^{1}and Pablo G. Debenedetti

^{1,a)}

### Abstract

We present an expression for the calculation of microscopic stresses in molecular simulation, which is compatible with the use of electrostatic lattice sums such as the Ewald sum, with the presence of many-body interactions, and which allows local stresses to be calculated on surfaces of arbitrarily complex shape. The ultimate goal of this work is to investigate microscopic stresses on proteins in glassy matrices, which are used in the pharmaceutical industry for the long-term storage and stabilization of labile biomolecules. We demonstrate the formalism's usefulness through selected results on ubiquitin and an α-keratin fragment, in liquid and glassy states. We find that atomic-level normal stresses on hydrophilic side-chains exhibit a similar fingerprint in both proteins, and protein-level normal stresses increase upon vitrification. Both proteins experience compressive stresses of the order of 10^{2} bar in the glassy state.

The financial support of Unilever U.K. Central Resources and the National Science Foundation Graduate Fellowship to H.W.H. is gratefully acknowledged. Computations were performed at the Terascale Infrastructure for Groundbreaking Research in Engineering and Science (TIGRESS) facility at Princeton University.

I. INTRODUCTION

II. LOCAL STRESS TENSOR

A. Local stress tensor derivation

B. Electrostatic potentials: The Ewald sum case

III. SIMULATION METHODS

A. Molecular dynamics

B. Inherent structures

C. Optimization of local stress algorithm

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

### Key Topics

- Proteins
- 65.0
- Tensor methods
- 35.0
- Electrostatics
- 14.0
- Carbohydrates
- 10.0
- Enthalpy
- 7.0

##### A61P

## Figures

Illustration of the local stress tensor calculation, Eq. (19) , for a group containing particles 2 and 5. The volume of the group, V g , is the sum of the Voronoi volumes of particles 2 and 5 shown by thick black lines. By definition of the group, Λ2 = Λ5 = 1 while Λ1 = Λ3 = Λ4 = 0. The dimensionless quantity, l 12 = 0.5, is the fraction of the line segment connecting particles 1 and 2 that is within V g , shown by the thick red line. Note that the double sum of Eq. (19) involves all pairs of particles, which may lead to contributions from particles not within the group (e.g., l 34 ≃ 0.4).

Illustration of the local stress tensor calculation, Eq. (19) , for a group containing particles 2 and 5. The volume of the group, V g , is the sum of the Voronoi volumes of particles 2 and 5 shown by thick black lines. By definition of the group, Λ2 = Λ5 = 1 while Λ1 = Λ3 = Λ4 = 0. The dimensionless quantity, l 12 = 0.5, is the fraction of the line segment connecting particles 1 and 2 that is within V g , shown by the thick red line. Note that the double sum of Eq. (19) involves all pairs of particles, which may lead to contributions from particles not within the group (e.g., l 34 ≃ 0.4).

The STRIDE algorithm ^{120} was used to make secondary structure assignments of ubiquitin, and a typical liquid configuration in water at ambient conditions is shown. The α-helix is the largest helical region shown in blue. The other two blue regions are 310 helices. β-strands are colored yellow. The C-terminus is the coil at the top right.

The STRIDE algorithm ^{120} was used to make secondary structure assignments of ubiquitin, and a typical liquid configuration in water at ambient conditions is shown. The α-helix is the largest helical region shown in blue. The other two blue regions are 310 helices. β-strands are colored yellow. The C-terminus is the coil at the top right.

The 1a coiled-coil region of α-keratin was simulated as a fragment solvated in water. The coiled-coil structure is stabilized by a stripe of hydrophobic side chains which run along the length of both α-helical monomers.

The 1a coiled-coil region of α-keratin was simulated as a fragment solvated in water. The coiled-coil structure is stabilized by a stripe of hydrophobic side chains which run along the length of both α-helical monomers.

(a) Atomic-level normal stresses on ubiquitin in glassy water at 1 bar. Colored squares represent the following hydrophilic side chain atoms: (red) nitrogen on ammonium (−N^{+}H3) groups in lysine and the N-terminus, and (green) carbon on carboxylate (−COO^{−}) groups in aspartic and glutamic acid. Backbone carbonyl carbons are represented by the dashed line, α-carbons by the solid line, and all other atoms appear as dots. Atoms are listed from N-terminus to C-terminus. (b) Atomic-level normal stresses on ubiquitin in liquid water at 1 bar and 300 K. Error bars are the standard deviation of the mean from 500 samples in both liquid and glass.

(a) Atomic-level normal stresses on ubiquitin in glassy water at 1 bar. Colored squares represent the following hydrophilic side chain atoms: (red) nitrogen on ammonium (−N^{+}H3) groups in lysine and the N-terminus, and (green) carbon on carboxylate (−COO^{−}) groups in aspartic and glutamic acid. Backbone carbonyl carbons are represented by the dashed line, α-carbons by the solid line, and all other atoms appear as dots. Atoms are listed from N-terminus to C-terminus. (b) Atomic-level normal stresses on ubiquitin in liquid water at 1 bar and 300 K. Error bars are the standard deviation of the mean from 500 samples in both liquid and glass.

Atomic-level normal stresses for keratin fragment in glassy water at 1 bar. Symbols are as in Figure 4 . Atoms are listed from the N-terminus to the C-terminus for both coils in succession. The first coil ends and the second coil begins in the middle of the figure, denoted by the vertical line. Error bars are the standard deviation of the mean from 500 samples.

Atomic-level normal stresses for keratin fragment in glassy water at 1 bar. Symbols are as in Figure 4 . Atoms are listed from the N-terminus to the C-terminus for both coils in succession. The first coil ends and the second coil begins in the middle of the figure, denoted by the vertical line. Error bars are the standard deviation of the mean from 500 samples.

(a) Residue-level normal stresses for ubiquitin in glassy water at 1 bar. Colored squares represent (blue) aspartic acid and (red) glutamic acid. Residues are listed from N-terminus to C-terminus. Note that the C-terminus residues also possess large, tensile normal stresses, perhaps due to the carboxylate group which is also present in aspartic and glutamic acid. (b) Residue-level normal stresses for keratin fragment in glassy water at 1 bar. The first coil ends and the second coil begins in the middle of the figure, denoted by the vertical line. Error bars are the standard deviation of the mean from 500 samples.

(a) Residue-level normal stresses for ubiquitin in glassy water at 1 bar. Colored squares represent (blue) aspartic acid and (red) glutamic acid. Residues are listed from N-terminus to C-terminus. Note that the C-terminus residues also possess large, tensile normal stresses, perhaps due to the carboxylate group which is also present in aspartic and glutamic acid. (b) Residue-level normal stresses for keratin fragment in glassy water at 1 bar. The first coil ends and the second coil begins in the middle of the figure, denoted by the vertical line. Error bars are the standard deviation of the mean from 500 samples.

The instantaneous normal stress on ubiquitin at 1 bar in the glass (blue) and liquid at 300 K (red). On average, ubiquitin is under compression in the glass and under tension in the liquid (see Table II for ensemble averages).

The instantaneous normal stress on ubiquitin at 1 bar in the glass (blue) and liquid at 300 K (red). On average, ubiquitin is under compression in the glass and under tension in the liquid (see Table II for ensemble averages).

Atomic-level normal stresses on ubiquitin in glassy water at 1 bar calculated with the Coulomb sum, U ^{ c }(ε s = ∞) = U ^{ c }(ε s = 1) + U ^{ p }. Symbols are as in Figure 4 . Error bars are the standard deviation of the mean from 100 samples.

Atomic-level normal stresses on ubiquitin in glassy water at 1 bar calculated with the Coulomb sum, U ^{ c }(ε s = ∞) = U ^{ c }(ε s = 1) + U ^{ p }. Symbols are as in Figure 4 . Error bars are the standard deviation of the mean from 100 samples.

## Tables

Summary of simulations performed in this work.

Summary of simulations performed in this work.

Summary of the normal stress on whole proteins in the glass and liquid.

Summary of the normal stress on whole proteins in the glass and liquid.

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