Conformation of a coarse-grained protein chain (an aspartic acid protease) model in effective solvent by a bond-fluctuating Monte Carlo simulation

In a coarse-grained description of a protein chain, all of the 20 amino acid residues can be broadly divided into three groups: Hydrophobic $\left(H\right)$, polar $\left(P\right)$, and electrostatic $\left(E\right)$. A protein can be described by nodes tethered in a chain with a node representing an amino acid group. Aspartic acid protease consists of 99 residues in a well-defined sequence of $H$, $P$, and $E$ nodes tethered together by fluctuating bonds. The protein chain is placed on a cubic lattice where empty lattice sites constitute an effective solvent medium. The amino groups (nodes) interact with the solvent $\left(S\right)$ sites with appropriate attractive $\left(PS\right)$ and repulsive $\left(HS\right)$ interactions with the solvent and execute their stochastic movement with the Metropolis algorithm. Variations of the root mean square displacements of the center of mass and that of its center node of the protease chain and its gyration radius with the time steps are examined for different solvent strength. The structure of the protease swells on increasing the solvent interaction strength which tends to enhance the relaxation time to reach the diffusive behavior of the chain. Equilibrium radius of gyration increases linearly on increasing the solvent strength: A slow rate of increase in weak solvent regime is followed by a faster swelling in stronger solvent. Variation of the gyration radius with the time steps suggests that the protein chain moves via contraction and expansion in a somewhat quasiperiodic pattern particularly in strong solvent.

Figure 1

(Color online) Coarse grained description of aspartic acid protease (1DIFA) with $H$ (pink), $P$ (gold), and $E$ (blue).

Figure 2

(Color online) Variation of the rms displacement of the center node with the time step $t$ for hydrophobic $\left(H\right)$ and polar $\left(P\right)$ chains (left) and $HP$ block-copolymer and aspartic acid (AS) chain (right) for various solvent media characterized by the interaction strengths $(\epsilon =0.0–2.0)$. Lattice size ${60}^3$ with 50 independent samples is used for each chain.

Figure 3

(Color online) Variation of the rms displacement of the center node $\left(R_n\right)$ and center of mass $\left(R_c\right)$ of the polar chain on a ${100}^3$ lattice at $\epsilon =2.0$ with 200 independent samples.

Figure 4

(Color online) Variation of the rms displacement of the center node $\left(R_n\right)$ and center of mass $\left(R_c\right)$ of the polar chain on a ${30}^3$ lattice at $\epsilon =2.0$ with 500 independent samples.

Figure 5

(Color online) Variation of the rms displacement of the center node ($R_n$, left) of aspartic acid chain and that of its center of mass ($R_n$, right) with the time step $t$ for various solvent media characterized by the interaction strengths $(\epsilon =0.0–3.2)$. Lattice size ${60}^3$ with 50–2000 independent samples is used in each solvent medium.

Figure 6

(Color online) Variation of the rms displacement of the center node $\left(R_n\right)$ and center of mass $\left(R_c\right)$ of the aspartic acid chain at $\epsilon =3.2$ on a ${100}^3$ lattice with 200 independent samples.

Figure 7

(Color online) Variation of the rms displacement of the center node $\left(R_n\right)$ and center of mass $\left(R_c\right)$ of the aspartic acid chain on a ${30}^3$ lattice at $\epsilon =3.2$ with 200 independent samples.

Figure 8

(Color online) Variation of the radius of gyration $\left(R_g\right)$ with the time step $t$ for hydrophobic $\left(H\right)$ and polar $\left(P\right)$ chains (left) and $HP$ block-copolymer and aspartic acid chain (right) for various solvent media characterized by the interaction strengths $(\epsilon =0.0–2.0)$. Lattice of size ${60}^3$ with 50 independent samples is used for each chain.

Figure 9

Equilibrium radius of gyration $\left(R_{gs}\right)$ vs interaction strength. Inset shows the variation of the radius of gyration $\left(R_g\right)$ of the aspartic acid chain and its end-to-end distance $\left(R_e\right)$ in the long time regime for the solvent interaction $(\epsilon =3.2)$. Statistics is the same as Figs. 2, 3, 4.

Figure 10

Variation of the radius of gyration $\left(R_g\right)$ with the time step $\left(t\right)$ on a ${100}^3$ sample in a single run with time interval of ${10}^2$ between consecutive data points at different $\epsilon =e$.

Figure 11

Variation of the radius of gyration $\left(R_g\right)$ with the time step $\left(t\right)$ on a ${100}^3$ sample in a single run with time interval of 10 between consecutive data points at different $\epsilon =e$.