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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.


Probability of two particles having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Results for the exact propagator are shown as black dashed lines [Eq. (12)], where the binding radius is σ = 1 nm. The probability of associating from FPR simulations is shown for DAB = 1 nm2/μs (blue), DAB = 20 nm2/μs (green), and DAB = 200 nm2/μs (orange). (a) Results for perfectly absorbing boundary (ka=∞) and a time step of Δt = 0.1 μs. Simulations are averaged over 106 trajectories each initialized to the prescribed r0. The inset plots the difference between exact and simulated results [pexact(r, t/r0) – psim(r; t/r0)]. The simulated results marginally underestimate the association rate due to incomplete trajectory sampling, with errors being largest for slow diffusion. (b) Same as (a) but with radiation BCs at σ = 1 nm with ka = 10 nm3/μs. For this lower rate (relative to absorbing where ka = ∞), the error is smaller for all DAB.
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Figure 1: Probability of two particles having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Results for the exact propagator are shown as black dashed lines [Eq. (12)], where the binding radius is σ = 1 nm. The probability of associating from FPR simulations is shown for DAB = 1 nm2/μs (blue), DAB = 20 nm2/μs (green), and DAB = 200 nm2/μs (orange). (a) Results for perfectly absorbing boundary (ka=∞) and a time step of Δt = 0.1 μs. Simulations are averaged over 106 trajectories each initialized to the prescribed r0. The inset plots the difference between exact and simulated results [pexact(r, t/r0) – psim(r; t/r0)]. The simulated results marginally underestimate the association rate due to incomplete trajectory sampling, with errors being largest for slow diffusion. (b) Same as (a) but with radiation BCs at σ = 1 nm with ka = 10 nm3/μs. For this lower rate (relative to absorbing where ka = ∞), the error is smaller for all DAB.

Mentions: In the FPR approach, we combine the use of the GF to evaluate particle interactions with positions that are sampled from the much simpler free-diffusion propagator to produce a method that extends readily to diverse systems while still maintaining high accuracy, as we detail below. The simple position updates also means our method could readily be incorporated into existing BD software. The error we introduce into the dynamics, and, therefore, the association rates of the particles by using the free propagator, can be measured at each time step and corrected for using a simple trajectory reweighting approach. With this trajectory reweighting, our FPR method will recover the exact association rates between reactants at each time step, as shown in Fig. 1, relative to the exact solution for various reaction conditions. By sampling positions from the free propagator, we dramatically simplify the position updates, which only require selecting Gaussian random numbers with an extra step to ensure the particles do not overlap. The simplicity of the particle position updates means we do not have to precalculate or store any large arrays and, therefore, allows us to readily expand the systems to a variety of reaction types. Additionally, this allows us to adapt to changes to the diffusion constant that might occur with complex formation during the simulations, or changes to reaction rates. Evaluating the association probability given by the GF simply requires the evaluation of the function at a particular position, so no inversion or sampling is necessary. The sampling efficiency of the FPR reweighting approach is optimized by ensuring that the average reweighting ratio upon exit from a reaction zone equals one by construction. While the particle dynamics is only approximate when approaching a reactive partner, by reproducing the exact association rates, we show that the correct behavior of both pairwise and many-body systems is still recovered for reaction types ranging from the reaction-rate limited to the diffusion-limited regime.


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)

Probability of two particles having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Results for the exact propagator are shown as black dashed lines [Eq. (12)], where the binding radius is σ = 1 nm. The probability of associating from FPR simulations is shown for DAB = 1 nm2/μs (blue), DAB = 20 nm2/μs (green), and DAB = 200 nm2/μs (orange). (a) Results for perfectly absorbing boundary (ka=∞) and a time step of Δt = 0.1 μs. Simulations are averaged over 106 trajectories each initialized to the prescribed r0. The inset plots the difference between exact and simulated results [pexact(r, t/r0) – psim(r; t/r0)]. The simulated results marginally underestimate the association rate due to incomplete trajectory sampling, with errors being largest for slow diffusion. (b) Same as (a) but with radiation BCs at σ = 1 nm with ka = 10 nm3/μs. For this lower rate (relative to absorbing where ka = ∞), the error is smaller for all DAB.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 1: Probability of two particles having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Results for the exact propagator are shown as black dashed lines [Eq. (12)], where the binding radius is σ = 1 nm. The probability of associating from FPR simulations is shown for DAB = 1 nm2/μs (blue), DAB = 20 nm2/μs (green), and DAB = 200 nm2/μs (orange). (a) Results for perfectly absorbing boundary (ka=∞) and a time step of Δt = 0.1 μs. Simulations are averaged over 106 trajectories each initialized to the prescribed r0. The inset plots the difference between exact and simulated results [pexact(r, t/r0) – psim(r; t/r0)]. The simulated results marginally underestimate the association rate due to incomplete trajectory sampling, with errors being largest for slow diffusion. (b) Same as (a) but with radiation BCs at σ = 1 nm with ka = 10 nm3/μs. For this lower rate (relative to absorbing where ka = ∞), the error is smaller for all DAB.
Mentions: In the FPR approach, we combine the use of the GF to evaluate particle interactions with positions that are sampled from the much simpler free-diffusion propagator to produce a method that extends readily to diverse systems while still maintaining high accuracy, as we detail below. The simple position updates also means our method could readily be incorporated into existing BD software. The error we introduce into the dynamics, and, therefore, the association rates of the particles by using the free propagator, can be measured at each time step and corrected for using a simple trajectory reweighting approach. With this trajectory reweighting, our FPR method will recover the exact association rates between reactants at each time step, as shown in Fig. 1, relative to the exact solution for various reaction conditions. By sampling positions from the free propagator, we dramatically simplify the position updates, which only require selecting Gaussian random numbers with an extra step to ensure the particles do not overlap. The simplicity of the particle position updates means we do not have to precalculate or store any large arrays and, therefore, allows us to readily expand the systems to a variety of reaction types. Additionally, this allows us to adapt to changes to the diffusion constant that might occur with complex formation during the simulations, or changes to reaction rates. Evaluating the association probability given by the GF simply requires the evaluation of the function at a particular position, so no inversion or sampling is necessary. The sampling efficiency of the FPR reweighting approach is optimized by ensuring that the average reweighting ratio upon exit from a reaction zone equals one by construction. While the particle dynamics is only approximate when approaching a reactive partner, by reproducing the exact association rates, we show that the correct behavior of both pairwise and many-body systems is still recovered for reaction types ranging from the reaction-rate limited to the diffusion-limited regime.

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.