Limits...
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

Simulations of reversible reactions A + B⇄ C with the FPR method (red and magenta) and with the Gillespie algorithm (black and blue). For both reactions, A and B particles are present at 52 μM concentrations and each have D = 1 nm2/μs with Δt = 0.5 μs. Reaction rates of ka = 1000 nm3/μs, kb = 1 s−1 and ka = 200 nm3/μs, kb = 10 s−1 are shown in red and magenta. The inset shows the relaxation R(t). The full relaxation is not known analytically, but is asymptotically exponential for well-mixed systems [R(t) ∼ exp{–t[kon(Aeq + Beq) + koff]}], represented by the solid gray lines with best-fit intercepts. The close agreement indicates that for these solution conditions, diffusion can accurately be represented as a constant contribution to the on rate. The transition to power-law relaxation for the RD systems happens at R(t) ∼ 10−5, beyond the statistical accuracy of these simulations.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4440698&req=5

Figure 11: Simulations of reversible reactions A + B⇄ C with the FPR method (red and magenta) and with the Gillespie algorithm (black and blue). For both reactions, A and B particles are present at 52 μM concentrations and each have D = 1 nm2/μs with Δt = 0.5 μs. Reaction rates of ka = 1000 nm3/μs, kb = 1 s−1 and ka = 200 nm3/μs, kb = 10 s−1 are shown in red and magenta. The inset shows the relaxation R(t). The full relaxation is not known analytically, but is asymptotically exponential for well-mixed systems [R(t) ∼ exp{–t[kon(Aeq + Beq) + koff]}], represented by the solid gray lines with best-fit intercepts. The close agreement indicates that for these solution conditions, diffusion can accurately be represented as a constant contribution to the on rate. The transition to power-law relaxation for the RD systems happens at R(t) ∼ 10−5, beyond the statistical accuracy of these simulations.

Mentions: For solutions of reversibly reacting particles that are at comparable concentrations, the full time-dependent relaxation has not been determined analytically. Only the long-time asymptotics on approach to equilibrium have been shown to be exponential in time for chemical kinetics, and power law for RD systems [34,35]. In Fig. 11, we compare simulations of moderate concentrations of A and B particles that reversibly react. For these solution conditions, the results agree quite well with the behavior of chemical kinetics, solved using the Gillespie algorithm [36] using kon and koff defined above. This indicates that the effects of diffusion on the association rate over the time scales shown can accurately be absorbed into an overall rate kon that accounts for both diffusion to contact and reaction at contact and equivalently for dissociation. The long-time asymptotics on approach to equilibrium for the RD system will eventually switch to obeying a power-law dependence on time. The switch to power-law relaxation occurs because spatial domains form as the system relaxes and the diffusion of particles to one another over these domains becomes the limiting step in the association between particles as equilibrium is approached. (Consequently, if the box is too small for domains to form, the relaxation will remain exponential.) For the solution conditions in Fig. 11, the simulation systems are not large enough to sensitively measure the onset of this power-law behavior, which begins at ∼0.15 s for ka = 200 nm3/μs and at ∼1 s for ka = 1000 nm3/μs, requiring high precision (10−5). Instead, in Fig. 12, we show simulations at the same moderate concentrations of A and B particles but with much faster equilibration (kb = 106 s−1). For these conditions, the power-law asymptotic behavior derived in Ref. [34] is visible and contrasts with the exponential decay observed in chemical kinetics (solved using the Gillespie algorithm [36]).


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)

Simulations of reversible reactions A + B⇄ C with the FPR method (red and magenta) and with the Gillespie algorithm (black and blue). For both reactions, A and B particles are present at 52 μM concentrations and each have D = 1 nm2/μs with Δt = 0.5 μs. Reaction rates of ka = 1000 nm3/μs, kb = 1 s−1 and ka = 200 nm3/μs, kb = 10 s−1 are shown in red and magenta. The inset shows the relaxation R(t). The full relaxation is not known analytically, but is asymptotically exponential for well-mixed systems [R(t) ∼ exp{–t[kon(Aeq + Beq) + koff]}], represented by the solid gray lines with best-fit intercepts. The close agreement indicates that for these solution conditions, diffusion can accurately be represented as a constant contribution to the on rate. The transition to power-law relaxation for the RD systems happens at R(t) ∼ 10−5, beyond the statistical accuracy of these simulations.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 11: Simulations of reversible reactions A + B⇄ C with the FPR method (red and magenta) and with the Gillespie algorithm (black and blue). For both reactions, A and B particles are present at 52 μM concentrations and each have D = 1 nm2/μs with Δt = 0.5 μs. Reaction rates of ka = 1000 nm3/μs, kb = 1 s−1 and ka = 200 nm3/μs, kb = 10 s−1 are shown in red and magenta. The inset shows the relaxation R(t). The full relaxation is not known analytically, but is asymptotically exponential for well-mixed systems [R(t) ∼ exp{–t[kon(Aeq + Beq) + koff]}], represented by the solid gray lines with best-fit intercepts. The close agreement indicates that for these solution conditions, diffusion can accurately be represented as a constant contribution to the on rate. The transition to power-law relaxation for the RD systems happens at R(t) ∼ 10−5, beyond the statistical accuracy of these simulations.
Mentions: For solutions of reversibly reacting particles that are at comparable concentrations, the full time-dependent relaxation has not been determined analytically. Only the long-time asymptotics on approach to equilibrium have been shown to be exponential in time for chemical kinetics, and power law for RD systems [34,35]. In Fig. 11, we compare simulations of moderate concentrations of A and B particles that reversibly react. For these solution conditions, the results agree quite well with the behavior of chemical kinetics, solved using the Gillespie algorithm [36] using kon and koff defined above. This indicates that the effects of diffusion on the association rate over the time scales shown can accurately be absorbed into an overall rate kon that accounts for both diffusion to contact and reaction at contact and equivalently for dissociation. The long-time asymptotics on approach to equilibrium for the RD system will eventually switch to obeying a power-law dependence on time. The switch to power-law relaxation occurs because spatial domains form as the system relaxes and the diffusion of particles to one another over these domains becomes the limiting step in the association between particles as equilibrium is approached. (Consequently, if the box is too small for domains to form, the relaxation will remain exponential.) For the solution conditions in Fig. 11, the simulation systems are not large enough to sensitively measure the onset of this power-law behavior, which begins at ∼0.15 s for ka = 200 nm3/μs and at ∼1 s for ka = 1000 nm3/μs, requiring high precision (10−5). Instead, in Fig. 12, we show simulations at the same moderate concentrations of A and B particles but with much faster equilibration (kb = 106 s−1). For these conditions, the power-law asymptotic behavior derived in Ref. [34] is visible and contrasts with the exponential decay observed in chemical kinetics (solved using the Gillespie algorithm [36]).

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