Periodic Reporting for period 1 - PomXYZ (The PomXYZ cluster in the bacterium Myxococcus xanthus: Self-assembly, translocation, and fission of an active protein complex that guides cell division)
Reporting period: 2022-10-01 to 2024-09-30
Subcellular self-organization is fundamentally necessary for cellular life, and in particular for cell division. There, the future cell division site at midcell needs to be marked for the cell division machinery. In the rod-shaped bacterium Myxococcus xanthus, the proteins PomX, PomY, and PomZ play a key role in this process. These proteins self-assemble to form a protein cluster on the cell nucleoid. Driven by a non-equilibrium reaction cycle of PomZ, the cluster then localizes at the cell midpoint, where it recruits the cell-division machinery. Upon division of the cell, also the cluster divides; on each of the respective daughter cells, the cluster then in turn translocates to the cell midpoint to again mark the cell division site.
A key aspect of the cluster translocation mechanism is the coupling of spatial dynamics with reactions. More specifically, PomZ proteins can exist in either an activated (dimeric, ATP-bound) state or a deactivated (monomeric) state. In the absence of PomX/PomY, PomZ is preferably in its activated form; the presence of PomX stimulates the deactivation of PomZ, whereby the bound ATP is hydrolized to ADP; the resulting PomZ monomers then diffusive in the cytosol, where after some time they again form ATP-bound dimers. This reaction cycle consumes energy, and the interaction of the ATP-bound PomZ dimers with the PomX/PomY cluster generates an effective force that drives the cluster towards midcell.
The objective of the project was to develop and study a mesoscopic model for the cell cycle of M. xanthus, and in particular to quantify the constraints imposed on the properties and interactions of the participating proteins. To account for the non-equilibrium PomZ reaction cycle described in the previous paragraph, the coupling of spatial dynamics with reactions is a key feature of our modeling. More explicitly, we consider overdamped stochastic dynamics of particles that can change their internal state (i.e. interaction properties) via reactions; to include the possibility of locally stimulated reactions, the reaction rates can depend on the local neighborhood of a particle. Therefore, in this particle-based reaction-diffusion model, both the spatial dynamics depends on the internal state of a particle, and the reaction rates at which the internal state changes depend on the spatial configuration.
A key aspect of the cluster translocation mechanism is the coupling of spatial dynamics with reactions. More specifically, PomZ proteins can exist in either an activated (dimeric, ATP-bound) state or a deactivated (monomeric) state. In the absence of PomX/PomY, PomZ is preferably in its activated form; the presence of PomX stimulates the deactivation of PomZ, whereby the bound ATP is hydrolized to ADP; the resulting PomZ monomers then diffusive in the cytosol, where after some time they again form ATP-bound dimers. This reaction cycle consumes energy, and the interaction of the ATP-bound PomZ dimers with the PomX/PomY cluster generates an effective force that drives the cluster towards midcell.
The objective of the project was to develop and study a mesoscopic model for the cell cycle of M. xanthus, and in particular to quantify the constraints imposed on the properties and interactions of the participating proteins. To account for the non-equilibrium PomZ reaction cycle described in the previous paragraph, the coupling of spatial dynamics with reactions is a key feature of our modeling. More explicitly, we consider overdamped stochastic dynamics of particles that can change their internal state (i.e. interaction properties) via reactions; to include the possibility of locally stimulated reactions, the reaction rates can depend on the local neighborhood of a particle. Therefore, in this particle-based reaction-diffusion model, both the spatial dynamics depends on the internal state of a particle, and the reaction rates at which the internal state changes depend on the spatial configuration.
An appropriate model for interacting proteins with localized binding sites are patchy colloids, where each patch corresponds to a binding site; to account for reactions, the binding behavior of a site can be dynamically altered (possibly via locally stimulated reactions). In the course of the project, we have implemented an open-source python module for particle-based reaction-diffusion dynamics, which comes with example files for simulating patchy colloids with locally stimulated binding-site reactions.
Using this module, we have performed extensive particle-based simulations of a finite system that represents the cell interior, and which contains a PomXY cluster that couples to reactive PomZ. From our numerical results we have identified a parameter regime for the (spatially constant) PomZ activation rate and the (locally stimulated) PomZ deactivation rate at which the continuous non-equilibrium turnover of PomZ leads to a significant midcell-localization effect for the Pom cluster.
To go beyond particle-based numerical simulations, we have performed an analytical coarse-graining of the PomZ dynamics, leading to a model in which the PomXY cluster couples to two fields (one field represents activated PomZ, the second field represents deactivated PomZ). Simulations of this coupled particle-field model quantitatively reproduce the confinement observed in our particle-only simulations.
In a second step we further reduced the coarse-grained particle-field description to an evolution equation for a single particle, which represents the PomXY cluster, in a potential. The potential describes the effective confinement due to the interaction of the PomXY cluster with PomZ, and we have obtained it by integrating out the PomZ field dynamics using a perturbation approach. The resulting effective potential is consistent with the results of the particle-only simulations.
The attached figure illustrates the two-step coarse-graining described in the above paragraphs.
An important practical aspect of working with particle-based simulations and coarse-graining is the parametrization of low-dimensional stochastic dynamics models from observed time series. This typically progresses by learning the model parameters of analytically calculated observables using observations thereof, which often requires knowledge of the propagator of the to-be-parametrized stochastic dynamics model. We made progress in this direction by perturbatively calculating the short-time evolution of overdamped Langevin dynamics to in principle arbitrary accuracy.
Using this module, we have performed extensive particle-based simulations of a finite system that represents the cell interior, and which contains a PomXY cluster that couples to reactive PomZ. From our numerical results we have identified a parameter regime for the (spatially constant) PomZ activation rate and the (locally stimulated) PomZ deactivation rate at which the continuous non-equilibrium turnover of PomZ leads to a significant midcell-localization effect for the Pom cluster.
To go beyond particle-based numerical simulations, we have performed an analytical coarse-graining of the PomZ dynamics, leading to a model in which the PomXY cluster couples to two fields (one field represents activated PomZ, the second field represents deactivated PomZ). Simulations of this coupled particle-field model quantitatively reproduce the confinement observed in our particle-only simulations.
In a second step we further reduced the coarse-grained particle-field description to an evolution equation for a single particle, which represents the PomXY cluster, in a potential. The potential describes the effective confinement due to the interaction of the PomXY cluster with PomZ, and we have obtained it by integrating out the PomZ field dynamics using a perturbation approach. The resulting effective potential is consistent with the results of the particle-only simulations.
The attached figure illustrates the two-step coarse-graining described in the above paragraphs.
An important practical aspect of working with particle-based simulations and coarse-graining is the parametrization of low-dimensional stochastic dynamics models from observed time series. This typically progresses by learning the model parameters of analytically calculated observables using observations thereof, which often requires knowledge of the propagator of the to-be-parametrized stochastic dynamics model. We made progress in this direction by perturbatively calculating the short-time evolution of overdamped Langevin dynamics to in principle arbitrary accuracy.
Our software for particle-based reaction-diffusion dynamics allows to simulate patchy colloids with reactive patches. Such models can be used to simulate proteins with complex and reactive binding behavior (such as the Pom proteins). We believe that the potential impact of the tools developed here will be maximized by first studying the general collective properties of patchy-colloids with reactive patches, and subsequently using those insights to study concrete biophysical protein complexes.
Our coarse-grained description of the PomXY cluster as a single particle in an effective potential quantitatively reproduces the PomXY cluster dynamics observed in our many-particle simulations. Since it is computationally significantly less expensive, our coarse-grained model allows us to evaluate the cluster confinement to midcell as a function of the system parameters orders of magnitude faster as compared to direct simulations; this in turn allows to determine more efficiently which system parameter ranges are consistent with experimental observations. Our coarse-graining procedure for many-particle systems is general, and as such will be applicable also to other biophysical many-particle systems in which spatial dynamics couple to reactions. To increase the potential impact, together with our research results we will release a simulation package for simulation coupled particle-field simulations.
Our results on the short-time propagator allow to practically evaluate the short-time dynamics of overdamped Langevin dynamics to unprecedented precision. These results will be useful for researchers that want to evaluate short-time expectation values analytically, or that want to parametrize stochastic dynamics models from data. To allow other researchers to easily build on our results, we have published an accompanying python module that allows to re-derive, evaluate, and extend the results presented in our manuscript.
Overall, our results constitute both general tools for stochastic (many-particle) systems, as well as insights into the specific PomXYZ system. In particular, we believe that further research into patchy-colloid models with reactions, as well as the application of such models to the fission of the cluster in the PomXYZ system, as well as to other biological non-equilibrium protein complexes, will be a fruitful avenue for future research. For our results to be as impactful as possible, we provide software that enables other scientists to reproduce and extend our research.
Our coarse-grained description of the PomXY cluster as a single particle in an effective potential quantitatively reproduces the PomXY cluster dynamics observed in our many-particle simulations. Since it is computationally significantly less expensive, our coarse-grained model allows us to evaluate the cluster confinement to midcell as a function of the system parameters orders of magnitude faster as compared to direct simulations; this in turn allows to determine more efficiently which system parameter ranges are consistent with experimental observations. Our coarse-graining procedure for many-particle systems is general, and as such will be applicable also to other biophysical many-particle systems in which spatial dynamics couple to reactions. To increase the potential impact, together with our research results we will release a simulation package for simulation coupled particle-field simulations.
Our results on the short-time propagator allow to practically evaluate the short-time dynamics of overdamped Langevin dynamics to unprecedented precision. These results will be useful for researchers that want to evaluate short-time expectation values analytically, or that want to parametrize stochastic dynamics models from data. To allow other researchers to easily build on our results, we have published an accompanying python module that allows to re-derive, evaluate, and extend the results presented in our manuscript.
Overall, our results constitute both general tools for stochastic (many-particle) systems, as well as insights into the specific PomXYZ system. In particular, we believe that further research into patchy-colloid models with reactions, as well as the application of such models to the fission of the cluster in the PomXYZ system, as well as to other biological non-equilibrium protein complexes, will be a fruitful avenue for future research. For our results to be as impactful as possible, we provide software that enables other scientists to reproduce and extend our research.