that adsorbs chemicals, we derive a formula for the force generated by a chemical potential gradient. In the derivation, we introduce the grand potential used in thermodynamics for an open system. The direction of motion of the element is such that the chemical potential increases. A formula is obtained by assuming that the reaction process reaches equilibrium faster than the motion of the element and by extending the minimization of free energy to include the contact with a particle bath with a given chemical gradient. We propose that the force leads to a general mechanochemical coupling; we call this force chemophoresis force. We show that the work done by the chemophoresis force can be much greater than the thermal energy fluctuations. By examining whether this statement holds true in the case of intracellular reaction processes, we discuss the possible role of this force in the partitioning dynamics of bacterial chromosomal locus/plasmids during cell division. By introducing a dynamical system in which equations of motion for plasmids and a reaction-diffusion equation for concentration of a chemical mutually determine each other, we demonstrate that the chemophoresis force can contribute to a regular positioning of plasmids observed in experiments.

The scheme of this paper is as follows. In Sec. 2 for results, we derive an expression for the chemophoresis force in Sec. 2-1, and in Sec. 2-2, we consider a simple toy model of chemophoresis. In Sec. 2-3, we show that conditions for chemophoresis are satisfied in the intracellular reaction process. Sec. 2-4 discusses applications of chemophoresis force in the dynamics of bacterial chromosomal locus/plasmid partitioning during cell division. First, we perform an order estimation to examine the validity of the application of chemophoresis to bacterial chromosomal locus/plasmid partitioning. Next, we demonstrate that the chemophoresis force can contribute to the regular positioning of plasmids, which has been observed in experiments. In Sec. 3, a brief discussion on the relevance of chemophoresis to intracellular organization is presented. Discussions of the chemophoresis on the basis of thermodynamics and statistical mechanics are given in Appendix A and B, respectively.

2. Results

2-1. Chemophoresis

Consider organelles or macromolecules that have a number of binding sites on their surface for reactions to occur; for example, nucleoprotein complexes (NCs) have several promoter sites to bind transcription factors. Let us model these biological elements simply as beads with several reaction sites to which molecules attach themselves, as shown in Figure 1. The bead is placed at = and moves in a d-dimensional space ∈ (d = 1, 2, 3). We consider an isothermal process that is homogeneous over space at a given temperature T. We also consider a chemical bath containing a chemical X with a spatially dependent concentration x() or, equivalently, the corresponding chemical potential μ(). This gradient is assumed to be sustained externally. A molecule of X is attached to a binding site B on the bead and forms a complex Y (Fig. 1), as given by the reaction

Figure 1

Schematic representation of our system. A bead is placed at = , and it moves in a d-dimensional space (d = 1, 2, 3). The adsorption reaction occurs on the surface of the bead.

The molecular number of the complexes on the bead is denoted by N. Note that we define the bead as a macroscopic entity relative to an X molecule. To consider local-equilibrium conditions, we make the following assumptions. The “adsorption” reaction on the bead is considered to be a macroscopic event and N is the average molecular number that is averaged over a much longer time scale than the microscopic time scale of the reaction. Further, the bead is assumed to move sufficiently slowly so that the above reaction is in local chemical equilibrium at the position = . In other words, the time scales of diffusion of the X molecules (τ) and the adsorption reaction (τ) are much smaller than the time scale of the motion of the bead (τ): τ, τ << τ. As long as the bead is in motion, it is in equilibrium with the reservoir at = .

With the assumption of the existence of local equilibrium, we can apply thermodynamics with spatially dependent thermodynamic variables. Indeed, at each position , the reaction process is described by the familiar classical Langmuir adsorption theory37,38; note that the theory has been successfully applied to DNA-protein binding equilibrium37. The grand potential at each position is given by Ω() = F() − yμ(), dΩ() = dF() − d(yμ()) = −ydμ(), where (V: volume of the bead), F() is the Helmholtz free energy, and dF() = μ()dy39.

Now, consider a virtual displacement of the bead. Under an infinitesimal displacement → + d, the change in the grand potential is dΩ() = −ydμ() = −y∇μ() · d. In other words, the position of the bead, , is adopted as an effective independent variable instead of the chemical potential μ(). Then, is the work coordinate, while −y μ() is the force exerted on the bead by the external world balanced by the force generated by the reservoir chemical potential distribution μ(); this force also acts on the bead. The reservoir-generated chemical gradient force per unit volume of the bead is

This expression is obtained as follows. Consider a quasistatic infinitesimal displacement d. Then, from the change in the grand potential, the maximum work done by the system on the external world through the reservoir is d’W = y ∇ μ() ·d. By considering that this work is done by the force that the reservoir exerts on the bead, i.e., d’W = ·d, the force formula in Eq. (2) is obtained. In other words, without an externally applied force, evolves spontaneously so that Ω() monotonically decreases, that is, dΩ() < 0. (see Appendix A for a detailed discussion in terms of the principle of maximum work). When we consider an overdamped system, where the kinetic energy of the bead is negligible, the phenomenological equation of motion is given by γ = − ∇Ω() = y ∇ μ(); here, it is assumed that the friction constant resulting in dissipation is proportional to the velocity with the proportionality constant γ — the friction coefficient per unit volume of the bead.

Now, consider the condition for chemical equilibrium. Because c = y + b = const., the dissociative constant is defined as , where y is given by the Langmuir isotherm (In statistical mechanics, the Langmuir isotherm is obtained as a function of the chemical potential μ() at = . We can derive the expression as a function of μ(). Also see Appendix B.). If we consider the cooperative adsorption of n chemicals on the bead (given by ), the isotherm for the adsorption is given as , where n denotes the Hill coefficient. Therefore, the equation of motion is written as

Eq. (3) is a general expression for the motion of an element that has a number of binding sites for chemical adsorption under the gradient of chemical potential. The theoretical description of the motion of the bead can be considered to be an extension of the Langmuir adsorption theory. We call this motion chemophoresis — similar to the nomenclature of typical ‘phoresis’ phenomena such as electrophoresis and thermophoresis. Because of the chemophoresis force, the direction of motion of the bead is such that the chemical potential is increased. Note that the force has an entropic origin from the viewpoint of statistical mechanics (also see Appendix B for an analysis of the force from the viewpoint of statistical mechanics).

When an additional external potential field E() such as an elastic energy is applied to the bead, the relation dF() = dE() + μ()dy() holds (from the first law of thermodynamics). Accordingly, the grand potential of the system given by Ω() = F() − y()μ() satisfies dΩ() = dF() − d(y()μ()) = dE() − y()dμ(). Further, by taking into account thermal fluctuations, the equation of motion of the system is given by

with < (t) > = 0 and < (t) ·(t′) > = 2dγkTδ(t − t′). Note that the equation is obtained as an extended form of the conventional one by considering the Helmholtz free energy as the thermodynamic potential for a closed system. It is straightforward to extend the present formula for multiple beads and multiple components, with interactions among the beads.

2-2. An example

We consider a simple toy model as an example of Eq. (4). We assume that a bead placed at r = ξ is tethered and balanced at r = 0 in a one-dimensional space r by a restoration force produced by a linear spring; the force is represented by the harmonic potential . The concentration of the chemical that reacts with the bead is assumed to be constant and is given by x(r) = x exp (λ(r − r)), where x = x(r) is the concentration of X at r = r, while the corresponding chemical potential is given by μ(r) = μ̄ + kT ln x(r). Here, μ̄ is the standard chemical potential. Then, by using Eq. (4) without considering thermal fluctuations, the equation of motion of the bead is obtained as . Here, γ, c, K and a are the frictional coefficient, maximum adsorption concentration, dissociative constant, and spring constant, respectively. This equation can be rewritten as

where A = λkTc, . The steady state solution of the equation shows bistability as it includes a sigmoid function. If x is considered as a control parameter, upon increasing x, Eq. (5) undergoes twofold bifurcation from monostable fixed points to bistable ones and then back to monostable states. A change in the chemical concentration leads to a change in the stable fixed position of the bead, which also shows bistability. This toy model can be used to explain how a stable position of an organelle undergoes bifurcation to a new location as the chemical concentration increases.

2-3. Conditions for chemophoresis

2-3-1. Condition (i): Relationships among time scales

The grand canonical description and local chemical equilibrium can be considered only under certain conditions. The following relationships must exist among time scales: (1) the time scale of adsorption τ must be shorter than that of bead motion τ in order for the adsorption reaction to reach equilibrium (approximately) before the bead moves, and (2) the time scale of bead motion must be much larger than that of the diffusion of the chemical adsorbed on the bead, i.e. τ >> τ. This condition is necessary to assume that the motion of the bead is macroscopic under the chemical gradient, which itself diffuses in space. In living cells of interest, the molecular number is not large enough for the instantaneous chemical concentration to show large fluctuations. Hence, the macroscopic concentration relevant to the force is determined by the temporal average of the concentration of molecules over the diffusion time scale40, and therefore, the above condition is important.

2-3-2. Condition (ii): Dominance of chemophoresis over thermal fluctuations

As long as the above conditions are satisfied, the direction of the gradient force, on an average, is such that the chemical potential increases, regardless of the magnitude of thermal fluctuations. However, the gradient force must be larger than the thermal noise in order for it to act effectively and to eliminate the need for long-time averaging to remove fluctuations. This leads to the following condition: the work done by the force must be greater than the thermal energy, i.e., for a directional motion from to (This inequality can also be derived from the steady-state solution of the Smoluchowski equation. For one-dimensional motion, the steady-state distribution of ξ is given by . Here, A is a normalized factor and . If the position ξ that satisfies W(ξ) ~kT is smaller than the system size L, the distribution is concentrated at higher chemical potentials and the gradient force dominates over the thermal fluctuations. In such a case, the above inequality is obtained.). The concentration and chemical potential of chemical X are obtained as and μ(r) = μ̄ + kT ln x(r), respectively. Here, x0 is the highest concentration and it corresponds to r = 0. In the case of one-dimensional motion from r = 0 to L, the above condition for the work is rewritten as follows: with N = V c. A straightforward calculation gives the inequality , which can be rewritten in the form

This inequality is accompanied by the additional condition . Note that the lower bound on decreases with and N, and the formula is valid over a wide range of values. Although this condition is valid for an exponentially decaying distribution, it is expected to be useful as an estimate of the order of magnitude.

2-4. An application of chemophoresis to plasmid/chromosomal locus partitioning

2-4-1. Estimate of the chemophoresis and the validity of the conditions

We consider applications of the chemophoresis force within a cell. As an example, consider the partitioning of chromosomal locus/plasmids in bacteria during cell division8–24. Here, the bead corresponds to a chromosomal locus/plasmid on which the relevant protein (X in the model) binds to form a NC (Y in the model) that is important for partitioning. For the application, the above-mentioned condition relating to the time scales (τ, τ << τ) has to be satisfied. We first examine this condition.

In general, the time scale of protein binding equilibrium on bacterial DNA, τ, is about the order of 1 (s). In Espeli et al., Fiebig et al. and Elmore et al., it has been suggested that the segregation of chromosomes is spatially restricted and the diffusion coefficient D of a chromosomal locus is estimated as 10−5~10−4 (μm2/s)41–43. The size of a NC, a, is assumed to be on the order of a~50 (nm), so that τ is estimated as . Similarly, τ is estimated as , where D is the diffusion coefficient of proteins within the cytoplasm and is roughly estimated as D ~ 3.0 (μm2/s) (according to Elf et al.44 and Xie et al.45). Therefore, τ << τ << τ, the condition for the chemophoresis is satisfied.

To examine the validity of condition (ii), we consider a specific example — the partitioning of bacterial chromosomal locus/plasmids to daughter cells by a Par system8–24. Here, the most ubiquitous partitioning system is the parABS partitioning loci (The Par system consists of three components: DNA binding protein ParB, ATPase ParA, and centromere-like site parS. ParB binds parS, spreads along the DNA, and forms a large NC around parS. ATP-bound ParA (ParA-ATP) can nonspecifically bind to DNA and interact with the ParB-parS NC8–10,16–19,22–24.). Before cell division, daughter chromosomal locus/plasmids are precisely segregated by ATP-bound ParA (ParA-ATP) dynamic localization along with the associated (ParA-ATP)-NC interaction at the parS site on chromosome/plasmid. It has been suggested that ParA-ATP localization pulls parS chromosomal locus10 or at least guides plasmids24, but the mechanism by which ParA-ATP generates the driving force for chromosomal locus/ plasmid segregation has not been elucidated yet. Now, by regarding ParA-ATP as X and the NC (to be precise, ParB-parS NC mediated by ParA-ATP) as Y in our model, we will show that this driving force is naturally explained by the chemophoresis force.

To confirm the validity of our formula, we simply consider the steady concentration distribution of ParA-ATP to be within [0, L]9,10. We then take n = 2 (because ParA-ATP cooperatively binds DNA)46,47 and choose K = 0.3 (μM) on the basis of recent data obtained in Leonard et al.46 and Castaing et al.47. Although the precise amount of ParA-ATP that binds DNA around the NC is unknown, N~10 is a natural estimate since it has been suggested that a large amount of ParA is required to bind DNA around an NC. The localization scale r of ParA-ATP localization is not precisely known either and could vary across bacterial species. By assuming from Toro et al.9 and Fogel et al.10, the condition in Eq. (6) can be rewritten as . Active protein binding generally occurs when a reasonable concentration x0~K is reached and x0 is larger than K so that the above inequality is satisfied. Therefore, the chemophoresis force dominates over the thermal fluctuations. Although the mechanism of ParA-ATP localization could depend on the species, the estimate suggests that the force plays a significant role in the partitioning of plasmids/chromosomal locus, independent of the specific molecular mechanisms. Further, we note that the above-mentioned “spring toy model” involving a gradient is applicable to the partitioning problem and may help explain the bifurcation of plasmids/chromosomal locus during cell division with a change in ParA-ATP localization.

2-4-2. Regular positioning determined by combining chemophoresis and the reaction-diffusion (RD) equation

Although the application mentioned here assumes a ParA-ATP steady distribution, the distribution sometimes shows pole-to-pole oscillation17–24 or the formation of a different spatial pattern by a few plasmids18,19,23. Nevertheless, our theory is applicable to time-varying distributions because the force derived here leads to directional motion toward the ParA foci (peaks). Indeed, the directional motion has been suggested in Hatano et al.24. Furthermore, the present chemophoresis force is applicable to a RD system so that the plasmid is positioned according to the chemical concentration pattern. We shall discuss this in detail.

As mentioned in the previous subsection, we hypothesized that the chemophoresis force generated by a ParA-ATP concentration gradient drives plasmids in the direction of increased concentration. On the other hand, it has been observed that ParB-bound plasmids can stimulate the ATPase activity of ParA16,19, which is similar to the Min system where MinE stimulates the ATPase activity of MinD homologous to ParA during cell division34–36. Hence, there exists feedback from plasmids to the ParA-ATP pattern dynamics. This means that the chemophoresis force generated by a ParA-ATP gradient acts on plasmids, whereas ParB on the plasmids can modulate the ParA-ATP distribution through chemical reactions; thus, there is a relation between the plasmids and the ParA-ATP dynamics. This interaction may lead to the regular spacing of plasmids, which is observed in experiments 18,19,23.

Here, we briefly study a dynamical system in which equations of motion for plasmids are coupled with a RD equation for ParA-ATP. Modeling studies in which biological details are taken into account will be reported elsewhere (Sugawara T. and Kaneko K., in preparation).

As was mentioned, ParA-ATP binds NC on a plasmid. ParA-ATP interacts with ParB which stimulates ParA ATPase activity at a catalytic rate k. Because ParA cannot bind the NC when it is not combined with ATP, free ParA products are released from the NC immediately after ATP hydrolysis. Thus the reaction formula can be written as

Through this reaction on the NC, a plasmid plays a role of a sink for ParA-ATP and induces a concentration gradient of this protein. Now, consider a one-dimensional space along the long cell axis and a plasmid i (1 ≤ i ≤ N) to be positioned at r = ξ ∈ [0, L]; Here, L is cell length. ParB is assumed to be collected and localized at the positions of NCs on plasmids. Denoting the concentration of ParA-ATP as u and that of ParB as b, the RD equation for the ParA-ATP concentration is given as follows:

where the first, second, and third terms represent the diffusion of ParA-ATP, its synthesis at a constant rate a, and linear degradation with a rate c, respectively; the last term denotes the inhibition by ParB on the NC.

On the other hand, from Eq. (3) and the representation for the chemical potential, μ = μ̄ + kT ln u, the equations of motion for plasmids are given as

with < η(t) > = 0 and < η(t) η (t′) > = 2γkTδδ(t − t′).

We adopt the Neumann boundary condition (NBC) for the RD equation: ∇u(0) = ∇u(L) = 0. To confine the plasmids to a cell r ∈ [0, L], we place reflection walls at r = 0 and r = L. This can be explicitly represented as

Here, we set a = 1, b = 10, c = 0.1, k = 1, D= 0.5, K = 0.5, n = 1, γ = 1000, and L = 10 and choose unit energy so that kT = 1 (note that is larger than K, so that the chemophoresis force on the plasmids dominates over thermal fluctuations).

Partitioning of two replicated plasmids

First, we examine the case of N = 1 and 2 to examine how two replicated plasmids are separated into daughter cells. Figure 2 shows the dynamics and distribution of u(r) and plasmid(s) along the long cell axis for the two N values. A plasmid is localized at the center of the cell for N = 1 whereas two plasmids are positioned approximately at distances of one-quarter and three-quarters of the cell axis for N = 2 (Fig. 2). These results are consistent with earlier reports on plasmid positioning during cell division20,21.

Figure 2

Dynamics and distribution along the long cell axis of u(r) and plasmids. We show the cases of N = 1 and 2. A plasmid is localized at the center of the cell for N = 1 whereas two plasmids are positioned approximately at distances of one-quarter and three-quarters of the cell axis for N = 2. Obtained from simulations of Eqs. (8) and (9) under the Neumann boundary condition, with the parameter values a = 1, b = 10, c = 0.1, k = 1, D = 0.5, K = 0.5, n = 1, γ = 1000, L = 10, and kT = 1.

Regular distribution of plasmids

Observation of such regular positioning of plasmids are not restricted to the cases of N = 1 and 2. For N > 2, stable regular distributions are formed as a result of the generation of the gradient of u(r) by each plasmid. Figure 3 shows the distributions of u(r) and plasmids for N = 3~6 along the long cell axis. Similarly, regular distribution of plasmids is achieved. A plasmid i is localized around a fixed position and they appear to get arranged at an interval determined by (Fig. 3).

Figure 3

Distribution of plasmids along the long cell axis. We show the cases for N = 3 ~ 6. The regular positioning of plasmids is achieved due to an effective inter-plasmid repulsive force and an effective repulsive force from the boundaries. A plasmid i is localized around a fixed position and they appear to get arranged at an interval determined by . Obtained from simulations of Eqs. (8) and (9) under the Neumann boundary condition, with the parameter values a = 1, b = 10, c = 0.1, k = 1, D = 0.5, K = 0.5, n = 1, γ = 1000, L = 10, and kT = 1.

The regular positioning of plasmids is due to an effective interplasmid repulsive force. A plasmid, which acts as a sink for ParA-ATP, contributes to the formation of a concentration gradient in which the concentration increases with an increase in distance from the plasmid. The other plasmids are subjected to the chemophoresis force by the gradient in the direction of increasing ParA-ATP concentration so that the latter plasmids are pushed away from the former. Because the former is also subjected to the chemophoresis force by the gradient derived from the latter, there is mutual repulsion among the plasmids. As a result, the regular positioning of plasmids is achieved. In fact, on the basis of a recent experiment14, the repulsive force between plasmids has been suggested to be cause of their regular positioning. The repulsive interaction between plasmids due to ParA-ATP and mediated by chemophoresis is consistent with such experimental results.

A concentration gradient of u(r) is maintained around a plasmid (sink), while it is vanished due to NBC at the boundaries r = 0 and L. On other words, the plasmid is reflected at the boundaries by NBC and confined within [0, L]. Hence, an effective repulsive force from the boundaries acts on the plasmid even though gradients at the boundaries are vanished under NBC.

Here, we adopted the Neumann boundary condition ∇u(0) = ∇u(L) = 0. Although this boundary condition may be appropriate for intracellular dynamics, we have confirmed that the formation of the regular distribution is independent of the choice of boundary conditions. In fact, simulations of Eqs. (8) and (9) under the Dirichlet boundary condition u(0) = u(L) = 0 have demonstrated the regular distribution of plasmids as a result of chemophoresis.

3. Discussion

In this paper, we have demonstrated that when a chemical concentration gradient exists, a macroscopic element that acts as a scaffold for the absorption of a chemical is generally subjected to a thermodynamic force; the force makes it move in the direction of increasing chemical potential. We term this force chemophoresis force. The force has an entropic origin and thus is independent of the specific molecular mechanism, and it is expected to provide clues to the mechanism underlying mechanochemical coupling. We have derived a general formula for the magnitude of the force and have presented a few conditions that should be satisfied for the validity of this formula. The conditions are shown to be satisfied for the intracellular motion of macromolecules or organelles under a suitable chemical gradient. Further, the force is shown to be greater than the thermal fluctuation force.

In particular, we have applied chemophoresis to bacterial partitioning systems during cell division. By introducing a dynamical system consisting of equations of motion for plasmids and a reaction-diffusion equation that are mutually coupled, we have explained the regular positioning of plasmids, which has been observed in experiments. Because it has been reported that other organelles are also organized by ParA or its homologue48,49, chemophoresis may contribute to these phenomena, too.

Here, we considered distributions of plasmids and ParA-ATP only along the long cell axis, and we studied equations of motion for plasmids coupled with the RD equation for ParA-ATP in one-dimensional space. It has been suggested that in reality, plasmids and ParA-ATP are restricted on a two-dimensional surface of the nucleoid of a host bacterial cell. A more realistic distribution can be obtained by taking into account two-dimensional plasmid motion under the gradient generated by ParB. In a future study, it is important to study a two-dimensional model by considering motions along both the longitudinal direction (parallel to the long cell axis) and transverse direction.

Furthermore, it is to be noted that our study does not exclude the possibility of the existence of other mechanisms for partitioning, such as the growth and shrinkage of ParA-ATP filaments19. To explain the regular positioning of the plasmids, the model used for the study of the growth and shrinkage of ParA-ATP filaments considers the polymerization of ParA-ATP along the long cell axis and the depolymerization of ParA-ATP by ParB on the plasmids19. In the model, it is sufficient to take into consideration only the polymerization-depolymerization in the longitudinal direction (parallel to the long cell axis), that is, in one-dimensional space. Indeed, chemophoresis is compatible with other possible mechanisms, and a combination of such mechanisms can enable the rapid and robust intracellular organization of macromolecules.

In applying chemophoresis to bacterial plasmid partitioning, we have focused particularly on the gradient generated and maintained by the regulation of ATP hydrolysis. Because the regulation of the gradient of a protein concentration by phospholylation-dephospholylation has been reported in Kholodenko1, Carazo-Salas et al.2, Caudron et al.3 and Kalab and Heald4 and suggested to be a general intracellular process, chemophoresis resulting from the regulation of phospholylation-dephospholylation, such as the regulation of the hydrolysis of ParA-ATP discussed in this paper, is expected to be ubiquitous in a variety of intracellular processes.

Apart from the chemophoresis reported here, there is another driving force generated by the concentration gradient; “diffusiophoresis”, which is well-known in the transport of colloid50–52. In the diffusiophoresis, hydrodynamic effect is essential, while in the chemophoresis, chemical adsorption reaction is important. However, if the hydrodynamic effect is necessary, we can simply add the term into our formulation for chemophoresis. In the organization of plasmids, it appears that macroscopic hydrodynamic flow is not necessary. Indeed, an entropic effect caused by a gradient generated and maintained by the regulation of phospholylation-dephospholylation can explain the plasmid partitioning well.

It is important to note that in contrast to the hydrodynamic effect or other entropic forces such as the excluded volume effect, the chemophoresis force acts only on macroscopic elements (such as plasmids) to which proteins can bind. Because specific binding of a chemical is one of the most fundamental intracelluar processes for proper and robust functions, the generation of the chemophoresis force by such specific binding is expected to be a universal phenomenon that is responsible for the intracellular positioning of macromolecules and organelles.