Figure 1

(Color online) (a) All-atom representation of the barnase-barstar complex surrounded by crowders. (b) Results of $\Delta \mu$ for the protein complex. Symbols are obtained in our previous work [26] by the insertion procedure, with results for crowder radii of 15, 20, 30, and $50\text{Å}$ shown as circles, squares, diamonds, and triangles, respectively, while curves are predictions of the GFMT. In all GFMT calculations, 10 000 directions were used to define the crowder-exclusion surface; the cubic grid representing the excluded volume had a spacing of $0.5\text{Å}$. At each crowder size, the calculation of $v_p$, $s_p$, and $l_p$ per protein conformation took $\sim 10\text{s}$ of CPU time on an AMD Opteron 2382 processor; subsequently using $v_p$, $s_p$, and $l_p$ in Eq. (1) to calculate $\Delta \mu$ at different crowder volume fractions required negligible amounts of time. In comparison, the aggregate CPU times for four crowder volume fractions at each crowder size were 80–160 s per snapshot of crowder configurations by the insertion procedure; to reduce statistical errors, the calculations for each protein conformation were repeated on 10–100 snapshots of crowder configurations, bringing the total CPU times per protein conformation at a given crowder size to 800–16 000 s. Overall the GFMT gained a speed up of $\sim {10}^2$ to ${10}^3$-fold over the insertion procedure.Reuse & Permissions

Figure 2

(Color online) Effects of crowding on the (a) folding free energy of cytochrome $b_{562}$ and (b) binding free energy of barnase and barstar. Symbols are obtained in our previous work [26] by the insertion procedure, while curves are predictions of the GFMT. The different curves represent the different crowder radii listed in Fig. 1b.Reuse & Permissions

Figure 3

(Color online) Effects of mixed crowding on the folding free energy of cytochrome $b_{562}$. Two species of crowders, with $15\text{-Å}$ and $50\text{-Å}$ radii, are mixed at different ratios. Symbols are obtained in our previous work [26] by the insertion procedure, while curves are predictions of the GFMT.Reuse & Permissions

Figure 4

GMFT predictions for the effects on the binding free energy of barnase and barstar by increasingly complex mixtures of crowders. The total volume fraction of crowders is fixed at 35%. $N_c$ denotes the number of crowder species. At $N_c=1$, all crowders have a $50\text{-Å}$ radius. At $N_c=2$, crowders with $50\text{-Å}$ and $45\text{-Å}$ radii are mixed. Each successive increase in $N_c$ introduces a new species that is $5\text{Å}$ smaller than the previous smallest crowders. At $N_c=9$, the crowder radii range from 10 to $50\text{Å}$. At each $N_c$, all the species have the same volume fraction.Reuse & Permissions