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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.


Related in: MedlinePlus

Probability of two particles with an additional Coulomb interaction, βV(r) = rc/r, having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Blue lines are for an attractive interaction with rc = −1, and pink lines are for a repulsive interaction with rc = +1. The binding radius is σ = 1 nm, DAB = 50 nm2/μs, ka = 50 nm3/μs, and Δt = 0.01 μs. The black dashed lines represent simulations propagated using the “exact” numerically solved GF to update the particle separations. The extremely close agreement between the reweighted probabilities and the exact GF updates demonstrates that trajectory overlap is not an issue for this time step. These simulations are averaged over 107 trajectories, each initialized to the prescribed r0. The gray lines represent the association probability calculated using the BD scheme of Zhou [31]. These BD simulations are much slower than the FPR simulations due to the very short time step necessary, and for r0 > 1 are propagated out to only t = 1 μs for 3 × 105 trajectories. For r0 = 1, we collected 105 trajectories out to 10 μs.
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Figure 7: Probability of two particles with an additional Coulomb interaction, βV(r) = rc/r, having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Blue lines are for an attractive interaction with rc = −1, and pink lines are for a repulsive interaction with rc = +1. The binding radius is σ = 1 nm, DAB = 50 nm2/μs, ka = 50 nm3/μs, and Δt = 0.01 μs. The black dashed lines represent simulations propagated using the “exact” numerically solved GF to update the particle separations. The extremely close agreement between the reweighted probabilities and the exact GF updates demonstrates that trajectory overlap is not an issue for this time step. These simulations are averaged over 107 trajectories, each initialized to the prescribed r0. The gray lines represent the association probability calculated using the BD scheme of Zhou [31]. These BD simulations are much slower than the FPR simulations due to the very short time step necessary, and for r0 > 1 are propagated out to only t = 1 μs for 3 × 105 trajectories. For r0 = 1, we collected 105 trajectories out to 10 μs.

Mentions: The size of the reaction zone, whether or not a potential is present, needs to be large enough to account for the effects of the reactive barrier at r = σ. Outside of the reaction zone, the particles will have essentially zero probability of reaching the reactive barrier (by construction) and, therefore, their dynamics will correctly be described by the solution to the diffusion equation without any reactive BC. For the case of no potential or a shorter-range or truncated potential that does not extend beyond the reaction zone, this solution is given by the free-diffusion propagator in Eq. (5). However, if the potential is long range, such that it acts even beyond the reaction zone separation, there are two additional considerations. First, if the force at these longer ranges still varies significantly over the length of the diffusional displacement, then sampling from the free propagator with drift [Eq. (24)] may not agree with the true irreversible propagator, and should therefore be reweighted. This could be accounted for by extending the reaction zone until the two distributions match and/or shortening the time step. Second, and more critically, accounting for these long-range forces means one must measure the effects of pairwise forces well beyond the reaction zone. This presents no issues for a single particle pair, and in Fig. 7, we show that trajectory reweighting works just as well in the presence of either a repulsive or an attractive Coulomb potential.


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 with an additional Coulomb interaction, βV(r) = rc/r, having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Blue lines are for an attractive interaction with rc = −1, and pink lines are for a repulsive interaction with rc = +1. The binding radius is σ = 1 nm, DAB = 50 nm2/μs, ka = 50 nm3/μs, and Δt = 0.01 μs. The black dashed lines represent simulations propagated using the “exact” numerically solved GF to update the particle separations. The extremely close agreement between the reweighted probabilities and the exact GF updates demonstrates that trajectory overlap is not an issue for this time step. These simulations are averaged over 107 trajectories, each initialized to the prescribed r0. The gray lines represent the association probability calculated using the BD scheme of Zhou [31]. These BD simulations are much slower than the FPR simulations due to the very short time step necessary, and for r0 > 1 are propagated out to only t = 1 μs for 3 × 105 trajectories. For r0 = 1, we collected 105 trajectories out to 10 μs.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 7: Probability of two particles with an additional Coulomb interaction, βV(r) = rc/r, having associated between times 0 and t, given they were separated by a distance r0 at t = 0. Blue lines are for an attractive interaction with rc = −1, and pink lines are for a repulsive interaction with rc = +1. The binding radius is σ = 1 nm, DAB = 50 nm2/μs, ka = 50 nm3/μs, and Δt = 0.01 μs. The black dashed lines represent simulations propagated using the “exact” numerically solved GF to update the particle separations. The extremely close agreement between the reweighted probabilities and the exact GF updates demonstrates that trajectory overlap is not an issue for this time step. These simulations are averaged over 107 trajectories, each initialized to the prescribed r0. The gray lines represent the association probability calculated using the BD scheme of Zhou [31]. These BD simulations are much slower than the FPR simulations due to the very short time step necessary, and for r0 > 1 are propagated out to only t = 1 μs for 3 × 105 trajectories. For r0 = 1, we collected 105 trajectories out to 10 μs.
Mentions: The size of the reaction zone, whether or not a potential is present, needs to be large enough to account for the effects of the reactive barrier at r = σ. Outside of the reaction zone, the particles will have essentially zero probability of reaching the reactive barrier (by construction) and, therefore, their dynamics will correctly be described by the solution to the diffusion equation without any reactive BC. For the case of no potential or a shorter-range or truncated potential that does not extend beyond the reaction zone, this solution is given by the free-diffusion propagator in Eq. (5). However, if the potential is long range, such that it acts even beyond the reaction zone separation, there are two additional considerations. First, if the force at these longer ranges still varies significantly over the length of the diffusional displacement, then sampling from the free propagator with drift [Eq. (24)] may not agree with the true irreversible propagator, and should therefore be reweighted. This could be accounted for by extending the reaction zone until the two distributions match and/or shortening the time step. Second, and more critically, accounting for these long-range forces means one must measure the effects of pairwise forces well beyond the reaction zone. This presents no issues for a single particle pair, and in Fig. 7, we show that trajectory reweighting works just as well in the presence of either a repulsive or an attractive Coulomb potential.

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.


Related in: MedlinePlus