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Free-Propagator Reweighting Integrator for Single-Particle Dynamics in Reaction-Diffusion Models of Heterogeneous Protein-Protein Interaction Systems.

Johnson ME, Hummer G - Phys Rev X (2014 Jul-Sep)

Bottom Line: FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone.With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events.Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Biophysics, The Johns Hopkins University, Baltimore, Maryland 21218, USA.

ABSTRACT

We present a new algorithm for simulating reaction-diffusion equations at single-particle resolution. Our algorithm is designed to be both accurate and simple to implement, and to be applicable to large and heterogeneous systems, including those arising in systems biology applications. We combine the use of the exact Green's function for a pair of reacting particles with the approximate free-diffusion propagator for position updates to particles. Trajectory reweighting in our free-propagator reweighting (FPR) method recovers the exact association rates for a pair of interacting particles at all times. FPR simulations of many-body systems accurately reproduce the theoretically known dynamic behavior for a variety of different reaction types. FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone. FPR applications include the modeling of pathways and networks of protein-driven processes where reaction rates can vary widely and thousands of proteins may participate in the formation of large assemblies. With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events. Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.

No MeSH data available.


Green's function for a pair of particles being propagated using Eq. (7) (dark solid lines) and propagated using Eq. (11) (blue dashed lines) with trajectory reweighting. Each curve represents the particle distribution after another time step, top to bottom at r = σ, with particles initially placed close to contact at r0 = σ + 0.05 nm. (a) ka =0.1 nm3/μs, DAB = 5 nm2/μs, and Δt = 0.1 μs. (b) ka = ∞, DAB = 20 nm2/μs, and Δt = 0.1 μs. The deviations between the exact and the free propagators varies depending on the reaction parameters. The FPR algorithm uses trajectory reweighting to recover the association rates that would result if positions were sampled using the exact propagator.
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Figure 2: Green's function for a pair of particles being propagated using Eq. (7) (dark solid lines) and propagated using Eq. (11) (blue dashed lines) with trajectory reweighting. Each curve represents the particle distribution after another time step, top to bottom at r = σ, with particles initially placed close to contact at r0 = σ + 0.05 nm. (a) ka =0.1 nm3/μs, DAB = 5 nm2/μs, and Δt = 0.1 μs. (b) ka = ∞, DAB = 20 nm2/μs, and Δt = 0.1 μs. The deviations between the exact and the free propagators varies depending on the reaction parameters. The FPR algorithm uses trajectory reweighting to recover the association rates that would result if positions were sampled using the exact propagator.

Mentions: Specifically, we seek to simplify the sampling of particle positions while still ensuring that the correct rates of association are preserved. Instead of sampling particle positions from the 3D version of Eq. (7), we sample positions from the free propagator, Eq. (5). Sampling 3D positions according to the free propagator [Eq. (5)] is much simpler. It amounts to adding independent Gaussian random numbers of mean zero and standard deviation to each of the three Cartesian coordinates of a particle with diffusion coefficient DA. The two distributions sampled by the free propagator and the reactive propagator are, in general, different until the particles move away from the reactive boundary (r ≫ σ), where the dynamics is accurately described by free diffusion [Eq. (5)]. In Fig. 2, we compare the propagator for the exact irreversible association with the propagator that uses the free-diffusion distribution to sample positions. At short times and close distances, we find the expected differences between the exact and simulated positional distributions; at longer times and separations, these differences diminish because the overall association rate is essentially exact. As shown below, the correct rates and equilibrium can thus be recovered by reweighting the association probability by the trajectory probabilities. Hence, although the dynamics of position updates will only be approximate at short separations, the rate of association will be exact. At larger separations, the dynamics becomes exact as well.


Free-Propagator Reweighting Integrator for Single-Particle Dynamics in Reaction-Diffusion Models of Heterogeneous Protein-Protein Interaction Systems.

Johnson ME, Hummer G - Phys Rev X (2014 Jul-Sep)

Green's function for a pair of particles being propagated using Eq. (7) (dark solid lines) and propagated using Eq. (11) (blue dashed lines) with trajectory reweighting. Each curve represents the particle distribution after another time step, top to bottom at r = σ, with particles initially placed close to contact at r0 = σ + 0.05 nm. (a) ka =0.1 nm3/μs, DAB = 5 nm2/μs, and Δt = 0.1 μs. (b) ka = ∞, DAB = 20 nm2/μs, and Δt = 0.1 μs. The deviations between the exact and the free propagators varies depending on the reaction parameters. The FPR algorithm uses trajectory reweighting to recover the association rates that would result if positions were sampled using the exact propagator.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4440698&req=5

Figure 2: Green's function for a pair of particles being propagated using Eq. (7) (dark solid lines) and propagated using Eq. (11) (blue dashed lines) with trajectory reweighting. Each curve represents the particle distribution after another time step, top to bottom at r = σ, with particles initially placed close to contact at r0 = σ + 0.05 nm. (a) ka =0.1 nm3/μs, DAB = 5 nm2/μs, and Δt = 0.1 μs. (b) ka = ∞, DAB = 20 nm2/μs, and Δt = 0.1 μs. The deviations between the exact and the free propagators varies depending on the reaction parameters. The FPR algorithm uses trajectory reweighting to recover the association rates that would result if positions were sampled using the exact propagator.
Mentions: Specifically, we seek to simplify the sampling of particle positions while still ensuring that the correct rates of association are preserved. Instead of sampling particle positions from the 3D version of Eq. (7), we sample positions from the free propagator, Eq. (5). Sampling 3D positions according to the free propagator [Eq. (5)] is much simpler. It amounts to adding independent Gaussian random numbers of mean zero and standard deviation to each of the three Cartesian coordinates of a particle with diffusion coefficient DA. The two distributions sampled by the free propagator and the reactive propagator are, in general, different until the particles move away from the reactive boundary (r ≫ σ), where the dynamics is accurately described by free diffusion [Eq. (5)]. In Fig. 2, we compare the propagator for the exact irreversible association with the propagator that uses the free-diffusion distribution to sample positions. At short times and close distances, we find the expected differences between the exact and simulated positional distributions; at longer times and separations, these differences diminish because the overall association rate is essentially exact. As shown below, the correct rates and equilibrium can thus be recovered by reweighting the association probability by the trajectory probabilities. Hence, although the dynamics of position updates will only be approximate at short separations, the rate of association will be exact. At larger separations, the dynamics becomes exact as well.

Bottom Line: FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone.With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events.Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Biophysics, The Johns Hopkins University, Baltimore, Maryland 21218, USA.

ABSTRACT

We present a new algorithm for simulating reaction-diffusion equations at single-particle resolution. Our algorithm is designed to be both accurate and simple to implement, and to be applicable to large and heterogeneous systems, including those arising in systems biology applications. We combine the use of the exact Green's function for a pair of reacting particles with the approximate free-diffusion propagator for position updates to particles. Trajectory reweighting in our free-propagator reweighting (FPR) method recovers the exact association rates for a pair of interacting particles at all times. FPR simulations of many-body systems accurately reproduce the theoretically known dynamic behavior for a variety of different reaction types. FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone. FPR applications include the modeling of pathways and networks of protein-driven processes where reaction rates can vary widely and thousands of proteins may participate in the formation of large assemblies. With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events. Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.

No MeSH data available.