Operational Principles for the Dynamics of the In Vitro ParA-ParB System

Adhesive Dynamics Simulations of Highly Polyvalent DNA Motors

Molecular motors, such as myosin and kinesin, perform diverse tasks ranging from vesical transport to bulk muscle contraction. Synthetic molecular motors may eventually be harnessed to perform similar tasks in versatile synthetic systems. The most promising type of synthetic molecular motor, the DNA walker, can undergo processive motion but generally exhibits low speeds and virtually no capacity for force generation. However, we recently showed that highly polyvalent DNA motors (HPDMs) can rival biological motors by translocating at micrometer per minute speeds and generating 100+ pN of force. Accordingly, DNA nanotechnology-based designs may hold promise for the creation of synthetic, force-generating nanomotors. However, the dependencies of HPDM speed and force on tunable design parameters are poorly understood and difficult to characterize experimentally. To overcome this challenge, we present RoloSim, an adhesive dynamics software package for fine-grained simulations of HPDM translocation. RoloSim uses biophysical models for DNA duplex formation and dissociation kinetics to explicitly model tens of thousands of molecular scale interactions. These molecular interactions are then used to calculate the nano- and microscale motions of the motor. We use RoloSim to uncover how motor force and speed scale with several tunable motor properties such as motor size and DNA duplex length. Our results support our previously defined hypothesis that force scales linearly with polyvalency. We also demonstrate that HPDMs can be steered with external force, and we provide design parameters for novel HPDM-based molecular sensor and nanomachine designs.

Optimizing Efficiency and Motility of a Polyvalent Molecular Motor

Molecular motors play a vital role in the transport of material within the cell. A family of motors of growing interest are burnt bridge ratchets (BBRs). BBRs rectify spatial fluctuations into directed motion by creating and destroying motor-substrate bonds. It has been shown that the motility of a BBR can be optimized as a function of the system parameters. However, the amount of energy input required to generate such motion and the resulting efficiency has been less well characterized. Here, using a deterministic model, we calculate the efficiency of a particular type of BBR, namely a polyvalent hub interacting with a surface of substrate. We find that there is an optimal burn rate and substrate concentration that leads to optimal efficiency. Additionally, the substrate turnover rate has important implications on motor efficiency. We also consider the effects of force-dependent unbinding on the efficiency and find that under certain conditions the motor works more efficiently when bond breaking is included. Our results provide guidance for how to optimize the efficiency of BBRs.

Theory of Active Intracellular Transport by DNA Relaying

The spatiotemporal organization of bacterial cells is crucial for the active segregation of replicating chromosomes. In several species, including Caulobacter crescentus, the ATPase ParA binds to DNA and forms a gradient along the long cell axis. The ParB partition complex on the newly replicated chromosome translocates up this ParA gradient, thereby contributing to chromosome segregation. A DNA-relay mechanism-deriving from the elasticity of the fluctuating chromosome-has been proposed as the driving force for this cargo translocation, but a mechanistic theoretical description remains elusive. Here, we propose a minimal model to describe force generation by the DNA-relay mechanism over a broad range of operational conditions. Conceptually, we identify four distinct force-generation regimes characterized by their dependence on chromosome fluctuations. These relay force regimes arise from an interplay of the imposed ParA gradient, chromosome fluctuations, and an emergent friction force due to chromosome-cargo interactions.

The structure of the bacterial DNA segregation ATPase filament reveals the conformational plasticity of ParA upon DNA binding

The efficient segregation of replicated genetic material is an essential step for cell division. Bacterial cells use several evolutionarily-distinct genome segregation systems, the most common of which is the type I Par system. It consists of an adapter protein, ParB, that binds to the DNA cargo via interaction with the parS DNA sequence; and an ATPase, ParA, that binds nonspecific DNA and mediates cargo transport. However, the molecular details of how this system functions are not well understood. Here, we report the cryo-EM structure of the Vibrio cholerae ParA2 filament bound to DNA, as well as the crystal structures of this protein in various nucleotide states. These structures show that ParA forms a left-handed filament on DNA, stabilized by nucleotide binding, and that ParA undergoes profound structural rearrangements upon DNA binding and filament assembly. Collectively, our data suggest the structural basis for ParA’s cooperative binding to DNA and the formation of high ParA density regions on the nucleoid.

ParA is an ATPase involved in the segregation of newly replicated DNA in bacteria. Here, structures of a ParA filament bound to DNA and of ParA in various nucleotide states offer insight into its conformational changes upon DNA binding and filament assembly, including the basis for ParA’s cooperative binding to DNA.

Burnt bridge ratchet motor force scales linearly with polyvalency: a computational study

Nano- and micro-scale burnt bridge ratchet motors, which translocate via "guide" molecules that bind to and degrade a field of "fuel" molecules, have recently emerged in several biological and engineering contexts. The capacity of these motors to generate mechanical forces remains an open question. Here, chemomechanical modeling suggests that BBR force scales linearly with the steady-state number of guide-fuel bonds.

Supercoiled DNA and non-equilibrium formation of protein complexes: A quantitative model of the nucleoprotein ParBS partition complex

ParABS, the most widespread bacterial DNA segregation system, is composed of a centromeric sequence, parS, and two proteins, the ParA ATPase and the ParB DNA binding proteins. Hundreds of ParB proteins assemble dynamically to form nucleoprotein parS-anchored complexes that serve as substrates for ParA molecules to catalyze positioning and segregation events. The exact nature of this ParBS complex has remained elusive, what we address here by revisiting the Stochastic Binding model (SBM) introduced to explain the non-specific binding profile of ParB in the vicinity of parS. In the SBM, DNA loops stochastically bring loci inside a sharp cluster of ParB. However, previous SBM versions did not include the negative supercoiling of bacterial DNA, leading to use unphysically small DNA persistences to explain the ParB binding profiles. In addition, recent super-resolution microscopy experiments have revealed a ParB cluster that is significantly smaller than previous estimations and suggest that it results from a liquid-liquid like phase separation. Here, by simulating the folding of long (≥ 30 kb) supercoiled DNA molecules calibrated with realistic DNA parameters and by considering different possibilities for the physics of the ParB cluster assembly, we show that the SBM can quantitatively explain the ChIP-seq ParB binding profiles without any fitting parameter, aside from the supercoiling density of DNA, which, remarkably, is in accord with independent measurements. We also predict that ParB assembly results from a non-equilibrium, stationary balance between an influx of produced proteins and an outflux of excess proteins, i.e., ParB clusters behave like liquid-like protein condensates with unconventional “leaky” boundaries.

In bacteria, faithful genome inheritance requires the two replicated DNA molecules to be segregated at the opposite halves of the cell. ParABS, the most widespread bacterial DNA segregation system, is composed of a centromere sequence, parS, and two proteins, the ParA ATPase and the ParB DNA binding protein. Hundreds of ParB assemble dynamically to form clusters around parS, which then serve as substrates for ParA molecules to catalyze the positioning and segregation events. The nature of these clusters and their interaction with DNA have remained elusive. Here, we propose a realistic minimal model that captures quantitatively the peculiar DNA binding profile of ParB in the vicinity of parS in Escherichia coli. From the viewpoint of DNA, the only fitting parameter is the in vivo supercoiling density resulting from the removal of DNA helices by toposiomerases, which is in accord with previous independent estimations. From the viewpoint of ParB clusters, we predict that they behave like liquid-like protein condensates with unconventional boundaries. Namely, we predict boundaries to be leaky (i.e. not sharp) as a result of the non-equilibrium protein production, diffusion and dilution. Altogether, our work provides novel insights into bacterial DNA organization and intracellular liquid-liquid phase separation.

Substrate stiffness tunes the dynamics of polyvalent rolling motors

Nature has evolved many mechanisms for achieving directed motion on the subcellular level. The burnt-bridges ratchet (BBR) is one mechanism used to achieve superdiffusive molecular motion over long distances through the successive cleavage of surface-bound energy-rich substrate sites. This mechanism has been associated with both nanoscale and microscale movement, with the latter accomplished through polyvalent interactions between a large hub (e.g. influenza virus) and substrate (e.g. cell surface receptors). Experimental successes in achieving superdiffusive motion by synthetic polyvalent BBRs have raised questions about the dynamics of their motility, including whether rolling or translation is better able to direct motion of microscale spherical hubs. Here we simulate the three-dimensional dynamics of a polyvalent sphere moving on and cleaving an elastic substrate. We find that substrate stiffness plays an important role in controlling both the motor's mode of motility and its directional persistence. As we tune lateral substrate stiffness from soft to stiff we find there exists an intermediate value that optimizes rolling behaviour. We also find that there is an optimal substrate stiffness for maximizing persistence length, while stiffness does not influence as strongly the superdiffusive dynamics of the particle. Lastly, we examine the effect of substrate density, and show that softer landscapes are better able to buffer against decreases in substrate occupancy, with the spherical motor maintaining superdiffusive motion more on softer landscapes than on stiff landscapes as occupancy drops. Our results highlight the importance of surface in controlling the motion of polyvalent BBRs.

Highly Polyvalent DNA Motors Generate 100+ pN of Force via Autochemophoresis

Motor proteins such as myosin, kinesin, and dynein are essential to eukaryotic life and power countless processes including muscle contraction, wound closure, cargo transport, and cell division. The design of synthetic nanomachines that can reproduce the functions of these motors is a longstanding goal in the field of nanotechnology. DNA walkers, which are programmed to "walk" along defined tracks via the burnt bridge Brownian ratchet mechanism, are among the most promising synthetic mimics of these motor proteins. While these DNA-based motors can perform useful tasks such as cargo transport, they have not been shown to be capable of cooperating to generate large collective forces for tasks akin to muscle contraction. In this work, we demonstrate that highly polyvalent DNA motors (HPDMs), which can be viewed as cooperative teams of thousands of DNA walkers attached to a microsphere, can generate and sustain substantial forces in the 100+ pN regime. Specifically, we show that HPDMs can generate forces that can unzip and shear DNA duplexes (∼12 and ∼50 pN, respectively) and rupture biotin-streptavidin bonds (∼100-150 pN). To help explain these results, we present a variant of the burnt-bridge Brownian ratchet mechanism that we term autochemophoresis, wherein many individual force generating units generate a self-propagating chemomechanical gradient that produces large collective forces. In addition, we demonstrate the potential of this work to impact future engineering applications by harnessing HPDM autochemophoresis to deposit "molecular ink" via mechanical bond rupture. This work expands the capabilities of synthetic DNA motors to mimic the force-generating functions of biological motors. Our work also builds upon previous observations of autochemophoresis in bacterial transport processes, indicating that autochemophoresis may be a fundamental mechanism of pN-scale force generation in living systems.

Can a Flux-Based Mechanism Explain Protein Cluster Positioning in a Three-Dimensional Cell Geometry?

The plane of bacterial cell division must be precisely positioned. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a large cluster, which is tethered to the nucleoid by the ATPase PomZ and moves in a stochastic but biased manner toward midcell where it initiates cell division. Previously, a positioning mechanism based on the fluxes of PomZ on the nucleoid was proposed. However, the cluster dynamics was analyzed in a reduced, one-dimensional geometry. Here, we introduce a mathematical model that accounts for the three-dimensional shape of the nucleoid, such that nucleoid-bound PomZ dimers can diffuse past the cluster without interacting with it. Using stochastic simulations, we find that the cluster still moves to and localizes at midcell. Redistribution of PomZ by diffusion in the cytosol is essential for this cluster dynamics. Our mechanism also positions two clusters equidistantly on the nucleoid, as observed for low-copy-number plasmid partitioning. We conclude that a flux-based mechanism allows for cluster positioning in a biologically realistic three-dimensional cell geometry.

DNA segregation under Par protein control

The spatial organization of DNA is mediated by the Par protein system in some bacteria. ParB binds specifically to the parS sequence on DNA and orchestrates its motion by interacting with ParA bound to the nucleoid. In the case of plasmids, a single ParB bound plasmid is observed to execute oscillations between cell poles while multiple plasmids eventually settle at equal distances from each other along the cell's length. While the potential mechanism underlying the ParA-ParB interaction has been discussed, it remains unclear whether ParB-complex oscillations are stable limit cycles or merely decaying transients to a fixed point. How are dynamics affected by substrate length and the number of complexes? We present a deterministic model for ParA-ParB driven DNA segregation where the transition between stable arrangements and oscillatory behaviour depends only on five parameters: ParB-complex number, substrate length, ParA concentration, ParA hydrolysis rate and the ratio of the lengthscale over which the ParB complex stimulates ParA hydrolysis to the lengthscale over which ParA interacts with the ParB complex. When the system is buffered and the ParA rebinding rate is constant we find that ParB-complex dynamics is independent of substrate length and complex number above a minimum system size. Conversely, when ParA resources are limited, we find that changing substrate length and increasing complex number leads to counteracting mechanisms that can both generate or subdue oscillatory dynamics. We argue that cells may be poised near a critical level of ParA so that they can transition from oscillatory to fixed point dynamics as the cell cycle progresses so that they can both measure their size and faithfully partition their genetic material. Lastly, we show that by modifying the availability of ParA or depletion zone size, we can capture some of the observed differences in ParB-complex positioning between replicating chromosomes in B. subtilis cells and low-copy plasmids in E. coli cells.

Subcellular Organization: A Critical Feature of Bacterial Cell Replication

Spatial organization is a hallmark of all living systems. Even bacteria, the smallest forms of cellular life, display defined shapes and complex internal organization, showcasing a highly structured genome, cytoskeletal filaments, localized scaffolding structures, dynamic spatial patterns, active transport, and occasionally, intracellular organelles. Spatial order is required for faithful and efficient cellular replication and offers a powerful means for the development of unique biological properties. Here, we discuss organizational features of bacterial cells and highlight how bacteria have evolved diverse spatial mechanisms to overcome challenges cells face as self-replicating entities.

Regulation of Pom cluster dynamics in Myxococcus xanthus

Precise positioning of the cell division site is essential for the correct segregation of the genetic material into the two daughter cells. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a cluster on the chromosome that performs a biased random walk to midcell and positively regulates cell division there. PomZ, an ATPase, is necessary for tethering of the cluster to the nucleoid and regulates its movement towards midcell. It has remained unclear how the cluster dynamics change when the biochemical parameters, such as the attachment rates of PomZ dimers to the nucleoid and the cluster, the ATP hydrolysis rate of PomZ or the mobility of PomZ interacting with the nucleoid and cluster, are varied. To answer these questions, we investigate a one-dimensional model that includes the nucleoid, the Pom cluster and PomZ proteins. We find that a mechanism based on the diffusive PomZ fluxes on the nucleoid into the cluster can explain the latter's midnucleoid localization for a broad parameter range. Furthermore, there is an ATP hydrolysis rate that minimizes the time the cluster needs to reach midnucleoid. If the dynamics of PomZ on the nucleoid is slow relative to the cluster's velocity, we observe oscillatory cluster movements around midnucleoid. To understand midnucleoid localization, we developed a semi-analytical approach that dissects the net movement of the cluster into its components: the difference in PomZ fluxes into the cluster from either side, the force exerted by a single PomZ dimer on the cluster and the effective friction coefficient of the cluster. Importantly, we predict that the Pom cluster oscillates around midnucleoid if the diffusivity of PomZ on the nucleoid is reduced. A similar approach to that applied here may also prove useful for cargo localization in ParABS systems.

Surfing on Protein Waves: Proteophoresis as a Mechanism for Bacterial Genome Partitioning

Efficient bacterial chromosome segregation typically requires the coordinated action of a three-component machinery, fueled by adenosine triphosphate, 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 nonlinear physics view point, this system explores the active separation of matter at micrometric scales with a dynamical instability between static positioning and traveling wave regimes triggered by the dynamical spontaneous breaking of rotational symmetry.

Reconstitutions of plasmid partition systems and their mechanisms

Bacterial plasmid and chromosome segregation systems ensure that genetic material is efficiently transmitted to progeny cells. Cell-based studies have shed light on the dynamic nature and the molecular basis of plasmid partition systems. In vitro reconstitutions, on the other hand, have proved to be an invaluable tool for studying the minimal components required to elucidate the mechanism of DNA segregation. This allows us to gain insight into the biological and biophysical processes that enable bacterial cells to move and position DNA. Here, we review the reconstitutions of plasmid partition systems in cell-free reactions, and discuss recent work that has begun to challenge long standing models of DNA segregation in bacteria.