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Minimizing memory as an objective for coarse-graining
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Image of FIG. 1.
FIG. 1.

Plot of λ = 2I 1/I 0 for β(t) = Ae tcos (ωt). λ is a good approximation for τ in the region ξ ≫ 1.

Image of FIG. 2.
FIG. 2.

Estimated memory timescale λ = 2I 1/I 0 as a function of the mapping parameters for a four-particle linear mass-spring model coarse-grained to a single effective particle using an internal coordinate style mapping. Coarse-grained variables corresponding to eigenmodes of the fine-grained system are marked by red upward triangles and optima of the estimated memory timescale are marked by green downward triangles. The function is plotted relative to its minimum attained value, 1.29. The memory-optimal coarse-grainings are quite close to the eigenmodes of the system, as conjectured, despite statistical errors in the estimation of the memory timescale and the sensitivity of the minima to noise.

Image of FIG. 3.
FIG. 3.

Coarse-grainings for the nonlinear molecular system. The angle terms involving the bond between atoms 7 and 8 (red) are varied as described in the text. Particles are shaded according to their groupings into effective particles. (a) Memory-optimal coarse-graining when the maps are required to divide the fine-grained particles evenly between the two effective particles and one of the particles must contain a contiguous segment of bonded atoms; (b) a coarse-graining with identical memory timescale to (a) when angles involving bond 7-8 are stiffer than others in the system; (c) an arbitrarily chosen coarse-graining used as a comparison to the memory-optimal ones.

Image of FIG. 4.
FIG. 4.

Comparison for the nonlinear molecular model of the memory heuristic λ = 2I 1/I 0 to the standard deviation of the fine-grained forces in each bin of the coarse-grained interaction, weighted by how often the system is observed in each bin. Each point corresponds to one possible mapping; data shown are for coarse-grainings of models with central bond angles that are more flexible (B 78k = 0.1) (red circles), as flexible (B 78k = 1) (green upward triangles), and less flexible (B 78k = 2) (blue downward triangles) than all other angles. (Inset) Same comparison for the case that the coarse-grained sites are not constrained to be the same size.

Image of FIG. 5.
FIG. 5.

Comparison between coarse-grained trajectories projected from the fine-grained nonlinear molecular model and generated directly from GLE models for mappings 1111111100000000 (a), 1110110000000000 (b), and 0001111111001100 (c), illustrated in Figs. 3(a)–3(c) , respectively.

Image of FIG. 6.
FIG. 6.

Comparison between the memory function and the autocorrelation of the force residual for the coarse-graining 1111111100000000 of the nonlinear molecular model. Note that λ = 2I 1/I 0 is independent of the vertical scale.

Image of FIG. 7.
FIG. 7.

Convergence of the Monte Carlo scheme used to search for memory-optimal mappings for heteroENM models of the actin monomer. In the contiguous case, effective particles are composed of atoms of residues that are contiguous in sequence, while in the discontiguous case, no such restriction is placed on the mappings.

Image of FIG. 8.
FIG. 8.

Structural comparison of coarse-grainings of an actin monomer: handpicked (a), memory-optimal sequence contiguous (b), and essential dynamics (c). Sequence ranges for colors are given in Fig. 9 .

Image of FIG. 9.
FIG. 9.

Sequential comparison of coarse-grainings of an actin monomer. Colors correspond to Fig. 8 . Effective particles in the memory-optimal coarse-graining that correspond to subdivisions of effective particles in the hand-picked coarse-graining are indicated in bold font and discussed further in the text.


Generic image for table
Table I.

Relative error in positions between projected fine-grained dynamics and coarse-grained model dynamics (χ) at two time intervals (δt) for the indicated coarse-grainings of the nonlinear molecular model.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Minimizing memory as an objective for coarse-graining