# Surfing on protein waves: proteophoresis as a mechanism for bacterial genome partitioning

###### Abstract

Efficient bacterial chromosome segregation typically requires the coordinated action of a three-component, fueled by adenosine triphosphate machinery called the partition complex. We present a phenomenological model accounting for the dynamic activity of this system that is also relevant for the physics of catalytic particles in active environments. The model is obtained by coupling simple linear reaction-diffusion equations with a proteophoresis, or “volumetric” chemophoresis, force field that arises from protein-protein interactions and provides a physically viable mechanism for complex translocation. This minimal description captures most known experimental observations: dynamic oscillations of complex components, complex separation and subsequent symmetrical positioning. The predictions of our model are in phenomenological agreement with and provide substantial insight into recent experiments. From a non-linear physics view point, this system explores the active separation of matter at micrometric scales with a dynamical instability between static positioning and travelling wave regimes triggered by the dynamical spontaneous breaking of rotational symmetry.

Controlled motion and positioning of colloids and macromolecular complexes
in a fluid, as well as catalytic particles in active environments, are
fundamental processes in physics, chemistry and biology with important implications
for technological applications Zottl (); Marko ().
In this paper, we focus on an active biological system for which precise experimental results are available.
Our work is fully inspired by studies of one of the most widespread
and ancient mechanisms of liquid phase macromolecular segregation and positioning known in nature:
bacterial DNA segregation systems. Despite the fundamental importance of these systems in the bacterial world
and intensive experimental studies extending over 30-years Gerdes (); Sanchez (); LeGalletal-1 (),
no global picture encompasses fully the experimental observations.

Partition systems encode only three elements that are necessary and sufficient for active partitioning:
two proteins ParA and ParB, and a specific sequence parS encoded on DNA.
The pool of ParB proteins is recruited as a cluster of spherical shape centered around the sequence parS,
forming the ParBS partition complex Sanchez ().
These ParBS cargos interact with ParA bound onto chromosomal DNA (ParA-slow) Leonard (); Bouet07 (), triggering unbinding
of ParA by inducing conformational changes through stimulation of adenosine triphosphate (ATP) hydrolysis and/or direct
ParB-ParA contact Vecchiarelli13 (), and thereby allowing ParA diffusion in the cytoplasm (ParA-fast) LeGalletal-1 ().
This process entails the oscillation of ParA from pole to pole and the separation of the ParBS partition complex
into two complexes with distinct sub-cellular trajectories and long-term localization.
Overall, these interactions result in an equidistant, stable positioning of the duplicated DNA molecules along the cell axis.

The specific modeling of ParABS systems falls into two categories:
either “filament” (pushing/pulling the cargos, similar to eukaryotic spindle apparatus Gerdes ())
or reaction-diffusion models Vecchiarelli10 (); Vecchiarelli13 (); Vecchiarelli14 (); Ietswaart (); Lim (); Surovtsev (); Jindal (); Frey ().
Recent superresolution microscopy experiments have been unable to observe filamentous structures of ParA LeGalletal-1 (); Lim (),
disfavoring polymerization-based models Ietswaart ().
Reaction-diffusion models have been mainly investigated numerically to describe experimental observations
like single or multiple ParBS complex positioning.
In most cases, these models require other assumptions - such as DNA elasticity Lim (); Surovtsev () -
as simple reaction-diffusion mechanisms are not sufficient to predict proper positioning.
Other reaction-diffusion models considered the dynamics of the partition complex
on the surface of the nucleoid Vecchiarelli10 (); Vecchiarelli13 (); Vecchiarelli14 (); Jindal ().
Recent experiments, however, demonstrate that partition complexes and ParA translocate through
the interior of the nucleoid, not at its surface LeGalletal-1 ().

Recently, in the context of the active colloids literature, there have been attempts to describe the ParABS system using models
inspired by the diffusiophoresis Anderson (); AndersonBC () of active colloidal particles in solute concentration gradients sugawara2011 (); Marko ().
These works have several important limitations for applications to ParABS, such as: rigid spherical particles (with surface reactions only),
the steady-state approximation, only one ParA population, or reproducing equilibrium positioning only.
The full dynamical behavior of the coupled system (ParBS cargo coupled to ParA) has thus not been elucidated.

Here we propose a general model of reaction-diffusion for ParA coupled to the overdamped motion of ParBS. Our continuum reaction-diffusion approach goes beyond the previous diffusiophoretic mechanisms sugawara2011 (); Ietswaart (); Jindal (); Surovtsev () by accounting for the finite diffusion of ParA-slow and ParA-fast, as well as the interaction of ParA-slow with the entire volume of ParBS partition complexes. Volumetric interactions are suggested by our recently developed “nucleation and caging” model Sanchez (); Parmeggiani-1 (), which accounts for both the formation of ParBS and the distribution of ParB in the spatial vicinity of parS specific DNA sites : the conformation of the plasmid is well described by a fluctuating polymer and the weak ParB-ParB interactions lead to foci of low density Sanchez (); Parmeggiani-1 (). The chromosome is thus likely to enter ParBS with bound ParA-slow thereby allowing for volumetric interactions. Such a volumetric interaction should also find useful applications in the field of porous catalytic particles. On the other hand, allowing for finite diffusion coefficients permits describing analytically the global dynamical picture of the model, contrary to previous numerical studies often restricted to a limited range of parameters. In particular this enables us to predict a dynamical transition between stable and unstable regimes. We observe that biological systems are generally close to the instability threshold. The ParABS system of the F-plasmid lies just below, enabling efficient positioning and precursor oscillations of ParA. Other ParABS systems (Surovtsev () and Refs. therein) could be just above, providing an explanation for the observed out-of-phase ParBS and ParA oscillations. Our model accounts for both these regimes.

The model. The ParA protein population is described by two coupled density fields: for the hydrolysed ParA-fast proteins, assumed to be unbound and diffusing rapidly within the nucleoid, and for the non-hydrolysed ParA-slow molecules, which are bound dynamically to the nucleoid and diffuse more slowly. These two species are coupled via a system of reaction-diffusion equations: the rapid species converts into the slow one with a constant rate , while the slow species is hydrolysed in the presence of the ParBS partition complexes located on DNA, with a rate (typically Vecchiarelli10 () and Ietswaart ()). The ParBS assemblies form 3D-foci complexes Sanchez () and interact with the ParA-slow proteins. The interaction probability is described by the profiles centered around the ParBS positions . These profiles play a double role: (i) they act as catalytic sources in the reaction-diffusion equations, triggering ParA-slow hydrolysis with the rate and (ii) they determine a feedback “proteophoresis” (volumetric) force, in contrast with chemophoresis forces that occur in general only at the complex surface. In what follows, the function , representing an idealized density profile of ParBS, is assumed to be symmetric with a compact support of width and a unit value at its maximum. The dynamics of the protein population is therefore described by the coupled reaction-diffusion equations:

(1) |

In these equations, in which we do not invoke the steady-state approximation (cf. Marko ()), and represent the diffusion constants of the fast and slow species, respectively and . The sum runs over the ParBS positions . The density fields are subjected to reflecting boundary conditions and , where is a unit vector normal to the cell boundary . The system described by Eqs.(1) together with these boundary conditions on and ensure total ParA protein number conservation. Note that ParA proteins can freely penetrate the partition complexes, which do not form barriers for diffusion.

The nonlinear coupling in the system is introduced by the forces driving the partition complexes, which are modeled as Brownian particles in an active medium. The back reaction on each complex is described by a “proteophoresis force” due to the ParA-slow concentration gradient acting on the whole volume of the complex. In the viscous medium prevailing in a cell, we do not expect inertial terms to be important. Neglecting in the first approximation the stochastic and confining forces, the dynamic equation for the complex then read

(2) |

Note that no direct coupling between complexes has been introduced.
The constant represents the energy of interaction
between a single ParA-slow protein and the ParBS partition complex. Hence, the order of magnitude of
is a fraction of the energy released by the ATP hydrolysis ().
The drag force coefficient is related to an effective diffusion constant of the complex by
the Einstein relation . Thanks to attractive protein-protein interactions (leading to hydrolysis energy
consumption) the interaction energy in (2) is positive,
and the corresponding proteophoresis force, and resulting motion, is in the direction of increasing ParA density gradient.
In the following, we will use the dimensionless coupling constant:
.
From numerical simulations it appears that the stochastic force does not affect crucially the main system dynamics.
Superresolution microscopy LeGalletal-1 () indicates that the partition complex motion is
confined to the cell symmetry axis, i.e. within the bacterial nucleoid. Therefore, in the minimal model we limit the study of its dynamics to one dimension and
denote by the coordinate along the cell axis, , where is the cell length.

Restoring proteophoresis force positions the partition complexes symmetrically along the nucleoid axis.
The model provides all the necessary ingredients for proper partition complex positioning.
We first look for stationary solutions when a single partition complex is present within the cell at position .
In order to keep the algebra simple, we approximate the profile function
by a Dirac-delta distribution footnote1 (), where the amplitude
is the typical interaction volume of the complex.
The stationary solutions of Eqs.(1) with reflecting boundary
conditions then reads

(3) |

where . The dependent constants and in (3) can (see SM) be easily computed by the gradient discontinuity at ,

(4) |

and by the conservation of the total number of ParA monomers. For a delta-like complex profile, the force acting on a static partition complex located at is proportional to the mean value of the ParA-slow density gradient at :

(5) |

This result shows that the unique equilibrium position of the complex is located at the cell center, i.e. . An important feature of the resulting force mediated by the ParA density distribution gradient is its finite range. Its screening length, given by , is illustrated in Fig.1, where the force is plotted for different values of . Clearly, the proteophoresis force, here estimated of the order of the picoNewton (nm), is sensed by the partition complex only if its distance to the cell boundary or to a neighboring complex is less than .

Note that the above quasistatic (adiabatic) analysis is valid only when the ParA distribution instantaneously adapts to the complex position (cf. Marko ()). The restoring character of the force, Eq.(5), then makes the symmetric position stable.

For bacterial cells containing several partition complexes, the sum over their positions in Eqs.(1) generates an effective indirect interaction among them that, together with the boundary conditions and protein number conservation, brings the system to an equilibrium state with highly symmetric complex positions. For instance, when two complexes are present within the cell (as would be the case after a DNA replication event) the equilibrium positions are found to be located at and , i.e. the “1/4” and “3/4” positions in terms of the cell axis length . A phase portrait of the system in the coordinates (see inset of Fig.1) clearly indicates the stable nature of these positions. This result is in excellent agreement with experimental observations LeGalletal-1 (); Glaser (), and can describe even more complex experimental situations with multiple ParBS, see some examples in Fig.2.

Interestingly, as we show below, when the evolution time scale of the ParA distribution is shorter than that of the partition complex, the symmetric static positions become unstable and the steady-state approximation breaks down, leading to oscillatory behavior of the complexes.

The translocation-segregation mechanism can become unstable with respect to ParA travelling waves. Analytical and numerical studies of Eqs. (1-2) show that stationary solutions (irrespective of the number of complexes) become unstable in cells where the ParA density profiles can develop large gradients. The concentration profiles and the partition complex start travelling together at a constant velocity , as if partition complexes were self-propelled by “surfing” on the ParA distribution wave they have themselves generated (see SM) to eventually bounce back and forth in presence of cell boundaries. This strongly suggests the existence of travelling waves (TW) in an unbounded system or in finite-size cells whose length is much larger than the screening length . For one complex, we look for solutions of Eqs. (1-2) in the TW form , where is the wave comoving reference coordinate, with the asymptotic conditions and when . The resulting system of ordinary differential equations admits analytical solutions for a Dirac partition complex profile . For more general shapes, solutions are easily obtained numerically. Typical TW-like snapshots of ParA distributions calculated for a rectangular complex profile are displayed in Fig.3(b,c). The equation of motion of the partition complex (2) takes the form and provides a nonlinear relation for determining the wave celerity .

The existence of travelling waves with nonzero velocity is concomitant with the loss of stability of the equilibrium positions of the partition complexes discussed above. Thus, we distinguish two dynamical regimes: (1) A stable regime without TWs (), with stable (equidistant, if more complexes are present) equilibrium complex positions independent of the initial conditions if the screening length is large with respect to the cell size, see Fig.3(a) and SM. This implies a transient translocation when the initial conditions do not correspond to stable positions. This regime occurs for small values both of the coupling constant (obtained, e.g., for large values of the limiting diffusion constant ) and the ParA concentration, . When the screening length is small, then ParBS cargos remain at their initial positions, not necessarily equidistant and without interaction between complexes. (2) A dynamical regime () with unstable equilibrium positions of the complexes and ParA density oscillations in the cell corresponding to TWs in an unbounded domain, see Fig.3(b,c) and SM. This occurs for large values of both and the initial ParA concentration . Since is large for small values of the diffusion constant , there results an apparently surprising phenomenon, namely that slower ParA-slow kinetics leads to faster complex dynamics. This regime occurs because the ParA-slow distribution variation in time is not rapid enough to follow the partition complex and trails behind it. Indeed, the stability threshold corresponding to the appearence of TWs at can be written as , where is the escape velocity of the complex and the speed of spatial rearrangement of the ParA-slow distribution (see SM for details). When the ParA distribution rapidly reequilibrates its symmetric profile with respect to the complex position and the system tends to the stable stationary regime, while in the opposite case spontaneous symmetry breaking and TW behavior occur. Using the expressions for and , we obtain the stability condition in the form: . This reveals that large complex sizes, interaction energies , and ParA densities, as well as low ParA-slow diffusion coefficients lead to the instability of the partition complex positioning. Importantly (see SM), a biologically reasonable choice of model parameters shows that the system is not far below the instability threshold, leading to a not only robust but also relatively fast segregation process, in agreement with experiment.

Discussion. Our model for bacterial DNA segregation is able to account for the whole of the experimental phenomenology of segregation and positioning of the replicated DNA molecules. This is possible because of the careful definition of reaction-diffusion equations for the two species of ParA (slow and fast), coupled to the overdamped motion of the ParBS cargo.

Our continuum reaction-diffusion approach significantly extends previous work
sugawara2011 (); Ietswaart (); Jindal (); Surovtsev (). Some of these Jindal (); Surovtsev ()
failed to observe a stable equipositioning regime because ParA-slow was not allowed to diffuse ():
thus diverges, setting the system in the unstable regime.
In Surovtsev (), relative positioning occurs only with multiple cargos as a crowding
effect, whereas it is known that positioning can occur even with a single plasmid AhSeng13 (), as predicted by certain modeling studies sugawara2011 (); Ietswaart ().
In line with the most recent experimental findings LeGalletal-1 (),
we assume that partition complexes evolve within the nucleoid volume near the axis of the rod-shaped bacterial cells,
in contrast with the translocation surface mechanism presented
in Vecchiarelli10 (); Vecchiarelli13 (); Vecchiarelli14 (); Jindal () performed on large surfaces coated by ParA,
lacking the confinement necessary for equipositioning.
Our proposed mechanism integrates explicitly a volumetric interaction Sanchez () with the partition complex (i.e. a length in 1D),
placing the system close to the stability threshold for the biological range of parameters.
In the case of a surface interaction, for which the volume is limited to the boundaries of the surface complex,
would thus take much higher values.
This argument can be easily generalized to higher dimensions .
Our approach also allows us to clarify analytically the physical mechanism at play, by going beyond the numerical
simulations usually performed in a limited range of parameters, and to show explicitly that other effects like
polymerization Ietswaart () and DNA elasticity Lim (); Surovtsev () are not needed to account for segregation.

These elements make the active system considered in our work unprecedented, with genuine size and bulk-dependent effects,
like the emergence of a critical coupling constant controlling the stability and the TW regimes.
Moreover, when multiple complexes are present, they generate indirect inter-complex interactions mediated solely by the “perturbed” medium.
This leads naturally to proper equilibrium partition complex positioning, as well as to spontaneous (left/right in 1D) symmetry breaking in the travelling wave regime.
To our knowledge this is the first model, in the context of active bacterial segregation via ParABS systems,
possessing very good qualitative and semi-quantitative agreement with all experimental observations,
including segregation and position control of single and multiple partition complexes (see also SM).
The model robustness also suggests its application to other biological processes, like macromolecule and organelle positioning in intracellular dynamics.

Beyond its biological inspiration, this model is a novel one for active particle dynamics (accounting for “proteophoresis”)
and nonlinear physics with a very rich phenomenology.
Indeed, our model falls in the class of active particles (partition complexes in the present case) which locally “perturb” a medium
(composed here of ParA proteins) that acts back on their dynamics and thus gives rise to particle self-propulsion.
Such a behavior also provides similarities with classical polaron systems Banyai ().
In contrast with previous works sugawara2011 (); Ietswaart (); Marko () on the subject, as well as on the self-propulsion of catalytic particles in active environments
under chemical gradients Zottl (), we do not invoke specifically the well-known mechanism of diffusiophoresis (or chemiphoresis)
Derjaguin (); AndersonBC (); Anderson () or autochemotaxis, which involve surface interactions and (possibly asymmetric) catalytic surface reactions sugawara2011 ()
coupled to surrounding hydrodynamic fluid flow relative to the particle surface (see Marko (); Zottl ()).
Future perspectives will include more refined comparisons with experimental observations and biological parameters and a generalization to higher dimensions.

###### Acknowledgements.

The authors acknowledge financial support from the Agence Nationale de la Recherche (IBM project ANR-14-CE09-0025-01) and from the CNRS Défi Inphyniti (Projet Structurant 2015-2016). This work is also part of the program “Investissements d’Avenir” ANR-10-LABX-0020 and Labex NUMEV (AAP 2013-2-005, 2015-2-055, 2016-1-024). We thank E. Frey for informing us that he and his collaborators have used a similar approach to model the positioning of the division plane in bacteria. We also thank John Marko and Ned Wingreen for interesting discussions, and Martin Howard for helpful comments on the manuscript.## References

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