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Chemophoresis engine: A general mechanism of ATPase-driven cargo transport [1]

['Takeshi Sugawara', 'Universal Biology Institute', 'The University Of Tokyo', 'Tokyo', 'Kunihiko Kaneko', 'Center For Complex Systems Biology', 'Meguro-Ku', 'Niels Bohr Institute', 'University Of Copenhagen', 'Copenhagen']

Date: 2022-08 

Abstract Cell polarity regulates the orientation of the cytoskeleton members that directs intracellular transport for cargo-like organelles, using chemical gradients sustained by ATP or GTP hydrolysis. However, how cargo transports are directly mediated by chemical gradients remains unknown. We previously proposed a physical mechanism that enables directed movement of cargos, referred to as chemophoresis. According to the mechanism, a cargo with reaction sites is subjected to a chemophoresis force in the direction of the increased concentration. Based on this, we introduce an extended model, the chemophoresis engine, as a general mechanism of cargo motion, which transforms chemical free energy into directed motion through the catalytic ATP hydrolysis. We applied the engine to plasmid motion in a ParABS system to demonstrate the self-organization system for directed plasmid movement and pattern dynamics of ParA-ATP concentration, thereby explaining plasmid equi-positioning and pole-to-pole oscillation observed in bacterial cells and in vitro experiments. We mathematically show the existence and stability of the plasmid-surfing pattern, which allows the cargo-directed motion through the symmetry-breaking transition of the ParA-ATP spatiotemporal pattern. We also quantitatively demonstrate that the chemophoresis engine can work even under in vivo conditions. Finally, we discuss the chemophoresis engine as one of the general mechanisms of hydrolysis-driven intracellular transport.

Author summary The formation of organelle/macromolecule patterns depending on chemical concentration under non-equilibrium conditions, first observed during macroscopic morphogenesis, has recently been observed at the intracellular level as well, and its relevance as intracellular morphogen has been demonstrated in the case of bacterial cell division. These studies have discussed how cargos maintain positional information provided by chemical concentration gradients/localization. However, how cargo transports are directly mediated by chemical gradients remains unknown. Based on the previously proposed mechanism of chemotaxis-like behavior of cargos (referred to as chemophoresis), we introduce a chemophoresis engine as a physicochemical mechanism of cargo motion, which transforms chemical free energy to directed motion. The engine is based on the chemophoresis force to make cargoes move in the direction of the increasing ATPase(-ATP) concentration and an enhanced catalytic ATPase hydrolysis at the positions of the cargoes. Applying the engine to ATPase-driven movement of plasmid-DNAs in bacterial cells, we constructed a mathematical model to demonstrate the self-organization for directed plasmid motion and pattern dynamics of ATPase concentration, as is consistent with in vitro and in vivo experiments. We propose that this chemophoresis engine works as a general mechanism of hydrolysis-driven intracellular transport.

Citation: Sugawara T, Kaneko K (2022) Chemophoresis engine: A general mechanism of ATPase-driven cargo transport. PLoS Comput Biol 18(7): e1010324. https://doi.org/10.1371/journal.pcbi.1010324 Editor: Alexandre V. Morozov, Rutgers University, UNITED STATES Received: November 23, 2021; Accepted: June 23, 2022; Published: July 25, 2022 Copyright: © 2022 Sugawara, Kaneko. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the manuscript and its Supporting information files. Funding: This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 19H05796,17H06386, 17K15050 and Novo Nordisk Foundation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Cell polarity regulates the direction of intracellular transport for cargos, such as organelles and macromolecules, by taking advantage of chemical gradients sustained with the aid of ATP or GTP hydrolysis [1]. For example, it is well known that eukaryotic cell polarity factors, such as Rho, GTPase, and Cdc42, regulate the orientation of cytoskeleton members so that molecular motors can carry cargo directionally on the cytoskeleton, contributing to cell movement [2], cell growth [3], and axon guidance [4]. Although the transport by the cytoskeleton is one of the most commonly observed mechanisms, the transport directly mediated by chemical gradient, if its existence is confirmed, should be of importance as a general mechanism for cargo transport as well, which we refer to as cargo chemotaxis here. A bacterial ParABS system [5–17] is a good candidate for cargo chemotaxis. It is the most ubiquitous bacterial polarity factor that regulates the separation of bacterial chromosome/plasmids into daughter cells by organizing their regular positioning along the cell axis [5–17]. Generally, it consists of three components as follows: The DNA binding protein ParB, ATPase, ParA, and the centromere-like site parS. ParB binds parS, spreads along the DNA, and forms a large partition complex (PC) around parS. ATP-bound ParA (ParA-ATP) can nonspecifically bind to DNA and interact with ParB-parS PC. Abundant ParA-ATP molecules are distributed on a nucleoid in a host cell. Their mobility is strongly restricted so that they are not homogenously distributed in the cell, thus enabling a sustained concentration gradient even within a micron-sized cell [13–15, 18–24]. Indeed, there are recent reports suggesting the existence of a concentration gradient in the in vivo experiments [25, 26]; They indicated that ParA-ATP gradient/localization can drive a parS site formation on a host genome/plasmid in the direction of the increased concentration, which can be a major candidate mechanism for plasmid partitioning and chromosome segregation [23–43]. ParA ATPase is an evolutionarily conserved protein which has many homologs [44–47]. Representative examples of its family are McdA/McdB ATPases controlling equidistribution of carboxysomes along a long cell axis in cyanobacteria [48–50], ParC/PpfA ATPases that regulate intracellular positions of chemotaxis protein clusters [51–53], MipZ ATPase that coordinates chromosome segregation in cell division [46, 54, 55], and MinD ATPase that determines a cell division plane [56–58]. These ATPase homologs, as well as ParA, work through a common mechanism essential to their function: Hydrolysis of an ATPase A by a partner protein B; A-ATP + B ⇄ C → A + ADP + B. By taking advantage of the free energy released by the reaction, a spatiotemporal pattern of the corresponding ATPase emerges [22, 46, 56–62], and cargo positions are coordinated [48–53]. One of the most renowned intracellular patterning systems is the MinCDE system that self-organizes the pole-to-pole oscillation of MinD, leading to the formation of a cell division plane at the cell center, upon stimulation of MinD ATPase activity induced by MinE at the inner cell membrane [57, 58]. In contrast, in the in vitro reconstitution of the Min system, traveling waves of MinD were observed [58–62]. Similar to the Min system, the pole-to-pole oscillation of ParA [14, 23, 63–67] also emerged in the ParABS system through the stimulation of ParA ATPase activity by ParB on the PC [13–15, 22–24]. Interestingly, a plasmid chases a ParA focus, following its oscillatory movement along the long host-cell axis [23], leading to oscillatory motion. In a recent in vitro experiment mimicking a ParABS system, Vecchiarelli et al. elegantly demonstrated the formation of directed motion of a cargo corresponding to a plasmid, referred to as “cargo surfing on ParA-ATP traveling wave” [28–30]. Hence, for both Min and Par systems, the emergence of traveling waves and the pole-to-pole oscillation of ATPase have been reported. The mechanism driving the plasmid motion, however, remains elusive [18], whereas the pattern dynamics of the Min system can be described by well-defined reaction-diffusion equations [58–60, 68–72]. Previously, we proposed a mechano-chemical coupling mechanism that enables directed movement of cargos, referred to as chemophoresis [1, 42, 43]. According to this mechanism, a macroscopic object with reaction sites on its surface is subjected to a thermodynamic force along an increasing concentration gradient. Cargo transport is possible via the chemophoresis force, [42, 43], and the possible role of the chemophoresis force in the separation dynamics of bacterial plasmids was discussed previously. By combining the plasmid motion driven by the chemophoresis force with a reaction-diffusion (RD) equation, we demonstrated that regular positioning of plasmids is possible in a ParABS system under ParA-ATP hydrolysis stimulated by ParB [16, 24, 64–67]. To date, however, spontaneous directed motion or pole-to-pole oscillation of plasmids [23, 64–67] has not been discussed in Ref [43], as was demonstrated by Vecchiarelli et al. [28–30] and theoretical studies [34–39]. In the present study, we extend our previous model and propose a chemophoresis engine as a general mechanism of cargo motion, which transforms chemical energy into directed motion via self-organization of the traveling wave, and then apply it to the plasmid motion in a ParABS system. In the previous study, the plasmid was assumed to be a point particle, where static equi-positioning and symmetric ParA-ATP distribution were robustly maintained [43]. However, such model with a zero-size limit is unrealistic, considering intracellular dynamics [26, 73] or reconstructing in vitro experiments performed by [28]. Here, by considering the finite size of plasmids explicitly, we show that organization of directed motion is possible via spontaneous symmetry breaking in the ParA-ATP pattern. We then recapitulate plasmid positioning to better describe the spatiotemporal profiles of ParA-ATP concentration and movement of plasmids. Actually, in the model presented here, the net chemophoresis force acts on the plasmid (PC) through the concentration difference between its ends, which is self-sustained by the high ATP hydrolysis stimulation. This self-driven mechanism leads to the directed motion of plasmids, as well as their equi-positioning [16, 24, 64, 67], and pole-to-pole oscillation as observed in bacterial cells and in vitro experiments [23, 64–67]. We mathematically show the existence and stability of the plasmid-surfing pattern, which allows cargo-directed motion through the symmetry-breaking transition of the ParA-ATP spatiotemporal pattern. We also indicate that plasmid size is a relevant parameter for the emergence of its directed movement. Finally, we quantitatively validate that the chemophoresis engine can work with parameters capturing in vivo conditions.

Models Chemophoresis force First, we briefly reviewed the chemophoresis force, a thermodynamic force acting on the cargo in the direction of the increased concentration of a chemical that can be bound on the cargo (See S1 Text for details.) We considered that a cargo was placed and moving in a d-dimensional space r ∈ Rd. The cargo had N molecular sites B, on each of which m molecules of chemical X was bound to form a complex Y at position r = ξ. At each site, the reaction mX(ξ) + B ⇄ Y occurred and was at chemical equilibrium. If a spatial gradient of chemical concentration X exists, the cargo is thermodynamically driven in the direction of the decreased free energy or the increased chemical potential of X [42, 43]. Here, such a gradient of the chemical potential μ(r) was assumed to be sustained externally through several active processes, supported by spatially distributed chemical gradients. We referred to the phenomenon as chemophoresis. The formula of the chemophoresis force was: (1) Here, , where x(r) is the concentration of X, K d is the dissociation constant, and m is the number of binding molecules corresponding to the Hill coefficient of the reaction. With this chemophoresis force, the cargo moved in the direction such that the concentration x(r) increased even under thermal fluctuation ([42, 43], S1 Text). For the force to work, the reaction mX + B ⇄ Y was required to reach chemical equilibrium fast enough for cargo motion. Therefore, we showed that the chemophoresis force was one of the fundamental thermodynamic forces driven by physicochemical fields. Note that the force had an entropic origin from the viewpoint of statistical mechanics. See also Ref. [43] for details of the derivation from the viewpoint of thermodynamics and statistical mechanics. To understand the origin of chemophoresis, it should be noted that microscopic binding events of X do not directly generate the force. Rather, the force works in the direction of larger frequency of the binding events (or larger time fraction of binding states) that was realized in the spatial location with a larger concentration of molecules in a chemical bath. The chemical gradient biases the binding frequency of X in a space-dependent manner. In other words, chemophoresis is driven by general thermodynamic force as a result of the free-energy (entropy) difference. It can also be derived by coarse-graining microscopic processes (S1 Text), whereas the macroscopic derivation implies its generality independent of specific microscopic models [35–37]. On the other hand, both the macroscopic (thermodynamic) and microscopic (statistical physics) theories are equivalent to each other, in that the force is generated with the aid of spatial asymmetry of molecule numbers bound on the bead, if its radius is finite. Further, for the chemophoresis force to act, X molecules do not necessarily have to bind cooperatively to the bead (as in the case of m = 1); if the concentration gradient of bound molecules is generated, the resultant free energy difference between its ends leads to the net chemophoresis force. Chemophoresis engine for plasmid partition We then applied the chemophoresis formula to plasmid motion. As the reaction on the cargo consumed chemical X, its concentration changed; therefore, studied its RD equation. It was introduced for ParA-ATP ([42, 43], S1 Text), which recapitulates the central- and equi-positioning of plasmids [42, 43]. It was also adopted successfully to explain the directed movement of beads in an in vitro experiment by Vecchiarelli et al. [28]. We considered a plasmid i(1 ≤ i ≤ M) placed into and moving in a d-dimensional space r ∈ Rd(d = 1 or 2) (Fig 1A). ParA-ATP dimers were bound to a PC on plasmid i at position r = ξ i . m ParA-ATP dimer molecules interacted with ParB, which stimulated ParA ATPase activity at a catalytic rate k [74]; N ParB molecules were assumed to be recruited to each PC at r = ξ i . Because ParA could not bind PC when it was not combined with ATP, free ParA products were released from the PC immediately after ATP hydrolysis. Thus the reaction was presented as follows: (2) PPT PowerPoint slide

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TIFF original image Download: Fig 1. Chemophoresis engine can recapitulate equi-positioning, directed movement, and pole-to-pole oscillation. (A) Schematic representation of the chemophoresis engine. A plasmid moves in a d-dimensional space r ∈ Rd(d = 1 or 2). ParA-ATP dimer (green sphere) binds a partition complex (PC, magenta sphere) on the plasmid at position r = ξ i . ParA-ATP dimer molecules interact with ParB molecules (white spheres), which stimulate ParA ATPase activity at a catalytic rate. Because ParA cannot bind PC when it is not combined with ATP, free ParA products (blue sphere) are released from the PC immediately after ATP hydrolysis. Through this reaction on PC i at r = ξ i , each plasmid acts as a sink for ParA-ATP and induces a concentration gradient of this protein. (B) One-dimensional case, on a nucleoid matrix along the long cell axis where a plasmid i(1 ≤ i ≤ M) is positioned at x = ξ i ∈ [0, L]. (C) The dynamics change among thermal motion, steady center-positioning, and directed movement followed by oscillatory mode as χ increases among χ = 0.5 (C1), χ = 2.5 (C2), and χ = 10 (C3) (two inner figures). (C1) The plasmid slightly tends to be localized at the cell center but it is still dominated by thermal fluctuations for M = 1 and χ ≔ kN/V = 0.5. (C2) It is stably localized at the cell center for M = 1 and χ = 2.5, and (C3) it shows directed movement, reflection at the end walls, and pole-to-pole oscillation for M = 1 and χ = 10. The corresponding ParA-ATP pattern dynamics also change among stochastic, steady center-positioning, and oscillatory waves (left). The oscillatory behavior of plasmids does not disrupt time-averaged center-positioning, but steady center-positioning of plasmids are sustained (Compare (C2) right and (C3), right). K d = 0.1, ε = 5, and L = 5. The distributions (right) were generated using 107 samples over 105 time step. https://doi.org/10.1371/journal.pcbi.1010324.g001 Through this reaction on PC i at r = ξ i , each plasmid acted as a sink for ParA-ATP and induced a concentration gradient of this protein. In the early model, the size of plasmids was assumed to be zero ([42, 43], S1 Text). However, the model is still too unphysical to better reconstruct the movement of plasmids with a finite size in bacterial cells [26, 73] as well as that of micro-sized beads in in vitro experiments [28]. To better describe spatiotemporal profiles of ParA-ATP concentration and directed movement of the plasmids, we considered plasmids (or PCs) as spheres with a radius of l b whose value is reported to be l b ∼ 0.075 μm in bacterial cells according to [26, 73] and l b = 1.0 μm in in vitro experiments [28]. Here, in order to discuss general situations, the derived equations were first rescaled by a dimensionless form and then numerical simulation was performed. Denoting the dimensionless concentration of ParA-ATP dimers on a nucleoid as u(r), the normalized RD equation was written as follows (see S1 Text for its derivation): (3) where the first and second terms represent the diffusion of ParA-ATP and its chemical exchange at a normalized constant rate with the cytoplasmic reservoir (denoted by its normalized concentration), respectively. The last term denotes the inhibition by ParB on the M PCs. K d is the normalized dissociation constant of the reaction mX + B ⇄ Y, and m is the Hill coefficient. V is the d-dimensional volume of the bead with a radius of l b . χ = kN/V is a maximum rate for ParA-ATP hydrolysis by ParB on each PC (S1 Text). Furthermore, θ(r) is a step function representing the space each PC occupies to describe the hydrolysis reaction space. Only within |r − ξ i | < l b , the reaction occurred. Without the last term (if χ = 0), u(r) reached a homogenous equilibrium state, u (r) = 1. In contrast, the normalized equations of motion for plasmids were represented as follows: (4) with thermal noise 〈η i (t)〉 = 0 and , and . Here, is the relative diffusion coefficient of the plasmid to that of ParA-ATP (see S1 Text for details). The parameters to be assigned to Eqs 3 and 4 are , and the system size L(= cell length).

Discussion In this study, to consider a generalized model of the plasmid partition ParABS system, a chemophoresis engine was introduced as a coupled dynamical system among the equations of motion for plasmids and the RD equation for ParA-ATP (Fig 1A). In the model, plasmid dynamics switched from static to dynamic mode with an increase in the maximum rate of ATP hydrolysis χ. The engine demonstrated equi-positioning, directed movement, and pole-to-pole oscillation, as observed in bacterial cells and in vitro experiments (Figs 1C, S1 and S2). Note that despite the plasmids’ oscillatory behavior, the regular positioning distributions were sustained (S1 and S2 Figs) due to an effective inter-plasmid repulsive interaction derived from the chemophoresis force, indicating the robustness of positional information generated by the chemophoresis engine. By simplifying Eqs 3 and 4, and introducing a space-time coordinate, we mathematically showed the existence (Fig 2B and 2C) and the stability (S5 Fig) of the plasmid-surfing pattern. The solution emerged through the symmetry-breaking transition of the ParA-ATP spatiotemporal pattern at a critical χ. We mathematically showed the directed movement emerges even in the limiting case of vanished plasmid-size l b → 0 (S6 Fig). Also, with an increase of the plasmid size l b , the solution for the directed movement disappeared as a result of an inverse pitchfork bifurcation (Fig 2D). By using parameters capturing in vivo conditions, we demonstrated that the chemophoresis engine can work even in bacterial cells (S7 Fig). The plasmid surfing also worked for a two-dimensional (2D) case (S8 and S9 Figs). The simulation results for the 2D case in Eqs 3 and 4 are shown for χ = 10 (S8A Fig) and χ = 50 (S8B Fig). In the former case, the cargo maintained its location, and u(r) had a symmetrical shape (S9A Fig), whereas directed motion by surfing on an asymmetrical traveling wave of u(r) was observed for the latter (S9B Fig), just like the 1D case. Although we analyzed the existence and stability of the surfing-on-wave pattern only in a noiseless situation (Fig 2), plasmids (or cargos) are always subjected to thermal fluctuations in cellular environments. Then, the plasmid motion was described by Langevin equation Eq 4. Further, we needed to elucidate that the plasmid-surfing-on-traveling-wave pattern remains robust against thermal fluctuations. For the chemophoresis force to act effectively, the force must be larger than the thermal noise, as discussed in a previous report [43]. Any force weaker than thermal noise cannot sustain even regular positioning [43]. We also examined how equi-positioning of plasmids can overcome thermal noise disturbances in a 1D case (S1 and S2 Figs). We confirmed that plasmid location dynamics shows a transition from stochastic switching to (freezing) steady equi-positioning (S1 and S2 Figs) as χ is increased, finally leading to persistent directed motion of the plasmids. This result suggested that the chemophoresis force dominates and directed movement of plasmids can overcome against thermal fluctuation for large hydrolysis rate. We propose a chemophoresis engine, a general mechano-chemical apparatus driving the self-motion of the intracellular cargo, as a means to elaborate the physical principles of ATPase-driven cargo transport [48–53]. The engine is based on 1) a chemophoresis force that allows motion along an increasing ATPase(-ATP) concentration and 2) an enhanced catalytic ATPase hydrolysis at the cargo positions. ATPase-ATP molecules are used as fuel to supply free energy by applying the chemophoresis force along the concentration gradient, whereas cargos generate a concentration gradient by catalyzing the hydrolysis reaction on their surface. Note that each cargo, as a catalyst, does not consume ATP, but only modulates the concentration pattern. Through the coupling and synergy between 1) and 2), directed movement of the cargo is self-organized, showing a “surfing-on-traveling-wave” pattern (Fig 2C). The chemophoresis engine is based only on these two general mechanisms and is expected to explain how the transportation of diverse cargos in bacterial and eukaryotic cells is organized. Although we have focused on the gradient generated by the regulation of ATPase, the regulation of the concentration gradient via phosphorylation-dephosphorylation reactions is ubiquitous. Therefore, the chemophoresis engine resulting from the regulation of the hydrolysis of other factors, such as GTPase, should work for a variety of intracellular processes [77–80]. Our theory is derived from macroscopic thermodynamics under nonequilibrium conditions, and although we have applied it here to the ParABS system, it is general enough to be independent of individual microscopic models constructed for each molecular mechanism. In the present study, the mathematical model of the chemophoresis engine is constructed only by extracting the essential parts of the phenomena, so it can be applied directly to other systems with common reaction mechanisms such as hydrolysis. Indeed, it has been reported in in vitro experiments that the directed motion of a micro-sized bead is self-driven by the RNA gradient which RNA hydrolysis on the bead generates [81]; The authors later termed this phenomenon “autochemophoresis” [82]. Furthermore, recent studies demonstrated substrate-driven chemotactic behaviors of metabolic enzymes (single-molecule chemotaxis) [83, 84], and discussed a mathematical model to recapitulate experimental results in the subsequent study [85]. Interestingly, the authors proposed the exact same thermodynamic mechanism as the chemophoresis force described previously [42, 43]. Therefore, we expect that the chemophoresis engine can also be applied to self-chemotactic behaviors even at a single-molecule level even though in the present study, self-chemotaxis is applied to a cargo size ranging from 50 nm to 1μm. However, to describe nanoscopic chemotaxis, we need to extend thermodynamics of chemophoresis to a stochastic one which is valid even under large thermal/chemical fluctuations. The merits of the chemophoresis engine are as follows: Self-generated chemical gradient for the chemophoresis force to apply; not requiring a large space to maintain the external chemical gradient. The chemophoresis engine can be effective in a moderate space. Therefore, the chemophoresis engine would work for eukaryotic intra-nuclear processes by restricting the mobility of chemicals on a nuclear membrane or a nuclear matrix functioning as a scaffold matrix. Based on the generality of the chemophoresis engine as well as suggestive reports in other systems [81, 82, 86], we can apply the mechanism to other hydrolysis events, RNAs, receptors, and others. We propose the chemophoresis engine as a general mechanism for hydrolysis-driven cargo transports in cells.

Methods Numerical methods for solving evolutionary equation, self-consistent equation, and eigenvalue equation Evolutionary equations, Eqs 3 and 4 were computationally solved as a hybrid simulation between reaction-diffusion equation and Langevin equation. Euler scheme for Eq 3 and Euler-Maruyama scheme for Eq 4 were used as numerical algorithms. Real-valued self-consistent equation, S9 Eq in S2 Text and complex-valued eigenvalue equation S16 Eq in S2 Text were solved by using Newton-Raphson method. Since ParA-ATP always exists as a dimer on a nucleoid, it is reasonable to simply consider a hydrolysis reaction without any cooperativity in a spatially limited space around a plasmid. Therefore, the Hill coefficient m was fixed as m = 1 in all the simulations. A normalized radius of plasmid l b was assigned to l b = 0.2 through the simulation except for Fig 2D.

Acknowledgments We thank Hironori Niki, Kazuhiro Maeshima, Akatsuki Kimura, Hiraku Nishimori, Akinori Awazu, Satoshi Sawai, Nobuhiko J. Suematsu, Satoshi Nakata, Sosuke Ito, and Yasushi Okada for their comments.

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